Flows on Compact Surfaces [1 ed.] 9783031329548, 9783031329555

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Birkhäuser Advanced Texts Basler Lehrbücher

Nelson G. Markley Mary Vanderschoot

Flows on Compact Surfaces The Weil–Hedlund–Anosov Program

Birkhäuser Advanced Texts Basler Lehrbücher A series of Advanced Textbooks in Mathematics

Series Editors Steven G. Krantz, Washington University, St. Louis, USA Shrawan Kumar, University of North Carolina at Chapel Hill, Chapel Hill, USA

This series presents, at an advanced level, introductions to some of the fields of current interest in mathematics. Starting with basic concepts, fundamental results and techniques are covered, and important applications and new developments discussed. The textbooks are suitable as an introduction for students and non– specialists, and they can also be used as background material for advanced courses and seminars.

Nelson G. Markley • Mary Vanderschoot

Flows on Compact Surfaces The Weil–Hedlund–Anosov Program

Nelson G. Markley North Potomac, MD, USA

Mary Vanderschoot Department of Mathematics and Computer Science Wheaton College (Illinois) Wheaton, IL, USA

ISSN 1019-6242 ISSN 2296-4894 (electronic) Birkhäuser Advanced Texts Basler Lehrbücher ISBN 978-3-031-32954-8 ISBN 978-3-031-32955-5 (eBook) https://doi.org/10.1007/978-3-031-32955-5 Mathematics Subject Classification: 37B99, 37B20, 37E35 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In Memory of My Mentors Charles Weeks Saalfrank Professor of Mathematics Lafayette College Gustav Arnold Hedlund Professor of Mathematics Yale University

Preface

There is an interesting mathematical story behind this book and its telling is a perfect beginning for the preface. It begins over 80 years ago at the 1935 Moscow Topology Conference. André Weil was one of the speakers and his lecture, Les familes de courbes sur le tori, was published in Matematicheskii Sbornik [77]. This paper begins with the statement of a striking theorem about the lifts of simple infinite curves on the torus to the usual Euclidean universal cover of the torus. The heart of the theorem being that if the lifted curve goes to infinity, then it does so in a particular direction. This result is potentially applicable to the lift of an orbit of a flow, a continuous real action (Section 5.1). There is no proof in the paper of the main theorem, but he does mention Magnier, presumably a student, who was working on generalizing it to surfaces of higher genus. Weil did, however, clearly have the idea that universal coverings provided a more efficient way to study flows on surfaces. This is very evident in an earlier paper [76] in which he duplicates the classic Poincaré analysis of flows on the torus using a covering flow and without using the Poincaré rotation number. Weil’s lecture and paper from the 1935 Moscow Topology Conference were largely forgotten for 25 years. Although Weil did many other things of note, he never published his proof of this theorem or returned to its subject. Also, any results obtained by Magnier are completely lost to the mathematical world. World War II more than likely played a role in these outcomes. The second key person in the story is G. A. Hedlund of Yale University. He was a student of Morse in classical differential geometry at Harvard. His work on the geodesic and horocycle flows led him over time to become increasingly interested in dynamical systems in general, and he regularly taught a graduate course on the subject. In early 1964, I asked him to be my thesis adviser in dynamical systems. The area he suggested I work on was a crossover from differential geometry to dynamical systems. The general idea was to use the Euclidean or hyperbolic geometry of the universal covering space to prove dynamical properties of flows on compact connected surfaces. For example, could I prove that the lift of an almost periodic orbit of a flow on the torus lay between two parallel lines of irrational slope? He also suggested that I start learning about Fuchsian groups. vii

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Preface

The inspiration for the thesis problem Hedlund suggested came from his thesis [36], where he studied geodesics on surfaces of genus 1 using Euclidean geometry. Morse [65] had done similar work on surfaces of genus greater than 1 using hyperbolic geometry. Hedlund had the idea that lifts of orbits of flows on the torus might have Euclidean geometric properties similar to his class A geodesics and yield new results about flows. Theorem VIII in [36] best illustrates the origin of Hedlund’s covering space ideas for flows on the torus. He obviously had hopes that this approach would also work for genus greater than 1, or he would not have suggested that I start reading about Fuchsian groups. Hedlund spent the summer of 1964 at IDA in Princeton, as he often did. One hot week in July he was back at Yale and looking for me. He wanted to find out if I knew anything about Weil’s 1936 paper because someone at Princeton had vaguely remembered that Weil’s paper from the Moscow Topology Conference might be connected to what I was doing. I did not, and went looking for the paper. I eventually found it in Yale’s beautiful rare book library, and then Hedlund had to personally check it out for me. Instead of being scooped, I had a second problem on my plate: prove Weil’s theorem. Needless to say, 1965 was a productive year for me, I graduated in the spring of 1966, and went to the University of Maryland as an assistant professor. Jumping forward another 25 years, in the spring of 1990 or 1991, Anosov visited the University of Maryland mathematics department. At that time, I was department chair and saw his name on my calendar. I was looking forward to a pleasant courtesy visit, but got a big surprise. He wanted to talk about Weil’s theorem and personally give me reprints of his recent papers on flows on surfaces. To my amazement, he had been interested in a covering space approach to flows on surfaces since the 1960s, but had only recently published his first paper on the subject. Then he discovered my early papers in much the same way that Hedlund had discovered Weil’s paper. There is another twist of fate in this story. On October 14, 1964, the day Khrushchev was ousted, Anosov gave the colloquium lecture at Yale and I was in the audience. As I remember, it was a long technical talk about hyperbolic diffeomorphisms. Although Hedlund was very interested in Anosov’s work, apparently flows on surfaces did not come up in their conversations. In Anosov’s first paper on flows on surfaces [3], published in 1988, he discusses Weil’s work and ideas about flows on surfaces in great detail and confirms my earlier comments about Weil’s paper in Sbornik. This paper also contains a theorem and generalizations of it that are particularly relevant to this book. His basic result shows that flows on compact connected orientable surfaces with a finite set of fixed points have the following property: every lifted positive orbit in the universal cover is either bounded or goes to infinity (Section 8.1). Thus, there are interesting classes of flows which naturally fit the covering space approach because Poincaré-Bendixson theory applies to the bounded positive orbits and the rest have properties that are amenable to covering space theory and planar geometric ideas. What led Anosov to Weil’s covering space approach? The following quote from Anosov’s 1995 survey paper [6] seems to answer this and other questions: “Weil never returned to this topic after the mid-1930s, and his approach was forgotten for

Preface

ix

a long time (although for the case of the torus it was presented in the well-known book [4]). Only in the mid 1960s did I resume his approach taking at the outset a more general point of view. It turns out that this approach can be applied in a more general situation than in item 4.” The reference cited by Anosov is Qualitative Theory of Differential Equations by V. V. Nemytskii and V. V. Stepanov, which is [67] in our bibliography, and, beginning on page 65 of Part 1 of Nemytskii and Stepanov, one can find Weil’s approach used and a reference to his first paper ([76] in our bibliography). After my meeting with Anosov, I began to refer to the covering space approach to flows on surfaces as The Weil-Hedlund-Anosov Program, which now seems very apt as the subtitle of this book. Several times in the past, I seriously thought about writing a book on this subject but always rejected it for the same reason. There are two fundamental theorems of Maier [46] that I felt had to be in such a book, but they were not yet in the fold. In 1995, Mary Vanderschoot, nee Jacobsen, decided to do her dissertation on flows on surfaces with me, and I was pleased to return to the subject with a student. After Mary graduated, we began collaborating to obtain a deeper understanding of lifted orbits going to infinity. Based on our individual earlier work, this turned out to be fertile ground for us and we published two papers, [61] and [62]. The second one was the motivation to write this book because it contains covering space proofs to Maier’s first and second theorems. My hope is that this monograph will inspire new research and advance the Weil– Hedlund–Anosov Program. There are plenty of interesting problems for the curious reader to discover. I have not attempted to list research questions, believing instead that the ones readers generate and solve will be more exciting. For my part I have tried to make the book as accessible as possible. Having the ideal modern reference for the topology of manifolds was an additional incentive to produce a manuscript depending heavily on the classification of compact connected surfaces and their covering spaces. Building on this material requires a solid consistent foundation on which the subject matter can be developed without ambiguity. Lee’s splendid book Introduction to Topological Manifolds [42] is perfect for this role and is referred to frequently. By carefully following Lee’s terminology and even notation in places, I have tried to maximize the usefulness of his book for the reader of this one. Today most graduate courses in topology have a large overlap with the content of [42]. Consequently, a working knowledge of a year course in graduate topology is the only essential background a reader should need, if they are willing to use [42] to expand their knowledge. It is not assumed, however, that the reader has any knowledge of dynamical systems. Rather, the intent is to provide the reader with a text that can serve as an introduction to dynamics or to expand their knowledge of the subject. A narrow treatment of flows on surfaces would not be appropriate for this book. On the universal covering space, we are constantly working with a real action and the action of a group of linear fractional transformations. Depending on the genus of the surface, we are using Euclidean or hyperbolic geometry and their rigid motions.

x

Preface

Many of the surface constructions are based on manifold ideas. Consequently, we take a little broader view and provide the reader with a richer mathematical context than absolutely necessary. Although we rely heavily on [42] for the topology of manifolds, all the dynamical systems material is contained in the book and can serve as an introduction to dynamical systems. As you might expect from the history of the subject, there were a number of independent discoveries of the same results. I have made mention of all the ones that I know. There are probably others. Anosov, Aranson, Grines, Nikolaev, and Zhuzhoma know an enormous amount about the subject, especially differentiable flows on surfaces. The book [68] by I. Nikolaev and E. Zhuzhoma is an extensive overview of the subject and contains a wonderful bibliography. My interest, however, has always been structural information about the larger class of continuous flows on surfaces. By structural information, I mean general theorems about what kinds of dynamical behavior can and cannot occur for flows on surfaces, broadly applicable tools that can be used to investigate specific flows on surfaces, and methods for constructing large families of flows on surfaces that show an interesting variation of dynamical behavior. The sections all fit into one of these three categories. I have tried to keep the notation as consistent as possible, but this is not possible in over 300 pages of mathematics. Setting aside some specific notation for some mathematical objects does help, such as .T2 for the torus. Some are used more frequently and others less frequently than .T2 , but the list is quite long. To help the reader quickly recall the meaning of these notations, there is also an index of special symbols beginning on page 355. The bibliography also reflects the broader view I have taken. In addition, a number of the books in the bibliography have excellent dynamical systems bibliographies. These items end with a note in bold face type about their bibliographies. In this way, the bibliography provides the reader with access to a wide range of both dynamical systems literature and related mathematics. Dynamical systems is a consumer of a wide range of other well-known and studied mathematical subjects. This treatment of flows on surfaces uses topology, especially covering spaces, the classification of compact connected surfaces, and geometry, especially Euclidean and hyperbolic rigid motions, to establish structural theorems that describe flows on surfaces generally. The basic concepts, however, like orbits, invariant sets, almost periodic points, and orbit spaces keep reappearing as you look across the many subareas of dynamical systems. I hope you find this particular mix of mathematical ideas interesting and challenging. North Potomac, MD, USA

Nelson G. Markley

Contents

1

Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Continuous Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Flows and Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Beck’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 14 21

2

Flows and Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lifting Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Compact Connected Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 43 51

3

A Family of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Suspension Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Cascades on the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Denjoy Flows on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 66 74

4

Local Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Bebutoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Whitney’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Classical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5

Flows on the Torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Weil’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Control Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Geometry of Recurrent Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 122 131

6

Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Poincaré Disk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Properties of Rigid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Groups of Rigid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146 161 173

7

Flows and Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Fuchsian Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Constructing Compact Dirichlet Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Lifts of Closed Curves Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 195 205 xi

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Contents

8

Lifts and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Anosov Dichotomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Omega Limit Points at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Geometry of Almost Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 218 232 245 260

9

Recurrent Orbit Closures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Covering Space Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Maier’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Counting Recurrent Orbit Closures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269 270 276 284

10

Existence of Transitive Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Constructing Rectangular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Constructing Flows from Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Transitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Locally Circular Cascades and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 296 313 330 339

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Index of Special Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Chapter 1

Dynamical Systems

The motion of the planets and moons in our solar system is a dynamical system that has been studied throughout recorded history. Although some of the early models of planetary motion were quite sophisticated, they were mostly based on incorrect premises. Eventually, using empirical data that is primitive by today’s standards, fundamental results were obtained. Copernicus (1543) demonstrated that the planets revolved around the sun and Kepler (1609) determined that these orbits were elliptical. The discovery of the gravitational force by Newton (1687) and calculus brought a new level of mathematical sophistication, activity, and results to celestial mechanics, but it remained the study of a single dynamical system as did many other problems in mechanics. The systematic study of differential equations was a positive force in the investigation of dynamical systems because its focus was on fundamental principles and techniques that could be applied to a large class of problems. The unified study of dynamical systems of all types is a twentieth century phenomenon built on a wide range of modern mathematical subjects. This will be evident even in this book’s examination of just flows on surfaces. Thus it seems appropriate to begin with a chapter on dynamical systems to provide the reader with a perspective of the subject and some basic results that apply in many different settings, including flows on surfaces. The work of Poincaré, Birkhoff, and others laid the foundation for the study of dynamical systems by asking broader, more qualitative questions. Many of the general ideas of the subject were already in papers before World War II. The midcentury publication of two books, one in Russia and one in the United States, provides a good marker of the birth of dynamical systems as a subject in its own right. Nemytskii and Stepanov [67] was first published in 1949 and the English translation was published by Princeton University Press in 1960 with the English title Qualitative Theory of Differential Equations. It is divided into two parts. The first part is devoted to differential equations and the second to general dynamical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_1

1

2

1 Dynamical Systems

systems. Modeled on differential equations, the dynamical systems considered in the second part are all continuous real actions on metric spaces. Gottschalk and Hedlund, the authors of the second book [31], which was published in 1955 by the American Mathematical Society, were influenced by the development of topological groups and differential geometry as much as by differential equations. They considered the dynamical behavior of continuous actions of topological groups on topological spaces and entitled their book Topological Dynamics. We will use aspects of both of these points of view. During the time these books were written, the communication between Russian and American mathematicians was rather limited. Unfortunate differences of terminology occurred in important overlapping areas of these books and can cause confusion even today. The terminology used here will be primarily American, but some of the differences will be pointed out. The first section, in the spirit of Gottschalk and Hedlund, provides an overview of continuous actions of topological groups on topological spaces, a topic that will play a role in the study of flows on surfaces. The discussion in this section ranges from continuous actions that have nice orbit spaces to the connection between almost periodic points and orbit closures. Of particular importance is the concept of a bitransformation group because it provides the general framework for applying Euclidean and hyperbolic geometry to the study of flows on surfaces. The second section is devoted to continuous actions of the real numbers (called flows) and the integers (called cascades) on metric spaces. The approach here is more like that of Nemytskii and Stepanov. It covers the classical notions of periodic orbits, limit sets, and recurrent points. On some spaces, there are no flows having an empty set of fixed points. For example, the 2-sphere is such a space. The Lefschetz fixed point theorem provides a criterion for every flow on a compact polyhedron, including the 2-sphere, to have at least one fixed point. This final topic in Section 1.2 applies to compact surfaces in a significant way. Flows on metric spaces have the property that they can be modified, almost at will, to add fixed points. The fixed points of a flow are always a closed set. Other than requiring that the expanded set of fixed points is also closed, the set of fixed points can be enlarged with new orbits that are subsets of the original orbits. This is the main theorem of the third section. Because surfaces are metric spaces, most of this book is set in the context of metric spaces. In this chapter, however, we will at least indicate the broader context in which dynamical systems can be studied and will see that more general groups than the integers and real numbers can play a role in the study of flows on surfaces.

1.1 Continuous Group Actions Let X be a topological space, and let G be a topological group with the group operation written multiplicatively. (For the reader unfamiliar with topological

1.1 Continuous Group Actions

3

groups, the material in Chapter 1 of [59] is adequate.) A continuous group action of G on X is a function .ϕ : X × G → X satisfying the following conditions: (a) The function .ϕ is continuous on .X × G with the product topology. (b) For all g and h in G and x in X, ϕ(x, gh) = ϕ(ϕ(x, g), h).

.

(c) For all x in X, .ϕ(x, e) = x, where e denotes the identity element of G. Although .ϕ will be the usual symbol for a continuous group action, it is often convenient to write xg instead of .ϕ(x, g). Then the equations in parts (b) and (c) of the definition can be written as .x(gh) = (xg)h and .xe = x. A transformation group is a triple .(X, G, ϕ) consisting of a topological space X, a topological group G, and a continuous group action .ϕ of G on X. Frequently, we simply write .(X, G) for a transformation group, leaving the continuous group action unnamed as it is in .(x, g) → xg. This definition of a transformation group can be and is considered in very general situations. However, full generality will not be necessary for our study of flows on surfaces. In particular, it can be and will be assumed that both the space X and the group G are metric spaces in the rest of the book, and hence both X and G are Hausdorff spaces. We will be meticulous about the necessary notation in this section before it is built into the notation in the future sections. The orbit of a point x in X is by definition .xG = ϕ(x, G) = {ϕ(x, g) : g ∈ G}. Given h in G, the orbit of xh is .(xh)G = x(hG) = xG. It follows that orbits are either disjoint or equal. Consequently, .x ∼ y if and only if .xG = yG defines an equivalence relation on X called the orbit equivalence relation. The resulting quotient space of .(X, G) is called the orbit space and is denoted by .X/G. Note that an orbit of the continuous action of G on X corresponds to precisely one point in .X/G and conversely. Technically, a continuous group action of G on X, as defined above, is a continuous right group action, and a continuous left group action of G on X is a continuous function .ψ : G × X → X such that .ψ(gh, x) = ψ(g, ψ(h, x)) and .ψ(e, x) = x. If .ψ : G × X → X is a continuous left action of G on X, then −1 , x) is a continuous right action of G on X and the two actions have .ϕ(x, g) = ψ(g the same orbits. For abelian groups, the distinction between left and right actions is unnecessary because .(xg)h = x(gh) = x(hg) = (xh)g. In the case of either a right or a left continuous group action, the orbit space will be denoted by .X/G. The points of .X/G are the orbits themselves and thus have no order. The function .π(x) = xG defines a surjective function from X to .X/G called the natural projection. In addition, a subset U of points in X is said to be a saturated set provided U = π −1 [π(U )].

.

(1.1)

4

1 Dynamical Systems

Defining a subset W of .X/G to be open if and only if .π −1 (W ) is an open set of X uniquely defines a topology on .X/G because .π is surjective. It follows that the natural projection .π : X → X/G is continuous. Exercise 1.1.1 Show that the natural projection .π : X → X/G is an open map. In general, given an equivalence relation .∼ on a topological space X, the equivalence classes of .∼ form a quotient space and .π mapping x to the equivalence class containing x is a quotient map. Then a subset W of .X/ ∼ is an open set of −1 (W ) is an open subset of X. As above .π is continuous. (For .X/ ∼ if and only if .π a full introduction to quotient spaces and additional examples of continuous group actions, see Lee [42].) The closure and interior of sets in metric spaces will be used frequently. Convenient definitions of the closure of Y and the interior of Y of a subset Y of a metric space X are as follows: Y− =

.

 {C : Y ⊂ C and C is closed in X}

and Yo=

.



{B : Y ⊃ B and B is open in X}.

Exercise 1.1.2 Prove that the closure of a connected set is connected, but the interior of a connected set need not be connected. In the context of metric groups, a homomorphism will always mean a continuous function .θ : G → H such that .θ (gh) = θ (g)θ (h), and an isomorphism will always mean a homomorphism .θ : G → H such that .θ is a homeomorphism of G onto .θ (G). Metric groups themselves provide a source of transformation groups. For a family of examples, let X and G be metric groups. If .θ : G → X is a homomorphism, then .ϕ(x, g) = xθ (g) defines a continuous right action of G on X. The orbits of .ϕ are the right cosets of .θ (G) in X. So the orbit of the identity e is just .θ (G). In particular, if G is a subgroup of X, then G is a group with the relative topology from X, and letting .θ be the inclusion homomorphism, .ϕ(x, g) = xg is a continuous right action of G on X. Of course one can define left actions in the same way. As a specific example, let .C denote the complex numbers with the usual metric topology given by .d(z, w) = |z − w|, let .S1 = {z ∈ C : |z| = 1} denote the unit circle, and let .Z denote the integers . Under complex multiplication, .S1 is compact metric group. If a is a real number, then .e2π ia = cos(2π a) + i sin(2π a) is in 1 2π ian is a homomorphism of .Z to .S1 . Thus .ϕ(z, n) = ze2π ian .S and .θ (n) = e is a continuous group action of .Z on .S1 . Note that .ϕ(z, n) = ze2π ian is just a counterclockwise or clockwise rotation of .S1 through .2π an radians when a is positive or negative, respectively.

1.1 Continuous Group Actions

5

It is easy to verify that .θ (Z) is a finite subgroup of .S1 if and only if a is a rational number. The closed subgroups of .S1 are either finite or equal .S1 ([59], p. 49). Since the closure of .θ (Z), denoted by .θ (Z)− , is a closed subgroup of .S1 ([59], p. 41), the 1 .ϕ-orbit of 1 is dense in .S if and only if a is irrational. Because the .ϕ-orbit of z is {ϕ(z, n) : n ∈ Z} = {ze2π ian : n ∈ Z} = zθ (Z),

.

every orbit of .ϕ is dense in .S1 when a is irrational, and every orbit is finite when a is rational, providing an interesting family of continuous .Z actions on .S1 . Moreover, they are all examples of cascades, that is, continuous actions of the topological group .Z. With the previous elementary concepts and examples of continuous group actions, it is time to introduce more substantive dynamical ideas and proofs starting with transitive points. Given a transformation group .(X, G), a point y in X is said to be a transitive point of a continuous (right) action of a group G on X provided that .(yG)− = X, that is, the orbit of y is dense. Obviously, if y is a transitive point, then every point of the orbit of y is also a transitive point. In the previous example, every point of .S1 is transitive when a is irrational, which is not true in general. Equivalently, a point y is a transitive point provided the orbit of y intersects every nonempty open set of X or, equivalently, y is in V G for every nonempty open set V of X. If y is a transitive point of .(X, G), then yG is a dense point of .X/G, that is, the point yG is in every nonempty open subset of the orbit space .X/G, which is strange. Recall that a residual set is, by definition, a countable intersection of open dense sets. Proposition 1.1.3 Let .(X, G) be a transformation group on a second-countable complete metric space X. The following are equivalent: (a) There exists a transitive point in X. (b) For every pair of open sets U and V of X, there exists .g ∈ G such that .U ∩V g = φ. (c) There exists a dense residual set of transitive points in X. Proof Because the orbit of a transitive point would intersect any two open sets of X, (a) implies (b) is trivial. Also (c) implies (a) is trivial. By hypothesis there exists a countable basis .Vj for X indexed by the positive integers denoted by .Z+ . Since .x ∈ X is a transitive point of .(X, G) if and only if x is in V G for every open set V of X, it follows that  .

j ∈Z+

is precisely the set of transitive points.

Vj G

6

1 Dynamical Systems

If, for every pair of open sets U and V of X, there exists .g ∈ G such that U ∩ V g = φ, then V G is an open dense set of X. Thus (b) implies (c) follows from the Baire category theorem (see [42], p. 85). 

.

A subset Y of X is an invariant set of a continuous group action of G on X provided that .yG ⊂ Y for all y in Y . So Y is invariant if and only if it is a union of orbits. For example, the set of transitive points is an invariant set, but it is also possibly the empty set. The closure and interior of an invariant set are routinely shown to be invariant (Exercise 1.1.4). The smallest closed invariant set containing .x ∈ X is obviously − .(xG) , which is referred to as the orbit closure. Exercise 1.1.4 Prove that the closure, interior, and boundary of an invariant set of X are invariant sets of X. Exercise 1.1.5 Prove that if .y ∈ (xG)− , then .(yG)− ⊂ (xG)− . The orbits of a continuous group action of a connected group G on a metric space X are connected sets. It follows that the components of X are invariant sets of the continuous group action of a connected group such as the real numbers .R. This is false for disconnected groups like the integers .Z. Given a continuous group action .ϕ of G on X, a nonempty subset Y of X is a minimal set of the continuous group action provided that .Y = (yG)− for all y in Y . Equivalently, a minimal set is a nonempty closed invariant set that does not properly contain a nonempty closed invariant set. Exercise 1.1.6 Prove that M is a minimal set if and only if M is a nonempty closed invariant set that does not properly contain a nonempty closed invariant set. In the example of the cascade (continuous .Z-action) on .S1 defined by ϕ(z, n) = ze2π ian ,

.

the set .S1 itself is a minimal set when a is irrational. In this case, the transformation group .(S1 , Z, ϕ) is said to be a minimal transformation group. When a is rational, each orbit is a minimal set. Proposition 1.1.7 If X is a compact metric space, then every continuous group action on X has a minimal set. Exercise 1.1.8 Prove Proposition 1.1.7 by applying Zorn’s lemma to the collection of nonempty closed invariant sets partially ordered by inclusion. Exercise 1.1.9 If .M1 and .M2 are minimal sets of .(X, G), then either .M1 = M2 or M1 ∩ M2 = φ.

.

Continuing with other useful concepts and terminology, let .(X, G, ϕ) = (X, G) be a transformation group. Given a point x in X, it is easy to see that .Px = {g ∈

1.1 Continuous Group Actions

7

G : ϕ(x, g) = xg = x} is a closed subgroup of G since X is Hausdorff. A point x is said to be a fixed point provided that .Px = G. When every point of a continuous group action is a fixed point, it is trivial. There is always a trivial continuous group action of G on X, namely, .ϕ(x, g) = x for all x and for all g. Given x, the function .g → xg is injective if and only if .Px = {e}. When .Px = {e} for all .x ∈ X, the continuous group action is said to be a free action. A subset A of a metric group G is said to be left syndetic provided there exists a compact subset C of G such that .G = AC and right syndetic provided .G = CA. Note that if A is a group, then A is left syndetic if and only if it is right syndetic. Exercise 1.1.10 Show that .{±n2 : n ∈ Z} is not syndetic. Given a right transformation group .(X, G, ϕ), a point x in X is said to be periodic provided that .Px is a syndetic subgroup of G and almost periodic provided that for every neighborhood U of x there exists a left syndetic subset A of G such that .xA ⊂ U . So fixed points are periodic points and periodic points are almost periodic points. (This agrees with the definition of periodic in [31] which does not define right and left continuous actions. In the current context, one simply uses right syndetic sets for left transformation groups.) Note that in the example of the cascade on .S1 defined by .ϕ(z, n) = ze2π ian , every point is periodic when a is rational. The connection between almost periodic points and their orbit closures is far more interesting than periodic points. Birkhoff first established this connection for .R-actions occurring in differential equations ([22], p. 199). Although it is now known far more widely, a metric space proof for continuous right group actions of metric groups suffices here. Theorem 1.1.11 Given a continuous group action .ϕ of a metric group G on a metric space X, suppose .(xG)− is a compact orbit closure in X. The point x is an almost periodic point if and only if .(xG)− is a minimal set. Proof First suppose that .(xG)− is a minimal set and let U be an open neighborhood of x. Set .A = {g ∈ G : xg ∈ U }. It suffices to show that A is left syndetic. Given any y in .(xG)− , by minimality there exists .g ∈ G such that .yg −1 is in U or equivalently y is in Ug. It follows that all the sets Ug with .g ∈ G cover .(xG)− , and by compactness there exist .c1 , . . . , cm in G such that (xG)− ⊂

m 

.

U cj .

j =1

Thus for any g in G, there exists .cj such that .xg ∈ U cj or .xgcj−1 ∈ U . Therefore,

gcj−1 is in A and .g = gcj−1 cj ∈ A{c1 , . . . , cm }. Thus .G = A{c1 , . . . , cm }. Since finite sets are always compact, A is left syndetic. Now suppose that x is almost periodic. Given .y ∈ (xG)− , it suffices to show that x is in .(yG)− . Let U be an open neighborhood of x. There exists an open neighborhood V of x such that .x ∈ V ⊂ V − ⊂ U because metric spaces are

.

8

1 Dynamical Systems

regular. Since x is almost periodic, there exists a syndetic subset A of G such that xA ⊂ V . Let C be a compact subset of G such that .G = AC. There exists a sequence .gn such that .xgn converges to y and a sequence .cn in C such that .gn cn−1 is in A. Because C is compact, we can assume without loss of generality that .cn converges to c in C. It follows that .cn−1 converges to .c−1 and −1 = (xg )c−1 converges to .yc−1 , which is in .V − ⊂ U . Because U was an .xgn cn n n arbitrary neighborhood of x, it follows that x is in .(yG)− . 

.

Exercise 1.1.12 Suppose x is an almost periodic point of a cascade .(X, ϕ) where X is compact. Show that x is also an almost periodic point of .(X, ϕ p ), .p > 1. Show that as p varies the orbit closures of .ϕ and .ϕ p may or may not be the same. There are a number of types of points and sets outside of dynamical systems which will be used in this section and in later parts of the book. We introduce them together so the reader encounters them in a comparative way. They are: (a) Given a topological space X and a subset A of X, a point q in X is an accumulation point of A provided every neighborhood of q contains a point of A other than q. (b) Given a topological space X and a subset A of X, a point q in X is a condensation point of A provided every neighborhood of q contains uncountably many points of A. (c) A set A in a topological space X is a perfect set provided each point of A is an accumulation point of A. (d) Given a topological space X, a subset A of X is a closed nowhere dense set of X provided A is closed and contains no open subsets of X. (e) A Cantor set is a perfect closed nowhere dense subset of X. Theorem 1.1.13 Let X be a compact connected metric space. If M is a minimal subset of a continuous group action of a metric group G on X, then one of the following holds: (a) .M = X. (b) M is a closed nowhere dense set such that for every open subset U of X either .U ∩ M is uncountable or empty. (c) M is a finite periodic orbit. Proof Suppose that M is a minimal set with nonempty interior .M ◦ . The interior of an invariant set must be invariant (Exercise 1.1.4). Let x be a point in .M ◦ . If y is any point in .M \ M ◦ , then there exists g such that yg is in .M ◦ by minimality. Thus ◦ ⊂ M and .M ◦ = M, making M an open and a closed set. Therefore, .M ⊂ M .M = X because X is connected. Suppose .M ◦ = φ, and hence M is a closed nowhere dense subset of X. Let C denote the set of condensation points of M. It will be shown that there are two possibilities: either .C = φ or .C = φ. Suppose the former. Because M is closed, .C ⊂ M. So there exists .x ∈ C ⊂ M. Note that C is an invariant subset of M. Let y be an element of M and U an open neighborhood

1.1 Continuous Group Actions

9

of y. By minimality, there exists .g ∈ G such that xg is in U and U must contain uncountably many points of M. Using the metric, U can be as arbitrarily small neighborhood of y as desired. Then there must also exist g such that xg is in U and U must contain uncountably many points of M. Since this process can be repeated infinitely often, .M = C. Suppose that .C = φ. Then for every .x ∈ M, there exists an open neighborhood .Ux such that .Ux ∩ M is finite or countable. Because M is compact, there exist .x1 , . . . , xm in M such that M=

m 

.

Uxi ∩ M.

i=1

It follows that M is finite or countable. Suppose M is countable. Then there are no finite orbits in M because it is assumed that M is minimal. So there exist .x ∈ M and a sequence .gn such that .xgn converges to y in M, which must be a fixed point in M, another contradiction. Therefore, M is finite and a single orbit. 

Corollary 1.1.14 Let f be a homeomorphism of .S1 onto itself. If M is a minimal set of a cascade .(S1 , f ), then one of the following holds: (a) .M = S1 . (b) M is a perfect closed nowhere dense subset of .S1 , that is, a Cantor set. (c) M is a periodic orbit. Corollary 1.1.15 Let X be a compact connected metric space. If G is a connected metric group, then the finite orbits of a continuous group action of G on X must be fixed points. Exercise 1.1.16 Prove Corollaries 1.1.14 and 1.1.15. Exercise 1.1.17 Consider the continuous action of .R on the unit disc .D2 = {z ∈ C : |z| ≤ 1} defined by .ϕ(z, t) = ze2π it . Show that the orbit of .z = 1/2 is a periodic orbit M that is not finite. Moreover, M is a minimal set that is nowhere dense in .D2 such that for every open subset U of .D2 either .U ∩ M is uncountable or empty. A metric space W is a discrete metric space provided that every point of W is an open subset of W or equivalently every subset of W is an open subset of W . Discrete topological spaces are all metric spaces ([42], p. 248). Any group with the discrete topology is naturally called a discrete metric group. A subset Y of metric space X is said to be a discrete subset of X provided that for every .y ∈ Y there exists an open set U of X such that .U ∩ Y = {y}. It follows that the relative topology on a discrete subset Y is precisely the usual discrete topology of Y independent of X. A continuous action of a discrete topological group G on a topological space X is said to be proper provided that for every point .(x, y) in .X × X there exist open neighborhoods U and V of x and y, respectively, such that

10

1 Dynamical Systems

{g ∈ G : U ∩ gV = φ}

.

is a finite subset of G. Proper actions will always be written on the left. The definition of proper given here is equivalent for discrete groups to the definition given by Proposition 12.9 in [42]. There are several variations of this definition that are called properly discontinuous actions or something similar. Following [42], we will avoid, as much as possible, the “discontinuous” terminology. (For discrete groups, the connection between proper actions and proper maps is explained in [42].) Proposition 1.1.18 If the continuous action of a discrete group G on a Hausdorff topological space X is proper, then every orbit is a discrete subset of X. Proof Given x in X, there exist open neighborhoods U and V of x such that .{g ∈ G : U ∩ gV = φ} is a finite subset of G. Letting .W = U ∩ V , it follows that .{g ∈ G : W ∩ gW = φ} is a finite subset of G and .W ∩ Gx is finite. Since X is Hausdorff, there exists an open neighborhood .W  of x such that .{x} = Gx ∩ W  . Hence .{gx} = gGx ∩ gW  = Gx ∩ gW  and every point of Gx is discrete. 

Proposition 1.1.19 If the continuous action of a discrete group G on a Hausdorff topological space X is proper, then the orbits of G are closed subsets of X and the orbit space .X/G is Hausdorff. Proof Let Gx and Gy be distinct points in .X/G or equivalently the orbits Gx and Gy are disjoint in X. Because the action is proper, there exist open neighborhoods .U0 and .V0 of x and y, respectively, such that {g ∈ G : U0 ∩ gV0 = φ} = {g1 , . . . , gm }.

.

Since .Gx ∩ Gy = φ, it follows that .x = gi y for .i = 1, . . . , m. Because X is a Hausdorff space and the action of G on X is continuous, there exist open neighborhoods .Ui and .Vi of x and y, respectively, such that .Ui ∩ gi Vi = φ. Set U=

m 

.

i=0

Ui and V =

m 

Vi .

i=0

Because .U ⊂ Ui and .V ⊂ Vi for .i = 0, . . . , m, it follows that .U ∩ gV = φ for all .g ∈ G and that .GU ∩ GV = φ. So U is an open neighborhood of x contained in .X \ Gy. Then .X \ Gy is a union of open sets indexed by the orbits Gx such that .Gx ∩ Gy = φ, and Gy is a closed set. Since .π : X → X/G is an open function, .π(U ) and .π(V ) are disjoint open neighborhoods of Gx and Gy. 

With a similar argument, we can also prove the following: Proposition 1.1.20 If the continuous action of a discrete group G on a Hausdorff topological space X is both proper and free, then for each x in X there exists an open neighborhood U such that .U ∩ gU = φ for all .g = e.

1.1 Continuous Group Actions

11

Proof Using the definition of a proper action with .x = y, there exist open neighborhoods .U1 and .V1 of x such that {g ∈ G : U1 ∩ gV1 = φ} = {g1 , . . . , gm }.

.

We can assume that .g1 = e. Then .gi x = x for .2 ≤ i ≤ m because the action is also free. Thus for .2 ≤ i ≤ m, as in the previous proof, there exist open neighborhoods .Ui and .Vi of x such that .Ui ∩ gi Vi = φ. Set U=

m 

.

(Ui ∩ Vi )

i=1

and verify that .U ∩ gU = φ implies that .g = e.



Corollary 1.1.21 If the continuous action of a discrete group G on a Hausdorff topological space X is both proper and free, then for each x in X there exists an open neighborhood U of x such that .π |gU , the restriction of the natural projection to .gU, is a homeomorphism of gU onto .π(U ) for all .g ∈ G. Exercise 1.1.22 Prove Corollary 1.1.21. Given a metric group G, suppose .ϕ and .ψ are continuous right actions of G on metric spaces X and Y . A homomorphism of .(X, G, ϕ) to .(Y, G, ψ) is a continuous function .θ : X → Y such that .θ (ϕ(x, g)) = ψ(θ (x), g) or more simply .θ (xg) = θ (x)g for all x in X and g in G. Homomorphisms of left actions are defined similarly. A homomorphism that is also a homeomorphism of X onto Y is called an isomorphism and an isomorphism of .(X, G, ϕ) onto itself is called an automorphism. The identity function of G is the function of G onto itself denoted by .ι and mapping g to g for all .g ∈ G. It is always an automorphism of .(X, G, ϕ), and consequently the automorphisms of .(X, G, ϕ) form a group. Let G and . be groups, and let X be a metric space. A bitransformation group .(, X, G) is a continuous left action of . on X and a continuous right action of G on X such that .(γ x)g = γ (xg) for all .γ ∈ , .x ∈ X, and .g ∈ G. In other words, the function .x → γ x is an automorphism of .(X, G) for each .γ in . and the function .x → xG is an automorphism of .(, X) for each g in G. Proposition 1.1.23 If .(, X, G) is a bitransformation group, then there is a unique continuous right action of G on the orbit space .X/  such that the natural projection .π : X → X/  is a homomorphism of .(X, G) onto .(X/ , G). Proof Since the natural projection .π : X → X/  is an open function, .π × ι : X × G → X/  × G, where .ι is the identity function of G onto itself, is an open function and thus a quotient map. Consider the function .(x, g) → π(xg). If .π(x) = π(y), then there exists .γ ∈  such that .x = γ y and π(xg) = π((γ y)g) = π(γ (yg)) = π(yg).

.

12

1 Dynamical Systems

Therefore, .(x, g) → π(xg) passes to the quotient space .X/  × G ([42], Corollary 3.30), and there exists a continuous function .ϕ  : X/  × G → X/  defined by .ϕ  (π(x), g) = π(xg). It follows readily that the function .ϕ  also satisfies conditions (b) and (c) in the definition of a continuous action. Obviously, .π is a homomorphism of .(X, G, ϕ) onto .(X/ , G, ϕ  ). 

A flow will always mean a continuous action of .R on a topological space. To analyze the dynamical behavior of a flow on a surface, we will frequently employ a bitransformation group .(, X, R) such that the flow given by Proposition 1.1.23 on .X/  is isomorphic to the flow being studied. Typically, . will be a discrete group and the continuous action of . on X will be both proper and free. The necessary manifold and covering space theory needed to make this strategy work will be discussed in Chapter 2. Before taking a closer look at flows in Sections 1.2 and 1.3, we can at least provide a simple example of a relevant bitransformation group. θ : R → R2 defined by . θ (t) = (at, bt) Let a and b be real numbers. The function . is a homomorphism of the topological group .R into the topological group .R2 . In fact, it is one to one unless .a = 0 = b. It can be used, as described earlier, to define a flow on .R2 by setting  ϕ ((x, y), t) = (x, y) +  θ (t) = (x + at, y + bt).

.

When .a = 0 = b, the flow . ϕ is the trivial flow, so we will assume that .a 2 + b2 = 0. Then the orbit of .(0, 0) is the closed subgroup .{t (a, b) : t ∈ R} and the other orbits are its cosets, that is, the orbits are a family of parallel lines. Similarly, there is a continuous left action .ψ of the discrete subgroup .Z2 of .R2 on 2 2 .R defined by .ψ((m, n), (x, y)) = (m+x, n+y). The .Z orbits are the cosets of the 2 2 subgroup .Z . The .Z action is both free and proper and produces a bitransformation 2 2 2 .(Z , R , R). By Propositions 1.1.19 and 1.1.23, there is a natural flow on the .Z orbit space, which is a Hausdorff topological space. Some readers may recognize this group as the universal covering group for the torus .S1 × S1 = T2 . Others will be introduced to these ideas in Section 2.2. Consequently, the .Z2 -orbit space is the same as the quotient topological group .R2 /Z2 and the quotient flow .ψ on .R2 /Z2 is given by ψ((x, y) + Z2 , t) = (x + at, y + bt) + Z2 .

.

To construct the above flow in a different way, define .e : R2 → T2 by .e(x, y) = and observe that e is a homomorphism of the topological group .R2 onto the topological group .T2 with kernel .Z2 . Now the homomorphism .θ = e ◦  θ: R → T2 can be used to define a flow .ϕ on .T2 by letting (e2π ix , e2π iy )

ϕ((z, w), t) = (z, w)θ (t) = (ze2π iat , we2π ibt ),

.

which is often called a straight line flow on the torus because the orbits are straight lines wrapped around the torus by e.

1.1 Continuous Group Actions

13

It follows from the first isomorphism theorem and the open mapping criterion for topological groups ([59], Theorems 1.4.14 and 1.5.18) that .f ((x, y) + Z2 ) = e(x, y) is an isomorphism of the topological group .R2 /Z2 onto the topological group .T2 . Because the flows .ψ and .ϕ are both defined algebraically, it is easily verified that f is an isomorphism of the flow .(R2 /Z2 , R, ψ) onto .(T2 , R, ϕ). So the standard straight line flows on the torus can also be obtained as quotient flows from bitransformation groups. The dynamical properties of the straight line flows on the torus can and will be analyzed directly. To study more general flows on compact connected surfaces geometrically, it will be particularly useful to know that the flow is a quotient flow of a bitransformation group. Continuing the discussion of the straight line flows on the torus, recall that we are assuming that .a 2 + b2 = 0 and that the orbit of .(z, w) is given by (z, w)R = {(ze2π iat , we2π ibt ) : t ∈ R}.

.

In particular, the orbit .(1, 1)R is the subgroup .θ (R) = {(e2π iat , e2π ibt ) : t ∈ R} and the other orbits are its cosets. If .b = 0, then it is easy to see that every orbit is periodic of period .1/a, that is, .1/a is the smallest positive number t such that 1 .(z, w)t = (z, w), and the orbits are the sets .S × {w}. Similar comments apply when .a = 0. The most interesting case is when .a/b is irrational. Theorem 1.1.24 If a and b are both nonzero and .a/b is irrational, then the straight line flow .ϕ((z, w), t) = (ze2π iat , we2π ibt ) on .T2 is a minimal flow. Proof Note that .S1 × {1} is a closed subgroup of .T2 isomorphic to .S1 and 2π ia/b , 1). Thus .ϕ 1 .ϕ1/b (z, 1) = (ze 1/b restricted to .S × {1} is an irrational rotation 1 on a copy of .S . It follows that (z, 1)R ∩ (S1 × {1}) = {(ze2π ika/b , 1) : k ∈ Z}

.

is dense in .S1 × {1}. Next observe that .ϕc/b (S1 × {1}) = S1 × {e2π ic }. Since .ϕc/b maps each orbit of .ϕ to itself,   ϕc/b (z, 1)R ∩ (S1 × {1} = (z, 1)R ∩ (S1 × {e2π ic })

.

and .(z, 1)R ∩ (S1 × {e2π ic }) is dense in .S1 × {e2π ic } for every c. Consequently, 2  .(z, 1)R is dense in .T . For every orbit .(z, w)R, there clearly exists .z such that  .(z, w)R = (z , 1)R. Therefore, every orbit is dense and the flow is minimal. 

If both a and b are nonzero and .a/b is rational, then there exist relatively prime integers c and d such that .a/b = c/d. Set .σ = c/a = d/b. In this case, the kernel of .θ is the discrete subgroup .H = {nσ : n ∈ Z}, which is obviously a syndetic subgroup of .R. Then one verifies that .P(z,w) = H for all .(z, w) in .T2 . This is an example of a periodic transformation group .(X, G), meaning that there is a syndetic subgroup H of G such that .xh = x for all .x ∈ X and all .h ∈ H . It follows from

14

1 Dynamical Systems

the next result that there exists a continuous action of .S1 on .T2 that has the same orbits as a straight line flow on .T2 with .a/b rational because .R/H is isomorphic to 1 .S ([59], Corollary 1.4.18). Proposition 1.1.25 If .ϕ : G × X → X is a continuous group action on a metric space X, then .GX = {g ∈ G : gx = x for all x ∈ X} is a closed normal subgroup of G and .ϕ(x, ˆ gGX ) = ϕ(x, g) is a continuous action of .G/GX on X with the same orbits as .ϕ. If .GX is syndetic, then .G/GX is compact. Proof It is easy to check that .GX is a closed normal subgroup of G, so .G/GX is a topological group ([59], Theorem 1.4.12). Following the proof of Proposition 1.1.23, let .π : G → G/GX be the natural projection and .ι : X → X the identity map. Clearly, .ι × π : X × G → X × G/GX is an open function, and the product topology on .X × G/GX equals the quotient topology given by .ι × π . Since   .π(g) = π(g ) if and only if .g = g h for some .h ∈ GX , it follows that .ϕ passes to the quotient and .ϕˆ is continuous. 

This completes a short basic introduction to continuous metric group actions on metric spaces. The two remaining sections of this chapter begin the focus on flows. As the book unfolds, however, the more general ideas of this first section will be essential in a number of chapters.

1.2 Flows and Cascades The real numbers .R and the integers .Z are especially useful topological groups because they are linearly ordered and are used to represent the passage of time. For example, the integers can be used to record infinite discrete time events such as an infinite sequence of coin tosses. The real numbers are regularly used to continuously parameterize the solutions of autonomous differential equations to describe their motion. These are examples of the initial ideas for cascades and flows. Recall from Section 1.1 that a cascade is simply a continuous action of the topological group .Z on a metric space and a flow is a continuous action of the topological group of real numbers .R on a metric space. The basic question for these dynamical systems is how do their orbits evolve both forward and backward over time. Since .Z is a discrete group, all subgroups of .Z are closed sets. Both .R and .{0} are closed subgroups of .R and are called the trivial subgroups of .R. If G is a nontrivial closed subgroup of .R, then there exists .a > 0 such that .G = {ka : k ∈ Z} and G is isomorphic to .Z ([59], Theorem 1.4.17). Since .Z is a closed subgroup of .R, the closed subgroups of .Z are .Z, .{0}, and sets of the form .G = {ka : k ∈ Z} with .a ∈ Z+ , the positive integers. It follows that the nontrivial closed subgroups of .R and .Z are syndetic subgroups of .R and .Z, respectively.

1.2 Flows and Cascades

15

Let .(X, R) = (X, R, ϕ) be a flow on a topological space. When needed, .ϕ will be the preferred symbol to denote the function defining a flow. For a flow, the definition from Section 1.1 of .Px is .Px = {t ∈ R : xt = x}. If .Px is a nontrivial subgroup of .R, then .Px = {ka : k ∈ Z} and is determined by the positive real number a, called the period of x. For the trivial subgroups of .R, the group .Px is either .R or .{0}. Therefore, given .x ∈ X, there are only three possibilities for the closed subgroup .Px = {t ∈ R : xt = x}: (a) .Px = R and x is a fixed point. (b) .Px = {ka : k ∈ Z} with .a > 0 and x is periodic. (c) .Px = {0} and the function .t → xt is injective. So the map .t → xt is not injective if and only if either (a) or (b) holds. For a cascade .(X, f ), a point x is periodic if and only if the orbit of x is finite. Then the period equals the smallest positive integer n such that .f n (x) = x, which is the number of points in the orbit of x. The analogous result for .R is as follows: Theorem 1.2.1 Given a flow .(X, R), the point .x ∈ X is periodic if and only if the orbit of x is compact. Moreover, when x is periodic but not a fixed point, the orbit of x is homeomorphic to .S1 . Proof The group .Px defined in Section 1.1 is essential in this proof. If .Px = R, then .xR = {x} which is compact. Suppose x is periodic with .Px = {ka : k ∈ Z} and .a > 0. Note that .xs = xt if and only if .s − t ∈ Px . Hence the function .t → xt passes to the quotient .R/Px and .Px + t → xt is a continuous injective function from .R/Px onto .xR. Since .R/Px is isomorphic to the compact topological group 1 .S ([59], Corollary 1.4.18), the orbit of x is the continuous image of a compact set and hence compact. Furthermore, the injective continuous function .Px + t → xt is a homeomorphism of .R onto the orbit of x, because .S1 is compact and the orbit of x with the relative topology is a Hausdorff space ([39], Theorem 5.8 or [42], Lemma 4.25). Therefore, the orbit of x is homeomorphic to .S1 . Suppose the orbit of x is compact. Without loss of generality, .X = xR. If x is not periodic, then the function .t → xt is an injective function of .R onto .xR. Using the compactness of .xR, we will show that it is an open function and hence a homeomorphism, which is impossible because .R is not compact. First, consider the compact set .x[−ε, ε] = {xt : −ε ≤ t ≤ ε} for a given .ε > 0. Then   ϕkε (x[−ε, ε]) = x[−ε, ε] (kε) = x[(k − 1)ε, (k + 1)ε],

.

and it follows that  .

k∈Z

ϕkε (x[−ε, ε]) = xR.

16

1 Dynamical Systems

The sets .ϕkε (x[−ε, ε]) are compact and hence closed. The Baire category theorem now implies that for some .k ∈ Z, the interior of .ϕkε (x[−ε, ε]) is not empty. Since .ϕkε is a homeomorphism for all .k ∈ Z, the interior of .x[−ε, ε] is not empty. Second, let U be an open neighborhood of 0 in .R and consider the set xU . There exists .ε > 0 such that .[−2ε, 2ε] ⊂ U . By the first step, there exists .σ ∈ [−ε, ε] such that .xσ is in the interior of .x[−ε, ε]. It follows that ϕ−σ (x[−ε, ε]) = x[−ε − σ, ε − σ ]

.

is a neighborhood of x and xU is a neighborhood of x because [−ε − σ, ε − σ ] ⊂ [−2ε, 2ε].

.

Third, if U is an open set of .R and .τ = 0 is in U , then .U − τ is a neighborhood of 0 and .x(U − τ ) is a neighborhood of x by the second step. Therefore, .ϕτ (x(U − τ )) = xU is a neighborhood of .xτ for all .τ = 0 ∈ U and .t → xt is an open function and hence a homeomorphism. We have obtained the contradiction that .R is homeomorphic to a compact space. Therefore, x must be periodic when its orbit is compact. 

This result holds for separable locally compact topological groups using the same basic ideas to prove it ([31], p. 21). We already understand how a periodic orbit of a flow evolves with time: it just keeps going around the same simple closed curve again and again. The evolution of the orbit of an almost periodic orbit of a flow is more subtle. It comes back arbitrarily close to its starting point infinitely often and there is only a bound on how long we must wait for the next return of the same degree of closeness. These are examples of recursive behavior, that is, the orbit of a point returns arbitrarily close to its starting point for a specified type of subset of the acting group. For the real numbers (and the integers), there are other natural recursive behaviors, for example, the orbit of a point returning arbitrarily close to its starting point for arbitrarily large positive numbers. This fits naturally into the concepts of alpha and omega limit sets, which generally help us understand the long-term behavior of an orbit. The nonzero elements of both .R and .Z divide naturally into positive and negative numbers. These groups are then naturally ordered by .a > b if and only if .a − b is positive, and by definition a sequence goes to infinity provided that it is eventually bigger than any prescribed positive number. For a flow or a cascade, a point x has a positive and a negative semi-orbit defined by .{xg : g > 0} and .{xg : g < 0}, respectively, where g is either an element of .R or .Z. In this context, we find the notation .O(x), .O+ (x), and .O− (x) for the orbit, positive semi-orbit, and negative semi-orbit slightly more convenient. For example, + − + − .O (x) seems clearer than .(xR ) for the closure of the positive semi-orbit, where + .R denotes the positive real numbers. And this notation works for both .R and .Z. The alpha and omega limit sets of a point x for a flow or a cascade are defined by

1.2 Flows and Cascades

17

α(x) =



.

O− (xg)−

g≤0

and ω(x) =



.

O+ (xg)− ,

g≥0

where g is in .R or .Z. The next two propositions contain a collection of basic properties of limit sets. Their straightforward proofs are left to the reader. Proposition 1.2.2 Let .(X, R) {.(X, Z)} be a flow {cascade} on a metric space and let x be an element of X. (a) A point y is in .ω(x) if and only if there exists a sequence .gi of positive real numbers {integers} increasing to infinity such that .

lim xgi = y.

i→∞

(b) A point y is in .α(x) if and only if there exists a sequence .gi of negative real numbers {integers} decreasing to minus infinity such that .

lim xgi = y.

i→∞

Exercise 1.2.3 Prove Proposition 1.2.2. Proposition 1.2.4 If .(X, R) {.(X, Z)} is a flow {cascade} on a metric space, then for all x in X the following hold: (a) (b) (c) (d) (e) (f) (g)

α(x) and .ω(x) are closed invariant sets of X. O(x)− = O(x) ∪ α(x) ∪ ω(x). .α(xg) = α(x) and .ω(xg) = ω(x) for all g in .R {.Z}. If .O− (x) is contained in a compact subset of X, then .α(x) = φ. If .O+ (x) is contained in a compact subset of X, then .ω(x) = φ. If x is a periodic point, then .α(x) = ω(x) = O(x). A compact subset M of X is a minimal set if and only if .α(y) = M for all .y ∈ M if and only if .ω(y) = M for all .y ∈ M. . .

Exercise 1.2.5 Prove Proposition 1.2.4. Exercise 1.2.6 Let .T : [a, b] → [a, b] be a homeomorphism of the closed interval [a, b] onto itself. Show that for every .x ∈ [a, b], .ω(x) consists of a single fixed point or an orbit of period 2.

.

Proposition 1.2.7 If .(X, R) is a flow on a compact metric space, then .α(x) and ω(x) are connected sets of X for all .x ∈ X.

.

18

1 Dynamical Systems

Proof Suppose .ω(x) is the disjoint union of two closed subsets E and F of .ω(x). It suffices to show that E or F is empty. Since .ω(x) is closed, E and F are closed sets of X and hence compact. Because X is a compact metric space, there exist disjoint open sets U and V containing E and F , respectively. Set .C = X \ (U ∪ V ), which is also a closed compact set. Now .ω(x) ∩ C = φ. Consider .O+ (xt)− for .t ≥ 0. It is the closure of the connected set .O+ (xt) and thus connected. If .O+ (xt)− ∩ C = φ, then .O+ (xt)− is contained in either U or V . If .O+ (xt)− ⊂ U , then .O+ (xτ )− ⊂ O+ (xt)− ⊂ U for all .τ ≥ t. Then .ω(x) ⊂ U , implying that F is empty. Similarly, E is empty when .O+ (xt)− ⊂ V . It remains to rule out the possibility that .O+ (xt)− ∩ C = φ for all .t ≥ 0. Because .O+ (xτ )− ⊂ O+ (xσ )− when .τ > σ ≥ 0, the sets .O+ (xτ )− satisfy the finite intersection hypothesis and then so do the sets .O+ (xτ )− ∩ C. Therefore, ω(x) ∩ C =

.

  O+ (xt)− ∩ C = φ t≥0

because compactness is equivalent to the finite intersection property ([39], Theorem 5.1), contradicting .ω(x) ∩ C = φ. Thus .ω(x) is connected. 

For an example of a disconnected omega limit set when X is not compact, see Example 3.10 (p. 343) in [67]. We can now define some other natural recursive behaviors for flows. Given a flow .(X, R), a point x in X is said to be positively recurrent {negatively recurrent} provided that x is in .ω(x) {.α(x)}. And a point x is said to be recurrent provided that x is negatively and positively recurrent, that is, x is in .α(x) ∩ ω(x). Using Proposition 1.2.4, observe that when x is positively recurrent or negatively recurrent, then every point in .O(x) is positively recurrent or negatively recurrent, respectively. And when x is recurrent, then every point in .O(x) is recurrent. Recall that in our terminology a fixed point is a periodic point and a periodic point is an almost periodic point. In this section we will extend that hierarchy to recurrent points and positively or negatively recurrent points. Theorem 1.2.8 Let .(X, R) {.(X, Z)} be a flow {cascade} on a compact metric space X. If y is a positively {negatively} recurrent point that is not periodic in X, then − with the relative topology contains a dense residual subset of recurrent .O(y) points such that .O(x)− = O(y)− . Proof Without loss of generality, we can assume that .X = O(y)− . Let d be a metric on X. For all m and n in .Z+ , set E(m, n) = {x ∈ X : d(x, xt) ≥ 1/m for all t > n},

.

and for all m and .−n in .Z+ set E(m, n) = {x ∈ X : d(x, xt) ≥ 1/m for all t < n}.

.

1.2 Flows and Cascades

19

If .xk is a sequence in .E(m, n) converging to x, then .1/m ≤ d(xk , xk t) → d(x, xt) as .k → ∞ for .t ≥ n. Thus x is in .E(m, n) and .E(m, n) is a closed set when .n > 0. Similarly, .E(m, n) is a closed set for .n < 0. Let U be an open set of X. Clearly there exists .τ such that .yτ ∈ U . When .n > 0, there exists .t > n such that .d(yτ, (yτ )t) < 1/m, because .yτ is also positively recurrent. Thus U cannot be contained in .E(m, n), and .E(m, n) is nowhere dense for m and n positive. Now consider .n < 0. There exists .t > −n such that .(yτ )t = y(τ + t) is in U and .d(yτ, (yτ )t) < 1/m because .yτ is positively recurrent. Consequently, .y(τ + t) is an element of U such that d(y(τ + t)(−t), y(τ + t)) = d(yτ, y(τ + t)) < 1/m

.

and .y(τ +t) is not in .E(m, n) because .−t < n, proving that .E(m, n) is also nowhere dense when .n < 0. Hence .E(m, n) is a closed and nowhere dense set and .D(m, n) = X \ E(m, n) is an open dense set for all .m ∈ Z+ and .n ∈ Z \ {0}. By the Baire category theorem, D=





.

D(m, n)

m∈Z+ n∈Z\{0}

is a dense residual set in X. Moreover, if x is in D and n is a positive integer, then x is not in .E(m, n) and .E(m, −n). It follows that there exist .s > n and .t < −n such that .d(x, xs) < 1/m and .d(x, xt) < 1/m. Therefore, x is recurrent. Since .X = O(y)− and a compact metric space is second-countable, Proposition 1.1.3 applies and there exists a dense residual set .D  of transitive points. Then  .D ∩ D is a dense residual set with the desired property. 

The construction of D in the proof is a special case of Theorem 3.31 in [31] and part of an overarching study of recursive behavior. Corollary 1.2.9 Let .(X, R) {.(X, Z)} be a flow {cascade} on a compact metric space X. If y is a transitive point such that the sets .{yt : |t| ≤ n} have no interior for .n ∈ Z, then X contains a dense residual subset of points that are both transitive and recurrent. Exercise 1.2.10 Prove Corollary 1.2.9. Classical fixed point theorems are often useful in obtaining global results. For example, there is no homeomorphism f of .D2 onto itself such that .(D2 , f ) is a minimal cascade because f has a fixed point by the Brouwer fixed point theorem. There is an important criterion for the existence of fixed points for a flow coming from a generalization of the Brouwer fixed point theorem. It is easy to show that for a flow .ϕ on topological space X, the map .ϕt (x) = ϕ(x, t) = xt is homotopic to the identity for each .t ∈ R. Just define .H : X × [0, 1] → X by .H (x, s) = ϕ(x, st). Then .H (x, 0) = ϕ(x, 0) = x and .H (x, 1) = ϕ(x, t) = ϕt (x). Consequently, the algebraic automorphisms induced on any homology groups of X by the functions

20

1 Dynamical Systems

ϕt are just the identity and generally not very useful. One exception to this is the Lefschetz fixed point theorem. After the introduction to simplicial complexes and polyhedrons in Chapter 5 of [42] are the following definitions: a topological space is a polyhedron provided that it is homeomorphic to the geometric realization of a simplicial complex. The homeomorphism of a geometric realization of a simplicial complex onto a topological space is called a triangulation of the space. The geometric realization of a simplicial complex is compact if and only if the simplicial complex is finite ([42], p. 114, Problem 5–4). And for a finite simplicial complex .K of dimension n, the Euler characteristic is defined by

.

χ (K) =

n 

.

(−1)k nk ,

k=0

where .nk is the number of k-dimensional simplices in .K. The Euler characteristic is actually a topological invariant of .|K|, the geometric realization of .K, that is, .χ (K) = χ (L), when .L is another finite simplicial complex such that .|K| and .|L| are homeomorphic [42], Theorem 13.32. Consequently, the Euler characteristic of a polyhedron is unambiguously defined. The next theorem appears in Algebraic Topology by Spanier ([72], p. 197) as an application of the Lefschetz fixed point theorem ([72], page 195) and is worth repeating because of its importance for flows on compact surfaces. Theorem 1.2.11 Let X be a compact polyhedron. If .χ (X) = 0, then every flow on X has a nonempty set of fixed points. Proof Because .ϕt is homotopic to the identity, its Lefschetz number is the Lefschetz number of the identity function and hence equals .χ (X). The Lefschetz fixed point theorem implies that the set of fixed points of each .ϕt is nonempty. Let .Fn denote the set of  fixed points of .ϕt when .t = 1/2n with .n ∈ Z+ . Then .Fn+1 ⊂ Fn for all n and .E = ∞ n=1 Fn = φ because X is compact. If x is in E, then .ϕt (x) = x for all t in .{k/2n : k ∈ Z and n ∈ Z+ }, which is dense in .R. Thus by the continuity of the flow, .ϕt (x) = x for all .t ∈ R and .x ∈ E. 

Corollary 1.2.12 Every flow on the 2-sphere .S2 = {(x, y, z) : x 2 + y 2 + z2 = 1} has a fixed point. Proof Let .γ be a 3-dimensional simplex, and let .K be the simplicial complex of all proper faces of .γ . Then .S2 is homeomorphic to .|K| and .χ (K) = 2. 

Exercise 1.2.13 Prove that a flow .(S2 , R) cannot be a minimal flow.

1.3 Beck’s Theorem

21

1.3 Beck’s Theorem This section is devoted to a result about fixed points of flows that is particularly valuable in the construction of examples of flows on surfaces. We begin with two preliminary elementary but useful facts about flows. Proposition 1.3.1 Let .ϕ be a flow on a metric space X with metric d, and let y be an element of X. Given .ε > 0 and real numbers .α and .β such that .α ≤ 0 ≤ β, there exists .δ > 0 such that .d(xt, yt) < ε for all t such that .α ≤ t ≤ β, whenever .d(x, y) < δ. Proof If the conclusion is false, there exists an .ε > 0, a sequence .xk converging to y, and a sequence .tk in the closed interval .[α, β] such that .d(xk tk , ytk ) ≥ ε. Since closed intervals are compact, it can be assumed without loss of generality that .tk converges to t in .[α, β]. Now the continuity of both the flow on X and the metric on .X × X implies that .d(yt, yt) ≥ ε > 0, a contradiction. 

Proposition 1.3.2 Let .ϕ be a flow on a metric space X with metric d, let .f : U → R be a continuous function on an open subset U of X, and let .α and .β be real numbers such that .α ≤ 0 ≤ β and .β − α > 0. If .{yt : α ≤ t ≤ β} ⊂ U and .ε > 0, then there exists .δ > 0 such that .{xt : α ≤ t ≤ β} ⊂ U and 

β β . f (xs) ds − f (ys) ds < ε, α  α whenever .d(x, y) < δ and .α ≤ α  < β  ≤ β.

β Proof If .U = X, then .{xt : α ≤ t ≤ β} ⊂ U and . α f (xs) ds is defined for all .x ∈ X. Suppose .U = X. The set .C = {yt : α ≤ t ≤ β} is a compact subset of X by the continuity of the flow. Hence d(C, X \ U ) = inf{d(p, q) : p ∈ C and q ∈ X \ U } = ρ > 0.

.

By Proposition 1.3.1, there exists .η > 0 such that .d(xt, yt) < ρ for all t such that .α ≤ t ≤ β when .d(x, y) < η. It follows that .{xt : α ≤ t ≤ β} ⊂ U and β . α f (xs) ds is defined when .d(x, y) < η. Since the function .f (xs) is defined and continuous on .{x : d(x, y) < η}×[α, β], the proof of Proposition 1.3.1 can be reused to show that given .ε > 0, there exists .δ > 0 such that .|f (xs) − f (ys)| < ε/(2(β − α)) for all s such that .α ≤ s ≤ β, whenever .d(y, x) < δ. Consequently, 

β β  β . f (ϕ(x, s)) ds − f (ϕ(y, s)) ds ≤ |f (ϕ(x, s)) − f (ϕ(y, s))| ds α α α

22

1 Dynamical Systems

≤

β

|f (ϕ(x, s)) − f (ϕ(y, s))| ds

α

≤

α

β

ε ds < ε, 2(β − α)

whenever .d(y, x) < δ and .α ≤ α  < β  ≤ β.



The main theorem in this section answers the following question: given a flow on a metric space can the set of fixed points be enlarged without otherwise altering the orbits? Theorem 1.3.3 (Beck) Let .ϕ be a flow on a metric space X, and let F be the set of fixed points of .ϕ. If E is a nonempty subset of X such that .E ∩ F = φ and .E ∪ F is a closed subset of X, then there exists a flow .ψ on X such that .E ∪ F is the set of fixed points of .ψ and .Oψ (x) ⊂ Oϕ (x) for all x in X. Beck’s original version of this theorem [18] required that the flow .ϕ has no fixed points. The point of his result was to prove the following theorem: for each closed set F of a metric space X, there exists a flow .ψF on X with F as its set of fixed points if and only if there is a flow on X with no fixed points. With only slight alterations, however, his proof can be used to modify a given flow to enlarge the set of fixed points. Proof Because the function .t → xt is continuous for each x in X, the set .{t : xt ∈ / E ∪ F } is an open subset in .R and hence the disjoint union of open intervals, which are the components of .{t : xt ∈ / E ∪ F }. If x is not in .E ∪ F , then one of these intervals contains 0. Let .(ax , bx ) be the component of .{t : xt ∈ / E ∪ F } containing / E ∪ F . The proof proceeds by constructing a flow .ψ on X such that 0 for each .x ∈ the .ψ-orbit of x is either .{xt : ax < t < bx } or .{x}. As usual d denotes a metric for the metric space X. Set g(x) =

.

inf{|t| : xt ∈ E ∪ F } if {t ∈ R : xt ∈ E ∪ F } = φ ∞

if {t ∈ R : xt ∈ E ∪ F } = φ.

Note that .g(x) = 0 if and only if .x ∈ E ∪ F because .E ∪ F is closed. Then set f (x) = inf{d(x, y) + g(y) : y ∈ X}.

.

Clearly, .f (x) ≤ g(x) and .f (x) ≤ inf{d(x, y) : y ∈ E ∪ F } = d(x, E ∪ F ). Obviously, .f (x) = 0 when x is .E ∪ F . Conversely, if .f (x) = 0, then there exists a sequence .yn such that both .d(x, yn ) and .g(yn ) converge to 0. It follows that there exists a sequence .tn of real numbers converging to 0 such that .yn tn ∈ E ∪ F . Then .yn tn converges to x because .ϕ is continuous, and x is in .E ∪ F because .E ∪ F is closed. So .f (x) = 0 if and only if x is in .E ∪ F . Next we will show that f is continuous. First observe that

1.3 Beck’s Theorem

23

f (x) ≤ d(x, y) + g(y) ≤ d(x, x  ) + d(x  , y) + g(y)

.

for all .x  , y ∈ X. Given .δ > 0, there exists y such that .d(x  , y) + g(y) ≤ f (x  ) + δ. Hence f (x) ≤ d(x, x  ) + f (x  ) + δ

.

for all .δ > 0, implying that .f (x) − f (x  ) ≤ d(x, x  ). Since the argument is symmetric in x and .x  , |f (x) − f (x  )| ≤ d(x, x  )

.

and f is continuous on X. Setting .ρ(x) = max{1, 1/f (x)} defines a continuous function on .X \ (E ∪ F ) such that .ρ(x)  σ ≥ 1/d(x, E ∪ F ) and .ρ(x) ≥ 1/g(x). We will use the function .h(x, σ ) = 

0 ρ(xt) dt to define the new flow on X. Exercise 1.3.4 Let .σ , .τ , and .σ + τ be in .(ax , bx ). Use the change of variable .t = σ + s to show .h(x, σ + τ ) = h(x, σ ) + h(xσ, τ ). The equation in Exercise 1.3.4, h(x, σ + τ ) = h(x, σ ) + h(xσ, τ ),

.

(1.2)

is called the cocycle equation and is intimately connected with the reparameterizations of flows. We will make a more systematic use of this connection than Beck did, but the essentials of the argument remain unchanged. (For an overview of cocycles and references, see [40].) If .bx < ∞, then .xbx ∈ E ∪ F and .bxt = bx − t when .t ∈ (ax , bx ). Since .g(x) ≤ bx , it follows that .g(xt) ≤ bx − t and that

τ

h(x, τ ) =

.

ρ(xt) dt ≥

0

0

τ

1 dt = − ln(bx − τ ) + ln bx . bx − t

Thus .

lim h(x, τ ) = ∞

τ →bx

and by similar reasoning .

lim h(x, τ ) = −∞

τ →ax

when .ax > −∞. Since .ρ(x) > 1 for all .x ∈ X \ (E ∪ F ),

24

1 Dynamical Systems .

lim h(x, τ ) = ∞ and

lim h(x, τ ) = −∞

τ →ax

τ →bx

when .bx = ∞ and .ax = −∞, respectively. Since .ρ(x) is a positive function on .X \ (E ∪ F ), the function .t → h(x, t) is increasing, obviously continuous, and a homeomorphism of .(ax , bx ) onto .R. Let .H (x, t) denote its inverse, that is, .H (x, h(x, t)) = t for all .(x, t) in .X \ (E ∪ F ) × R and .h(x, H (x, t)) = t for all x in .X \ (E ∪ F ) and t such that .ax < t < bx . So in this situation .H (x, τ ) is the unique real number .σ such that

τ=

σ

(1.3)

ρ(xt) dt.

.

0

Using the cocycle equation (1.2),   h x, H (x, s) + H [xH (x, s), t] =     h x, H (x, s) + h xH (x, s), H (xH (x, s), t) = .

s+t = h(x, H (x, s + t)). Since there is exactly one real number .σ such that .h(x, σ ) = s + t, H (x, s + t) = H (x, s) + H (xH (x, s), t).

(1.4)

.

To show that H is continuous on .X \ (E ∪ F ) × R, it suffices, because .X × R is a metric space, to prove that given a sequence .(xn , τn ) converging to .(x, τ ) in X \ (E ∪ F ) × R,

.

the sequence .H (xn , τn ) converges to .H (x, τ ). Let .σn = H (xn , τn ) and .σ = H (x, τ ). Because .1 ≤ ρ(y) for all .y ∈ X, we have

.|b − a| ≤

a

b

ρ(xt) dt .

(1.5)

Using equation (1.3) and inequality (1.5), it follows that for large n σn |σn − σ | ≤ ρ(xn t) dt σ σn

= ρ(xn t) dt −

.

0



≤ |τn − τ | +

0

σ

σ



σ

ρ(xn t) dt +

0

0



σ

ρ(xt) dt − 0



σ

ρ(xt) dt −

ρ(xn t) dt .

0

ρ(xt) dt

1.3 Beck’s Theorem

25

Now the right-hand side goes to zero because .τn converges to .τ and Proposition 1.3.2 applies to the last term. Thus H is continuous on .X \ (E ∪ F ) × R. Define .ψ : X \ (E ∪ F ) × R → X \ (E ∪ F ) by .ψ(x, t) = xH (x, t). Then .ψ is continuous because .ϕ(x, t) = xt is continuous by hypothesis and it was just shown that .H (x, t) is continuous. In addition, .ψ(x, s + t) = ψ(ψ(x, s), t) follows from equation (1.4), and .ψ(x, 0) = x because obviously .H (x, 0) = 0. Thus .ψ defines a flow on .X\(E ∪F ). If we extend the definition of .ψ to .X×R by setting .ψ(x, t) = x for all .t ∈ R when x is in .E ∪ F , then clearly .ψ(x, s + t) = ψ(ψ(x, s), t) and .ψ(x, 0) = x remain valid. So it remains to prove that .ψ is continuous on .X × R. Using sequences again, let .(xn , τn ) be a sequence converging to .(x, τ ) in .X × R. We will consider three special cases that together imply the general case. If x is in .X \ (E ∪ F ), then .(xn , τn ) is in .X \ (E ∪ F ) for large n because .X \ (E ∪ F ) is an open set of X. Hence .ψ(xn , τn ) converges to .ψ(x, τ ) because .ψ is continuous on .X \ (E ∪ F ). If .(xn , τn ) is in .(E ∪ F ) × R for all n, then x is in the closed set .E ∪ F and .ψ(xn , τn ) = xn converges to .x = ψ(x, τ ). That leaves the third and substantive case that x is in .E ∪ F and .(xn , τn ) is in the open set .X \ (E ∪ F ) for all n. Let .σn = H (xn , τn ). Since .ψ(xn , τn ) = xn H (xn , τn ) = xn σn , it suffices by the continuity of .ϕ to show that .σn converges to 0. Because .τn is a convergent sequence of real numbers, there exists .M > 0 such that .|τn | < M for all n. There exists a decreasing sequence .δk of positive real numbers converging to 0 such that

δk

.

δk+1

1 dt > M. t

For each .δk , there exists a positive integer .Nk such that .d(xn t, xt) < δk+1 for .|t| ≤ δk when .n > Nk by Proposition 1.3.1 (with .ε = δk+1 , .α = −δk , and .β = δk ). To complete the proof, it will be shown that .|σn | < δk when .n > Nk . Suppose that .|σn | ≥ δk and .n > Nk . Observe that f (xn t) ≤ d(xn t, xt) + g(xt) < δk+1 + |t|

.

because .xt (−t) = x ∈ E ∪ F , and thus ρ(xn t) ≥

.

1 . δk+1 + |t|

The following estimation provides a contradiction when .σn > 0:

τn =

σn

ρ(xn t) dt

.

0

≥

δk −δk+1

ρ(xn t) dt 0

26

1 Dynamical Systems

≥

δk −δk+1

δk+1 + t

0

=

1

δk

δk+1

dt

1 ds s

> M. A similar argument provides a contradiction for .σn < 0. Thus .|σn | < δk when n > Nk and .σn converges to 0 to complete the proof that .ψ is a flow such that .E ∪ F is the set of fixed points of .ψ and .ψ(x, R) ⊂ ϕ(x, R) for all x in X. 

As a simple example of the usefulness of this theorem, consider again the straight line flow .(T2 , R) on the torus given by .ϕ((z, w), t) = (ze2π iat , we2π ibt ) such that a and b are not both zero. It has no fixed points. Since points are closed subsets of metric spaces, the theorem implies there exists a flow .ψ on the torus with the set of fixed points equal to .E = {(1, 1)} such that the .Oψ (z, w) = Oϕ (z, w) if and only if .(z, w) is not an element of the .Oϕ (1, 1) which is just the one-parameter subgroup 2π iat , e2π ibt ) : t ∈ R}. .{(e If .a/b or .b/a is rational, then every orbit of .ϕ is periodic with period .σ > 0. So .ϕ is a periodic flow. Then .Oϕ (1, 1) decomposes into two .ψ-orbits: .{(1, 1)} and 2π iat , e2π ibt ) : 0 < t < σ }. Clearly, .{(e .

ω((e2π iaσ/2 , e2π ibσ/2 )) = α((e2π iaσ/2 , e2π ibσ/2 )) = {(1, 1)}.

.

The other orbits are unchanged. Hence they are compact and periodic by Theorem 1.2.1. Applying Proposition 1.3.1 at .(1, 1) to .α = −k and .β = k with + shows that the .ψ-periods of these orbits go to infinity as the .ψ-periodic .k ∈ Z orbits approach .(1, 1). Now consider a and b, both nonzero, such that .a/b is irrational so that .ϕ is a minimal flow. In this case, .Oϕ (1, 1) decomposes into three .ψ-orbits: .{(1, 1)}, {(e2π iat , e2π ibt ) : t > 0} = Oψ (e2π ia , e2π ib ),

.

and {(e2π iat , e2π ibt ) : t < 0} = Oψ (e−2π ia , e−2π ib ).

.

All the other orbits are unchanged and hence dense. Since .Oϕ+ (1, 1) and .Oϕ− (1, 1) are dense in .T2 , all the .ψ-orbits except .{(1, 1)} are dense in .T2 and αψ (e2π ia , e2π ib ) = {(1, 1)} = ωψ (e−2π ia , e−2π ib ).

.

Exercise 1.3.5 Explain why the following properties hold for the flow .ψ when .a/b is irrational:

1.3 Beck’s Theorem

27

(a) The point .(e2π ia , e2π ib ) is a positively recurrent point of the flow .ψ which is not negatively recurrent, .(e−2π ia , e−2π ib ) is a negatively recurrent point of the flow .ψ which is not positively recurrent, and the other .ψ−orbits except .{(1, 1)} are recurrent and not periodic. (b) The only minimal set of .ψ is the set .{(1, 1)} consisting of only the fixed point. (c) The point .(1, 1) is the only almost periodic point of the flow .ψ. Thus Theorem 1.3.3 easily provides an example showing that recurrent points include a more general type of recurrence than almost periodic points. Other interesting examples will follow from Theorem 1.3.3 in subsequent chapters.

Chapter 2

Flows and Covering Spaces

The thesis of this book is that covering spaces provide a suitable modern setting for a unified presentation of the structure of flows on compact surfaces. This chapter lays the foundation for using covering spaces to study flows on compact surfaces. Using the universal covering space, flows on a compact connected surface can be studied with a bitransformation group similar to the discussion of the straight line flow on the torus (Section 1.1). This approach allows the use of Euclidean and hyperbolic geometries to study the behavior of orbits. The present chapter will provide a common foundation on which to develop these ideas. Since surfaces are just two-dimensional manifolds, covering space theory for manifolds is an appropriate setting. Introduction to Topological Manifolds by John M. Lee ([42]) is an ideal reference for this approach. It will be the consistent choice of reference for covering spaces. Flows on surfaces are part of the larger subject of flows on manifolds. To provide a broader perspective, results that are only needed for surfaces but are no harder to prove for manifolds are included, primarily in Section 2.1. It begins with a review of definitions and basic properties of manifolds, including manifolds with boundary. The two main topics are connected sums of manifolds and filling a boundary component homeomorphic to a sphere. The discussions of these topics include applications to the construction of flows. After reviewing the essential definitions and fundamental covering space theorems for this book from [42], Section 2.2 establishes the primary connection between covering spaces and flows by proving that flows lift to covering spaces. For finite-sheeted normal coverings, it is then shown that important basic dynamical properties also lift. The third section contains a review of the classification of surfaces from [42] and its extension to bordered surfaces [63]. The last result of the section is that every nonorientable compact surface has a 2-sheeted orientable covering which will be used repeatedly to obtain results about flows on nonorientable surfaces.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_2

29

30

2 Flows and Covering Spaces

2.1 Topological Manifolds Manifolds are second-countable Hausdorff topological spaces that locally look like Rn . So, not surprisingly, the n-dimensional Euclidean spaces defined by

.

Rn = {u = (u1 , . . . , un ) : ui ∈ R for i = 1, . . . , n}

.

are manifolds and play a significant role in the theory of manifolds. Given typical points .u = (u1 , . . . , un ) and .v = (v1 , . . . , vn ), of course, u + v = (u1 + v1 , . . . , un + vn )

.

and .au = (au1 , . . . , aun ) for .a ∈ R is the standard vector space structure of .Rn . The Euclidean norm is defined by ⎛ u = ⎝

n 

.

⎞1/2 u2j ⎠

.

j =1

The usual metric used on .Rn is the Euclidean metric defined by .d(u, v) = u − v. With this metric, .Rn is also a topological group under the addition of vectors and .(a, u) → au is continuous. Any open Euclidean ball Enr (u) = {x ∈ Rn : x − u < r}

.

with .r > 0 and .n ≥ 0 is homeomorphic to .Rn and the unit ball Bn = {x ∈ Rn : x < 1}.

.

An n-dimensional topological manifold or more simply an n-dimensional manifold is by definition a second-countable Hausdorff topological space X such that every point has an open neighborhood homeomorphic to an open set of .Rn . The condition that every point of an n-dimensional manifold has an open neighborhood homeomorphic to an open set of .Rn is equivalent to each of the following conditions: (a) Every point has an open neighborhood homeomorphic to an open Euclidean ball .Enr (u). (b) Every point has an open neighborhood homeomorphic to the unit ball .Bn . (c) Every point has an open neighborhood homeomorphic to .Rn . A homeomorphism of an open subset U of an n-dimensional manifold onto an open subset of .Rn is called a chart or sometimes a coordinate chart on U . An open subset U of an n-dimensional manifold X will be called a Euclidean ball of X provided that U is homeomorphic to .Enr (u) for some .r > 0.

2.1 Topological Manifolds

31

Exercise 2.1.1 Prove that an open subset of a manifold is a manifold. The simplest examples of n-dimensional manifolds are .Rn itself and any open subset of .Rn . In particular, every .Enr (u) is a manifold. The unit spheres .Sn−1 = {x ∈ Rn : x = 1} are .(n − 1)-dimensional manifolds. To see this, first note that .Sn−1 is a compact metric space and hence second-countable and Hausdorff. Then given n−1 with .y > 0, for example, it follows that .U = {x ∈ Rn : x > 0} ∩ Sn−1 .y ∈ S n n is an open neighborhood of .y and .f (x) = (x1 , . . . , xn−1 ) is a homeomorphism of U onto .Bn−1 with  −1 .f (w) = (w, 1 − w2 ). Clearly, .h : Sn → Sn defined by .h(x) = −x is a homeomorphism of .Sn onto itself such that .h−1 = h or .h2 = ι, the identity map. Thus we have a .Z2 continuous action on .Sn that is easily seen to be free and proper. The orbit space n .S /Z2 is Hausdorff by Proposition 1.1.19 and second-countable because the natural projection .π : Sn → Sn /Z2 is open and onto. Because .Sn is an n-dimensional manifold, it follows from Corollary 1.1.21 that .Pn = Sn /Z2 is an n-dimensional manifold called the n-dimensional projective space. If X and Y are n-dimensional and m-dimensional manifolds, respectively, then .X × Y with the product topology is an .(n + m)-dimensional manifold. Since topologically .C and .R2 are identical, so are .S1 = {z ∈ C : |z| = 1} and the 1dimensional manifold .{x ∈ R2 : x = 1}, which we will also denote .S1 . So by induction .Tn is an n-dimensional manifold called an n-dimensional torus. Using multiplication of complex numbers makes .S1 a topological group. It follows that .Tn is a topological group for all .n ∈ Z+ . Proposition 2.1.2 If X is an n-dimensional manifold, then X is a locally compact metric space. Proof Let x be a point in X. There exist an open neighborhood U of x and a homeomorphism f of U onto .Bn such that .f (x) = 0 = (0, . . . , 0). If V is any open neighborhood of x, then .f (U ∩ V ) is an open neighborhood of .0 and there exists .r > 0 such that C = {v ∈ Rn : v ≤ r} ⊂ f (U ∩ V ).

.

Since f is a homeomorphism, .f −1 (C) is a compact neighborhood of x contained in .U ∩ V , proving that X is locally compact. Because X is Hausdorff, .f −1 (C) is also a closed neighborhood of x, proving that X is also a regular topological space. Finally, the Urysohn metrization theorem ([39], Theorem 4.16) implies that X is metrizable. 

The proof that the dimension of a manifold is a topological invariant can be found in [35], Theorem 2B.3, [42], Theorem 13.22, and [72], p. 199. We will use the following theorem from Spanier:

32

2 Flows and Covering Spaces

• Brouwer invariance of domain: If U and V are homeomorphic subsets of .Sn and U is open in .Sn , then V is open in .Sn . Because .Sn is the one-point compactification of .Rn (see, for example, [59], p. 253), it follows that if U and V are homeomorphic subsets of .Rn and U is open in .Rn , then V is open in .Rn . (For a more general discussion of compactifications, see Chapter 5 of [39] or Chapter 6, Section 19 of [81].) Proposition 2.1.3 Let x be a point in a topological space X. If there exist open neighborhoods .U1 and .U2 of x homeomorphic to open subsets of .Rm and .Rn , respectively, then .m = n. Proof Clearly, .U = U1 ∩ U2 is an open neighborhood of x homeomorphic to open subsets .V1 and .V2 of .Rm and .Rn , respectively. Assume .m < n. Under this assumption, .Rm is homeomorphic to {x ∈ Rn : xj = 0 for m < j ≤ n},

.

which is a closed nowhere dense subset of .Rn . Consequently, .V1 is homeomorphic to a subset .V3 of .Rn with empty interior. Since both .V2 and .V3 are homeomorphic to U , the open subset .V2 of .Rn is homeomorphic to the subset .V3 of .Rn that is not open, contradicting invariance of domain. 

A connected 1-dimensional manifold is homeomorphic to .S1 if it is compact and to .R if it is not compact ([42], Theorem 6.1). So there is only one compact connected 1-dimensional manifold. A surface is just another name for a 2-dimensional manifold. There are a countable number of compact connected surfaces. We have already seen three of them, namely, .S2 , .P2 , and .T2 . The rest will be discussed in more detail in Section 2.3. The results in the remainder of this section are more easily pictured for surfaces, but the larger context is worth understanding. There is also the concept of a manifold with boundary. The prototype for a manifold with boundary is the upper half space .Hn = {v ∈ Rn : vn ≥ 0}. Given .r > 0, the set {v ∈ Hn : v < r and vn ≥ 0} = Enr (0) ∩ Hn

.

is an open neighborhood of .0 in .Hn that is not homeomorphic to an open subset of n .R by invariance of domain. The collection of these sets is a neighborhood base of n n .H at .0. The same comments apply to every point of .{v ∈ R : vn = 0}, which is n−1 an .(n − 1)-dimensional manifold homeomorphic to .R . It is, however, also clear that .{v ∈ Rn : vn > 0} is an n-dimensional manifold. An n-dimensional manifold with boundary is by definition a second-countable Hausdorff topological space X such that every point has an open neighborhood homeomorphic to an open set of .Hn . A point x lies in the boundary of X provided that there exist an open neighborhood U of x and a homeomorphism f of U onto an open subset of .Hn such that .fn (x) = 0, where .fn is the .nth coordinate function

2.1 Topological Manifolds

33

of f . Observe that .f (U ) ∩ {v ∈ Rn : vn = 0} = V is homeomorphic to an open subset of .Rn−1 and every point in .f −1 (V ) is a boundary point of X. Furthermore, when x is in the boundary of X, an n-dimensional manifold with boundary, there exist an open neighborhood U of x and a homeomorphism f of U onto .Enr (0) ∩ Hn for some .r > 0 such that .f (x) = 0. As stated, the definition of an n-dimensional manifold with boundary includes n-dimensional manifolds, that is, a manifold is a manifold with boundary such that the boundary is empty. Proposition 2.1.4 Let X be a manifold with nonempty boundary denoted by .∂X, and let x be a boundary point of X. A homeomorphism f of an open neighborhood U of x onto an open subset of .Hn has the following properties: (a) Letting .fn (x) denote the .nth coordinate function, .fn (x) = 0. (b) There exist an open neighborhood V of x contained in U and .r > 0 such that .f |V ∩ ∂X is a homeomorphism onto .{v : vn = 0 and v − f (x) < r}. Proof For part (a), suppose .fn (x) > 0. It can be assumed that .f (U ) is an open subset of .{v ∈ Rn : vn > 0}. By hypothesis, there exists a homeomorphism g of an open neighborhood V of x onto .Wr = Enr (0) ∩ Hn such that .g(x) = 0. By replacing V with .g −1 (Wρ ) for a sufficiently small .ρ < r, we can assume that .V ⊂ U , and hence V is also homeomorphic to an open subset of .Rn . This a contradiction because an open subset of .Rn cannot be homeomorphic to .Wρ by invariance of domain, completing part (a). From part (a), we know that .f (x) is in .{v ∈ Rn : vn = 0}. Because .f (U ) is an open subset .Hn , there exists .r > 0 such that .Enr (f (x)) ∩ Hn is contained in .f (U ). Then .V = f −1 (Enr (f (x))) is an open neighborhood of x contained in U . It follows from part (a) that f (V ∩ ∂X) ⊂ Enr (f (x)) ∩ {v ∈ Rn : vn = 0} = {v : vn = 0 and v − f (x) < r}.

.

Conversely, if .u is in .{v : vn = 0 and v − f (x) < r}, then .y = f −1 (u) is in V and U is an open neighborhood of y such that .fn (y) = 0. Therefore, .y = f −1 (u) is in .V ∩ ∂X by the definition of a boundary point and f (V ∩ ∂X) = {v : vn = 0 and v − f (x) < r}.

.

It follows that V is an open neighborhood of x such that f maps .V ∩ ∂X homeomorphically onto .{v : vn = 0 and v − f (x) < r}. 

Corollary 2.1.5 Let X be an n-dimensional manifold with nonempty boundary. (a) .X \ ∂X is an open subset of X and with the relative topology an n-dimensional manifold. (b) .∂X is a closed subset of X. (c) .∂X with the relative topology is an .(n − 1)-dimensional manifold.

34

2 Flows and Covering Spaces

Exercise 2.1.6 Prove Corollary 2.1.5. Clearly, .Hn is an n-dimensional manifold with .∂Hn = {x ∈ Hn : xn = 0}. It follows that .∂Hn is homeomorphic to .Rn−1 and that .Hn \ ∂Hn is homeomorphic to n .R . A simple example of a compact manifold with nonempty boundary is Dn = (Bn )− = {u : u ≤ 1},

.

called the closed unit ball or unit disk. Its boundary is obviously .Sn−1 . Similarly, for real numbers a and b such that .0 < a < b, the annular regions n .{u : a ≤ u < b} and .{u : a ≤ u ≤ b} in .R are connected manifolds with nonempty boundary. Again the boundary component(s) are homeomorphic to .Sn−1 . The inversion map .ξ(u) = u/u2 is a homeomorphism of .Rn \ {0} onto itself that maps .{u : a ≤ u < b} onto .{u : 1/b < u ≤ 1/a}. Since .ξ(u) = u if and only if .u = 1, it follows that .ξ({u : 1 ≤ u < a}) = {u : 1/a < u ≤ 1} for 2 .0 < a < 1. Clearly, .ξ = ι. Proposition 2.1.7 If .ϕ is a continuous action of a connected topological group G on X, an n-dimensional manifold with boundary, then the components of .∂X are G-invariant sets of X. Proof If .∂X = φ, there is nothing to prove. If h is a homeomorphism of X onto itself, then it follows from the definition of boundary points that .h(∂X) = ∂X. Consequently, the boundary is G-invariant even when G is not connected. Since the G-orbit of every point in X is a connected set containing x, the G-orbit of .x ∈ ∂X must be in the component of .∂X containing x. 

Proposition 2.1.8 If X is an n-dimensional manifold with boundary, then X is a locally compact metric space. Exercise 2.1.9 Prove Proposition 2.1.8 using Urysohn’s metrization theorem and the argument in the proof of Proposition 2.1.2. Proposition 2.1.10 If X is a connected manifold with boundary, then .X \ ∂X is connected. Proof Since .X \ ∂X is an open set of X, the components of .X \ ∂X are open sets. Let x be in .∂X and let f be a homeomorphism of an open neighborhood V of x onto .Enr (0) ∩ Hn . Then .f −1 (Enr (0) ∩ {v ∈ Rn : vn > 0}) is an open connected set of .X \ ∂X and hence is contained in exactly one component of .X \ ∂X. Therefore, −1 (En (0) ∩ {v ∈ Rn : v = 0}) is an open connected neighborhood of x in .∂X that .f n r is contained in the closure of a unique component of .∂X. It follows that the closures of the components of .X \ ∂X are disjoint. Let C be a component of .∂X. Then C must be contained in the closure of a unique component of .X \ ∂X by the previous paragraph. Now let U be a component of .X \ ∂X. Clearly, .U − \ U ⊂ ∂X. Either .C ⊂ U − or .C ∩ U − = φ. Thus every point in .U − has an open neighborhood contained in .U − , making .U − an open and closed subset of X. Therefore, .U − = X and .U = X \ ∂X. 

2.1 Topological Manifolds

35

In general the closure of a Euclidean ball in a manifold is not a manifold with boundary, but this can be remedied by the choice of the Euclidean ball. A Euclidean ball B is regular provided that .B − is contained in a Euclidean ball .B with a chart h such that .h(B ) = Enr (0) for some .r > 1 and .h(B) = Bn . Because h is a homeomorphism, .h(B − ) = (Bn )− = Dn . Thus, when B is a regular Euclidean ball, .B − is a compact connected manifold homeomorphic to the closed unit ball n − = B − \ B homeomorphic to .Sn−1 . Moreover, .B \ B is .D with boundary .∂B homeomorphic to the annular region .{u : 1 ≤ u < r} of .Rn , and .B \ B − is homeomorphic to the annular region .{u : 1 < u < r} of .Rn . Every manifold has a countable basis of regular Euclidean balls ([42], Lemma 4.31, p. 83). In the next result, .∂B − is the boundary of .B − not .(∂B)− . Proposition 2.1.11 If X is a connected n-dimensional manifold and B is a regular Euclidean ball in X, then .X \ B − is a connected manifold and .X \ B is a connected manifold with boundary .∂B − , which is homeomorphic to .Sn−1 . Proof Since .X \ B − is an open subset of a manifold, it is a manifold. Because X \ B = (X \ B − ) ∪ (B − \ B),

.

to show that .X \ B is a manifold with boundary, it suffices to show that the points of .(B − \ B) are boundary points of .X \ B. Note that .B \ B is an open subset of .X \ B containing .(B − \ B) because .B \ B = B ∩ (X \ B). Thus .B − \ B is the boundary of .X \ B if and only if it is the boundary of .B \ B. Since .B \ B is homeomorphic to .{u : 1 ≤ u < r}, the points in − = B − \ B are the boundary points of .X \ B. Thus .X \ B is a manifold with .∂B boundary .∂(X \ B) = ∂B − , which is homeomorphic to .Sn−1 . If .X \ B − is the disjoint union of two open sets U and V , then one of them, say V , contains the connected set .B \ B − . It follows that .V ∪ B = V ∪ B − is an open set of X and that U and .V ∪ B − are disjoint open subsets of X whose union is X. Since X was assumed to be connected, this is impossible and .X \ B − is connected. Finally, .X \ B is connected because .X \ B = (X \ B − )− . 

Regular Euclidean balls in compact manifolds can also be collapsed to points. For the continuous action of a connected topological group, in particular, an invariant regular Euclidean ball can be collapsed into a fixed point. The next two results will show how this can be accomplished. Proposition 2.1.12 If B is a regular Euclidean ball in a compact connected manifold X and D is a compact set in X such that .D ∩ B − = φ, then there exists a continuous surjective quotient map .θ : X → X such that .θ (B − ) is a point, − − .θ |(X \ B ) is a homeomorphism onto .X \ θ (B ), and .θ (x) = x for all .x ∈ D. Proof By hypothesis, there exists a Euclidean ball .B containing .B − with a chart h such that .h(B ) = Enr (0) with .r > 1 and .h(B) = Bn . As noted earlier, .h(B − ) = Dn . Let .ρ be a real number such that .r > ρ > 1 and set .Uρ = h−1 (Enρ (0)). Then h maps − n − .Uρ homeomorphically onto .Eρ (0) .

36

2 Flows and Covering Spaces

Because D and .B − are disjoint compact sets in X, there exists .ρ such that .D ∩ Uρ = φ. Define a function .f : [0, ρ] → [0, 1] by

f (s) =

.

⎧ s−1 ⎪ ⎪ ⎨ ρ−1

for ρ ≥ s ≥ 1

⎪ ⎪ ⎩0

for 1 ≥ s ≥ 0

and verify that f is continuous and surjective. Then .F (x) = f (x)x defines a continuous function of .Enρ (0)− onto itself such that .f (Dn ) = {0}. Define .θ by θ (x) =

h−1 (F (h(x)))

.

for x ∈ Uρ− for x ∈ X \ Uρ .

x

Then .θ is continuous because .h−1 (F (h(x))) = x on .Uρ− \ Uρ = Uρ− ∩ (X \ Uρ ). Routine arguments show that .θ (B − ) is a point in X and that .θ maps .X \ B − bijectively to .X \ θ (B − ). Since X is compact, every closed set C of X is compact, so .θ (C) is compact and hence closed because X is Hausdorff. Therefore, .θ is a closed function and the topology on X is the quotient topology. (The fact that a function with a compact domain and Hausdorff range is a quotient map will be used regularly.) It follows that .θ maps open sets of .X \ B − to open subsets of .X \ θ (B − ) because they are saturated. Thus .θ maps .X \ B − homeomorphically onto .X \ θ (B − ). 

Exercise 2.1.13 Verify the following details in the proof of Proposition 2.1.12: (a) the function f is continuous and surjective, (b) .θ (B − ) is a point in X, and (c) .θ maps .X \ B − bijectively to .X \ θ (B − ). Theorem 2.1.14 Let B be a regular Euclidean ball in a compact connected manifold X, let D be a compact set in X such that .D ∩ B − = φ, and let .ϕ : X ×G → X be a continuous action of a metric group on X. If B is a .ϕ-invariant set, then there exist a continuous action .ψ : X × G → X and a homomorphism − .θ : (X, G, ϕ) → (X, G, ψ) such that .θ (B ) is a .ψ-fixed point y and .θ is an isomorphism of the flow .ϕ restricted to .X \ B − onto the flow .ψ restricted .X \ {y} such that .θ (x) = x for all .x ∈ D. Proof Let .θ : X → X be given by Proposition 2.1.12. Letting .y = θ (B − ), define .ψ by ψ(x, g) =

θ ϕ(θ −1 (x), g)

when x = y

y

when x = y.

.

Because B is .ϕ-invariant, the sets .B − and .X \ B − are .ϕ-invariant.

2.1 Topological Manifolds

37

To prove that .ψ is continuous, let .(xn , gn ) be a sequence in .X × G converging to (x, g). It must be shown that .ψ(xn , gn ) in X converges to .ψ(x, g). (G-metric avoids the use of nets or filters.) If .x = y, then it follows from Proposition 2.1.12 and the definition of .ψ that .ψ(xn , gn ) converges to .ψ(x, g). Suppose .x = y and let V be an open neighborhood of y. Then .θ −1 (V ) is an open set of X containing .B − . It suffices to assume that .xn = y for all n and to prove that −1 (x ), g ) is in .θ −1 (V ) for large n. .ϕ(θ n n If this is not the case, then by passing to subsequence it can be assumed that −1 (x ), g ) is in .X \ θ −1 (V ) for all n and converges to w in the closed set .X \ .ϕ(θ n n −1 θ (V ) because X is compact. By passing to a subsequence one more time, we can assume that .θ −1 (xn ) converges to z. Obviously, .θ (z) = y and z is in .B − . Therefore, − because it is .ϕ-invariant, but w is in .X \ θ −1 (V ), a .w = ϕ(z, g) which is in .B contradiction. Thus .ψ is continuous. The remaining details are left to the reader. 

.

Let X be an n-dimensional manifold with boundary, .n ≥ 2, that consists of two components .X1 and .X2 both homeomorphic to .{u : 1 ≤ u < r} for some .r > 1. So X has two boundary components .∂X1 and .∂X2 both homeomorphic to .Sn−1 and to each other. There also exist homeomorphisms .hj of .Xj onto .{u : 1 ≤ u < r}.

 Let .σ be any homeomorphism of .∂X1 onto .∂X2 . Setting .g(x) = h2 σ [h−1 1 (x)] for .x ∈ Sn−1 defines a homeomorphism of .Sn−1 onto itself. It can be extended to a homeomorphism G of .{u : 1 ≤ u < r} onto itself such that .G(u) = u by setting  G(u) = u g

.

 u , u

(2.1)

a property of spheres that will also be used again in Section 2.3. Then .G ◦ h1 is also a homeomorphism of .X1 onto .{u : 1 ≤ u < r}. Letting .ξ(u) = u/u2 as before defines a map f from X to .{u : 1/r < u < r} by f (x) =

.

ξ ◦ G ◦ h1 (x)

for x ∈ X1

h2 (x)

for x ∈ X2 .

Given .x ∈ ∂X1 , both .h1 (x) and .h2 (σ (x)) are in .Sn−1 and f (x) = ξ ◦ G ◦ h1 (x)

.

= ξ ◦ g ◦ h1 (x) = ξ ◦ h2 ◦ σ ◦ h−1 1 ◦ h1 (x) = ξ ◦ h2 ◦ σ (x) = h2 (σ (x)) = f (σ (x)).

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2 Flows and Covering Spaces

Because .h1 and .h2 are surjective, it follows that .f −1 (u) = {x, σ (x)} for some −1 (u) = {y, σ −1 (y)} for some .y ∈ ∂X . Obviously, .y = σ x. Thus .x ∈ ∂X1 or .f 2 −1 .f (u) consists of 1 or 2 points of X depending on whether .u = 1 or .u = 1. If C is a closed set of X, then .C = (C ∩ X1 ) ∪ (C ∩ X2 ) and both .C ∩ X1 and .C ∩ X2 are closed sets of X. Observe that f maps both .C ∩ X1 and .C ∩ X2 homeomorphically onto closed subsets of .{u : 1/r < u < r}. Consequently, .f (C) = f (C ∩ X1 ) ∪ f (C ∩ X2 ) is a closed subset of .{u : 1/r < u < r} and f is a closed function. Therefore, f is a quotient map, and the quotient topology coincides with the usual topology on .{u : 1/r < u < r}. The function f determines an equivalence relation .∼σ on X with equivalence classes .{x, σ (x)} for .x ∈ ∂X1 and .{x} for .x ∈ / ∂X1 ∪ ∂X2 . Note that .∼σ depends only on the choice of .σ . Thus forming the quotient space .X/ ∼σ uses the homeomorphism .σ to glue .∂X1 and .∂X2 together. The uniqueness of the quotient space ([42], Corollary 3.32) implies that .X/ ∼σ is homeomorphic to the manifold .{u : 1/r < u < r} and therefore a manifold. This establishes the following proposition: Proposition 2.1.15 Let X be an n-dimensional manifold with boundary that consists of two components .X1 and .X2 both homeomorphic to .{u : 1 ≤ u < r} for some .r > 1. Let .σ be any homeomorphism of .∂X1 onto .∂X2 . If .x ∼σ y if and only if ±1 (y), then .X/ ∼ with the quotient topology is an n-dimensional .x = y or .x = σ σ manifold homeomorphic to .{u : 1/r < u < r}. Let X and Y be connected n-dimensional manifolds, and let .BX and .BY be and .B with regular Euclidean balls in X and Y contained in Euclidean balls .BX Y n charts .hX and .hY such that .hX (BX ) = hY (BY ) = Er (0) for some .r > 1, respectively. By Proposition 2.1.11, .X\BX and .Y \BY are connected manifolds with − boundaries .∂BX and .∂BY− , respectively, and both boundaries are homeomorphic to − n−1 .S . So there exists a homeomorphism .σ : ∂BX → ∂BY− . Set .X = X \ BX and .Y = Y \ BY . Define a topology on the disjoint union .X  Y of .X and .Y by a subset V is open if and only if .V ∩X and .V ∩Y are open in .X and .Y , respectively. This topology is called the coherent topology. It follows that .X  Y is a manifold, but not a connected one. As before, define an equivalence relation .∼σ on .X  Y by .x ∼σ y if and only if .x = y or .x = σ ±1 (y). The connected sum of X and Y is .X  Y / ∼σ with the quotient topology and is denoted by .X#Y . For example, see Figure 2.1. Fig. 2.1 The connected sum of two copies of a torus

2.1 Topological Manifolds

39

Proposition 2.1.16 Let .(X, ϕ) and .(Y, ψ) be flows on connected n-dimensional manifolds, and let .BX and .BY be regular Euclidean balls in X and Y contained in and .B . If .σ : ∂B − → ∂B − is a surjective flow isomorphism Euclidean balls .BX Y X Y − − of .∂BX to .∂BY , then ϕ#ψ(w, t) =

.

ϕ(w, t)

if w ∈ X

ψ(w, t) if w ∈ Y

is a flow on .X#Y . Exercise 2.1.17 Prove Proposition 2.1.16 by verifying that the flows .ϕ and .ψ consistently define a flow on .X  Y / ∼σ with the quotient topology. Proposition 2.1.18 If X and Y are connected n-dimensional manifolds, then any connected sum .X#Y of X and Y is a connected n-dimensional manifold. − Proof Let .π : X  Y → X#Y be the quotient map. Observe that .X \ BX = X \ − ∂BX is an open connected saturated subset of .X  Y and that .π maps the manifold − .X \ ∂B X homeomorphically onto an open connected set .WX of .X#Y . Likewise, − .π is a homeomorphism of .Y \ ∂B Y onto the open set .WY of .X#Y . Furthermore, .(B \ BX )  (B \ BY ) is a saturated open set of .X  Y , and .π (B \ BX )  (B \ X Y X Y BY ) = W is an open set of .X#Y homeomorphic to .{u : 1/r < u < r} for some .r > 0 by Proposition 2.1.15. Since .X#Y = WX ∪ WY ∪ W , it follows that every point of .X#Y has an open neighborhood homeomorphic to an open subset of .Rn and that .X#Y is second− countable. Note that .WX and .WY are connected because .X \ BX and .Y \ BY− are connected by Proposition 2.1.11 and that neither .WX ∩ W nor .WY ∩ W is empty. Consequently, .X#Y is connected because the open set W is homeomorphic to an annular region and hence connected. To prove that .X#Y is Hausdorff, first note that if x and y are distinct points in .WX , .WY , or W , then there exist disjoint open neighborhoods U and V of x and y because .WX , .WY , and W are open manifolds contained in .X#Y . Otherwise, x is in .WX or .WY and y is not in .WX or .WY , respectively. Consider the case that x is in .WX and y is not in .WX . Then .π −1 (x) is a − point in .X \ ∂BX and there exists an open neighborhood U of .π −1 (x) such that − − − .U ⊂ X \ ∂BX because .X \ ∂BX is an open subset of .X , which is a metric space (Proposition 2.1.2) and hence a regular topological space. So U and .U − are saturated open and closed sets, respectively. Therefore, .π(U ) and .π(U − ) are open and closed sets of .X#Y , respectively. It follows that .π(U ) and .X#Y \ π(U − ) are disjoint open neighborhoods of x and y, respectively. The same reasoning works for the other case to complete the proof. 

Proposition 2.1.19 Any continuous group action .ϕ of a metric group G on the unit sphere .Sn−1 extends to a continuous group action .ψ of G on the unit disk .Dn . Proof Building on equation (2.1), set

40

2 Flows and Covering Spaces

ψ(x, g) = xϕ(x/x, g)

.

for .x = 0 and .g ∈ G, and set .0g = 0. Note that .ψ(x, g) = x for all .x and g. Obviously, .ψ(x, e) = x and .ψ(0, gh) = ψ(ψ(0, g), h). Observe that for .x = 0   x   xϕ x ,g x ψ(x, g) = =ϕ ,g , . ψ(x, g) ψ(x, g) x and then 

ψ(x, g) ,h ψ(x, g)     x ,g ,h = xϕ ϕ x   x , gh = xϕ x



ψ(ψ(x, g), h) = ψ(x, g) ϕ

.

= ψ(x, gh).

Let .xk be a sequence in the unit ball converging to .x and .gk a sequence in G converging to g. If .x = 0, then clearly  ψ(xk , gk ) = xk ϕ

.

xk , gk xk 



converges to .ψ(x, g) and .ψ is continuous at .(x, g). The continuity of .ψ at .(0, g) follows from .ψ(x, g) = x. 

Given topological spaces X and Y , a continuous function .h : X → Y is called an embedding provided that h is a homeomorphism from X to .f (X) with the relative topology from Y . When .h : X → Y is an embedding, X is said to be embedded in Y by h and .h(X) is an embedded copy of X in Y . If X is a compact manifold with nonempty boundary, then there exists an open set U containing .∂X such that U is homeomorphic to .∂X ×[0, 1) by a homeomorphism taking .x ∈ ∂X to .(x, 0) ([35], Proposition 3.42). The open set U is called a collar neighborhood of .∂X. By making the radius of the collar, it can be assumed that the components of U are collar neighborhoods of the components of .∂X. The ideas used to prove Proposition 2.1.18 can now be used to remove a boundary component homeomorphic to a sphere by gluing a closed unit ball to it. This construction will be denoted by .X . Theorem 2.1.20 Let X be an n-dimensional compact connected manifold with nonempty boundary, and let C be a component of .∂X. If C is homeomorphic

2.1 Topological Manifolds

41

to .Sn−1 , then there exists .X , a compact connected manifold with boundary, and embeddings h and .h of X and .Dn , respectively, into .X with the following properties: (a) (b) (c) (d)

∂X = h(∂X \ C). n−1 ) = h(C). .h (S n  \ h(X). .h (B ) = X n  .h (B ) is a regular Euclidean ball in .X . .

Furthermore, if .ϕ is a continuous group action of a connected metric group G on X, then there exists a continuous group action .ψ of G on .X such that h is an isomorphism of .(X, G, ϕ) into .(X , G, ψ). Proof There exists a collar neighborhood .UX of C such that .UX is homeomorphic to the annular region .{u ∈ Rn : 1 ≤ u < 2} by a homeomorphism f that maps C onto .Sn−1 . Let .UDn = {u : 1/2 < u ≤ 1}, which is a collar neighborhood of .∂Dn = Sn−1 . Let .σ = f |C, so .σ is a homeomorphism of C onto .Sn−1 . Form n .X  D with the coherent topology and, as before, define .∼σ by .x ∼σ y if and only if .x = y or .x = σ ±1 (y). In particular, .σ −1 (u) ∼σ u when .u = 1. It will be shown that .X = X  Dn / ∼σ has the desired properties. Letting .π : X  Dn → X be the quotient map, the sets .WX = π(X \ C) and .WDn = π(Bn ) are open in .X because .X \ C and .Bn are open saturated sets of .X  Dn . And .W = π(UX  UDn ) is an open subset of .X homeomorphic to .{u : 1/2 < u < 2} by Proposition 2.1.15. Now the final three paragraphs of the proof of Proposition 2.1.18 can be used almost verbatim to show that .X is an n-dimensional connected manifold with boundary and .C ∩ ∂X = φ. The compactness of X and .Dn imply that .X  Dn is compact and hence so is .X . Since .X is Hausdorff, the functions .h = π |X and .h = π |Dn are homeomorphisms into .X because they are continuous and injective and both X and .Dn are compact. It follows from the first two paragraphs of the proof that the only points in .X that have open neighborhoods homeomorphic to .Enr (0) ∩ Hn for some .r > 0 are precisely the points in .π(∂X \ C) = h(∂X \ C), proving property (a). Properties (b) and (c) are consequences of the definition of .∼σ . For property (d), it must be shown that θ (u) =

h (u)

.

π

◦ f −1 (u)

for u ≤ 1 for 1 ≤ u < 2

is a homeomorphism of .{u : u < 2} onto an open set of .X . When .u = 1, π ◦ f −1 (u) = π(σ −1 (u)) = θ (u)

.

because .σ −1 (u) ∼σ u, and it follows that .θ is a continuous injective function.

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2 Flows and Covering Spaces

To show that .θ is an open function, it suffices to show that the saturated set π −1 (θ (V )) is an open set of .X  Dn when V is an open subset of .{u : u < 2}. Set .V1 = V ∩ {u : 1 ≤ u < 2} and .V2 = V ∩ Dn . Then .f −1 (V1 ) is an open set of X because f is a homeomorphism and .UX is an open subset of X. It follows that −1 (θ (V )) ∩ X = f −1 (V ) is an open set of X. Similarly, .π −1 (θ (V )) ∩ Dn = V .π 1 1 2 2 is an open set of .D n . Therefore, .

π −1 (θ (V )) = [π −1 (θ (V1 )) ∩ X] ∪ [π −1 (θ (V2 )) ∩ Dn ] = f −1 (V1 ) ∪ V2 ,

.

which is an open subset of .X  Dn . Thus .θ is an open function and .θ is a homeomorphism, completing the proofs of (a) through (d). Let .ϕ be continuous group action of a connected topological group G on X. By Proposition 2.1.7, C is a .ϕ-invariant set, and .σ can be used to define a continuous action of G on .Sn−1 by setting ϕ (u, g) = σ (ϕ(σ −1 (u), g)).

.

Its extension to .Dn given by Proposition 2.1.19 will also be denoted by .ϕ . Together .ϕ and .ϕ define a continuous action . ϕ of G on .X  Dn in the obvious way. Since h and .h are homeomorphisms of the . ϕ -invariant sets X and .Dn of .X  n  D into .X , they can be used to define flows .ψ1 and .ψ2 on .h(X) and .h (Dn ), respectively, as .σ was used to define .ϕ on .Sn−1 . Observe that h(X) ∩ h (Dn ) = h(C) = h (Sn−1 )

.

and that .h(C) and .h (Dn ) are invariant sets of .ψ1 and .ψ2 . Therefore, ψ(y, g) =

.

ψ1 (y, g)

if y ∈ h(X)

ψ2 (y, g)

if y ∈ h (Dn )

defines a flow on .X if .ψ1 and .ψ2 agree on their common invariant set. If .x ∼σ u, then x is in C, .u is in .Sn−1 , and .u = σ (x). Hence ϕ (u, g) = σ (ϕ(σ −1 (u), g)) = σ (ϕ(x, g))

.

and .ϕ(x, g) ∼σ ϕ (u, g) or equivalently .h(ϕ(x, g)) = h (ϕ (u, g)). Consequently, ψ1 (h(x), g) = h(ϕ(x, g)) = h (ϕ (u, g)) = ψ2 (h (u), g)

.

and .ψ is the required continuous group action on .X .



2.2 Lifting Flows

43

2.2 Lifting Flows The primary purpose of this section is to explore the interconnection between covering spaces and flows. This necessitates a certain amount of review of covering space facts. The source of the covering space material presented here is Chapters 11 and 12 of [42], but other books on algebraic topology such as [35] and [72] contain most of the same information but with some variation. When a topological space X has a basis of open connected sets, the space X is said to be locally connected. Hence, the components, that is, the maximal connected subsets of X, of a locally connected space are open sets of X. A topological space X is path-connected provided that for every pair of distinct points x and y in X there exists a continuous function .f : [0, 1] → X, called a curve or path, such that .f (0) = x and .f (1) = y. When X has a basis of open path-connected sets, the space X is said to be locally path-connected. For example, manifolds are all locally path-connected. Clearly, a path-connected topological space is connected, and a locally pathconnected topological space is locally connected. It is readily shown that the path components, that is, the maximal path-connected subsets of X, of a locally pathconnected space are open sets and coincide with the components of X. In particular, a manifold is connected if and only if it is path-connected. Let X and Y be two topological spaces. Recall that two continuous functions f and g from Y to X are homotopic provided there exists a continuous function .H : Y × [0, 1] → X, called a homotopy, such that .H (y, 0) = f (y) and .H (y, 1) = g(y) for all .y ∈ Y . Two paths .f0 , f1 : [0, 1] → X are path-homotopic provided there exists a homotopy .H : [0, 1]2 → X such that .H (s, 0) = f0 (s), .H (s, 1) = f1 (s), .H (0, t) = f0 (0) = f1 (0), and .H (1, t) = f0 (1) = f1 (1) for all .s, t ∈ [0, 1]. A closed curve or loop is a continuous function .α : [0, 1] → X such that .α(0) = α(1). Given q in X, path-homotopy is an equivalence relation on all the loops .α such that .α(0) = α(1) = q, and the equivalence classes are the elements of the fundamental group . 1 (X, q). The identity element is the class of null-homotopic loops, that is, the loops that are path-homotopic to the constant path .s → q for all .s ∈ [0, 1]. A continuous function .g : X → Y induces a homomorphism .g∗ :

(X, q) → (X, g(q)) by mapping the equivalence class of .f0 to the equivalence class of .g ◦ f0 . When X is path-connected, . 1 (X, q) is independent of the base point q and is denoted simply by . 1 (X). When the fundamental group . 1 (X) of a path-connected space X is trivial, the space X is said to be simply connected. A space X is said to be locally simply connected provided it has a basis of open simply connected sets. Since a simply connected space is path-connected, locally simply connected spaces must also be locally path-connected. Every manifold is locally simply connected because .Rn is simply connected.  → X, a connected open subset U of X is Given a continuous function .π : X evenly covered by .π provided that each component of .π −1 (U ) is an open subset

44

2 Flows and Covering Spaces

 that is mapped homeomorphically onto U . Thus the cardinality of the fiber of .X π −1 (x) is constant on an evenly covered connected open set.  → X is a covering map Following [42], a continuous surjective function .π : X  is a covering space provided that the space .X  is path-connected and locally and .X path-connected and that every point of X is in an open connected set of X that is evenly covered by .π . The classical example of a covering map is .π : R → S1 with 2π ix or generally .π : Rn → Tn with .π(x) = e .

  π(x) = e2π ix1 , . . . , e2π ixn .

.

It is also easy to verify that the natural projection .π : Sn → Pn = Sn /Z2 is a covering map and an example of the following result: Proposition 2.2.1 If the continuous action of a discrete group G on a connected  → X/G   is proper and free, then the natural projection .π : X manifold .X is a  covering map and .X/G is a connected manifold.  is a connected manifold, it is locally path-connected and pathProof Since .X   is is Hausdorff. Because .X connected. Proposition 1.1.19 implies that .X/G  second-countable and .π is an open function by Exercise 1.1.1, the space .X/G is  second-countable. Now Corollary 1.1.21 implies that .X/G is a manifold and every   is in an open connected set of .X/G that is evenly covered by .π . point in .X/G   is connected. Finally, .X/G is connected because .X 

It is easily shown that a covering map is always an open function and hence a quotient map. If X has a covering space, then X is path-connected, locally pathconnected, and has a basis of open connected sets that are evenly covered.  → X is a covering map, then the connectivity of X implies that the If .π : X cardinality of .π −1 (x) is the same for all .x ∈ X. When the constant cardinality of the fibers is finite, the covering map is said to be finite-sheeted. For example, k 1 1 .π(z) = z is a k-sheeted covering of .S by .S .  Given a covering map .π : X → X, a topological space Y , and a continuous  such that function .f : Y → X, a lift of f is a continuous function .f : Y → X  .f = π ◦ f . Lifts have a number of extremely useful properties that will be used  → X is a covering map, then the following hold: repeatedly. If .π : X • Unique lifting property: Let Y be a connected space and let .f1 and .f2 be lifts of .f : Y → X. If .f1 equals .f2 at some point in Y , then .f1 = f2 ([42], Proposition 11.9). • Path lifting property: If .f : [0, 1] → X is a path in X and .p  is a point in −1 (f (0)), then there exists a unique lift .f of f such that .f(0) = p .π  ([42], Proposition 11.10). • Homotopy lifting property: Suppose .f0 , f1 : [0, 1] → X are path-homotopic. If .f0 and .f1 are lifts of .f0 and .f1 , respectively, such that .f0 (0) = f1 (0), then .f0 and .f1 are path-homotopic ([42], Proposition 11.11).

2.2 Lifting Flows

45

• Lifting criterion: Let Y be a connected locally path-connected topological  and .q ∈ Y space, and let .g : Y → X be a continuous function. Given .p ∈ X such that .π( p) = g(q), there exists a unique lift . g of g such that . g (q) = p  if and only if the subgroup .g∗ ( 1 (Y, q)) of . 1 (X, g(q)) is contained in the subgroup  p .π∗ ( 1 (X, )) of . 1 (X, π( p )) = 1 (X, g(q)) ([42], Theorem 11.15). Let .HY denote the group of homeomorphisms of a space Y onto itself. Let  → X be a covering map. A covering transformation is by definition a π : X  onto itself such that .π ◦ T = π . homeomorphism T of the covering space .X Covering transformations are sometimes called deck transformations or covering  is a covering space automorphisms. The identity map  .ι of a covering space .X transformation. It is readily checked that the set of covering transformations is a group under the composition of functions and hence a subgroup of .HX . The group of covering transformations will be denoted by . or .X  if necessary and often called the covering group. The covering group will always be endowed  will be written on the left. with the discrete topology, and its continuous action on .X  but the .-orbit of . Note that the orbit . x is a subset of .π −1 (π( x )) for all . x ∈ X, x is not in general equal to .π −1 (π( x )).  → X. If S and T  can be viewed as lifts of .π : X Covering transformations on .X  then .S = T by the are two elements of . such that .S( p ) = T ( p) for some .p  ∈ X,  unique lifting property. Thus . acts freely on .X. Let U be an open connected subset of X that is evenly covered by the covering  is also a  is a component of .π −1 (U ). If T is in ., then .T U map .π . Suppose .U −1 component of .π (U ). Consequently, T permutes the components of .π −1 (U ), and ∩U  = φ if and only if T is the identity map of .X.  It is now an exercise to verify .T U  that . acts properly on .X. .

 Exercise 2.2.2 Verify that . acts properly on .X.  → X be a covering map, and let .ϕ be a continuous Theorem 2.2.3 Let .π : X (right) action of a topological group G on X. If G is connected, locally pathconnected, and simply connected, then there exists a unique continuous group action  such that . ϕ on .X π( ϕ ( x , g)) = ϕ(π( x ), g) or π( x g) = π( x )g

.

 and all .g ∈ G. Furthermore, if T is a covering transformation for for all . x ∈ X  → X, then π :X

.

T ( ϕ ( x , g)) =  ϕ (T ( x ), g) or T ( x g) = (T  x )g

.

 and all .g ∈ G. for all . x∈X   Proof Consider the function .f = ϕ◦(π ×ι) mapping .X×G to X and a point . y of .X. Let e denote the identity of G. y , e) = π( y ) = y, an element of X. Because

Then .f (  × G, (   G is simply connected, . 1 X y , e) = 1 (X, y ), that is, every element of

46

2 Flows and Covering Spaces

  × G, ( ×G y , e) can be represented by a closed curve or loop .α : [0, 1] → X

1 X  of the form .α(s) = (β(s), e) and .β represents an element of . 1 (X,  y ). It follows that .f ◦ α = π ◦ β. Therefore,

.



  × G, (   f∗ 1 X y , e) = π∗ 1 X, y

.

and the lifting criterion applies to f . So there exists a unique continuous function ×G → X  such that .π ◦   and  ϕ:X ϕ ( x , g) = f ( x , g) = ϕ(π( x ), g) for all . x∈X .g ∈ G and such that . ϕ ( y , e) =  y . It remains to show that . ϕ is a group action. Clearly, .π ◦  ϕ ( x , e) = ϕ(π( x ), e) = π( x ) and the function . x →  ϕ ( x , e) is a lift ϕ ( y , e) =  y = ι( y ), the unique lifting property implies that . ϕ ( x , e) =  x of .π . Since .  for all . x ∈ X. Given x in X, consider the continuous function .

 f (x, g, h) = ϕ(x, gh) = ϕ ϕ(x, g), h

.

mapping .X × G × G into X. If . x is in .π −1 (x), then an easy calculation shows that π( ϕ ( x , gh)) = π( ϕ ( ϕ ( x , g), h)) = f (x, g, h)

.

and they are both lifts of f . Because . ϕ ( x , e) =  x , it follows that . ϕ ( x , ee) =  x and . ϕ ( ϕ ( x , e), e) =  x . The unique lifting property now implies that . ϕ ( x , gh) =  ϕ ( ϕ ( x , g), h) for all .g, h ∈ G. Since x was an arbitrary point in X and . x was an arbitrary point in .π −1 (x),  ϕ ( x , gh) =  ϕ ( ϕ ( x , g), h)

.

 × G × G. Similarly, one shows that .T ( is valid on .X ϕ ( x , g)) =  ϕ (T ( x ), g).



Exercise 2.2.4 In the proof of Theorem 2.2.3, verify that (a) .π( ϕ ( x , gh)) = ϕ ( x , g)) =  ϕ (T ( x ), g). π( ϕ ( ϕ ( x , g), h)) = f (x, g, h) and that (b) .T ( Theorem 2.2.3 appears in a 1964 paper [45] by Lima, who used it to prove that commuting vector fields on a compact surface have a common fixed point. Prior to 1964, it may well have been a folklore theorem. → Of course, Theorem 2.2.3 applies to a flow .(X, R) with a covering map .π : X  X. The resulting flow on .X will be called the lifted flow and will be denoted by  R). Given . .(X, x in .π −1 (x), the path lifting property implies that .t →  x t is the unique lift of .t → xt such that .0 →  x . If .π( x1 ) = π( x2 ) and . x1 =  x2 , then . x1 t =  x2 t for all .t ∈ R by the unique lifting property. Because the unique lift of a  R) if and only if constant function is a constant function, . x is a fixed point of .(X,  .π( x ) is a fixed point of .(X, R). Moreover, .(, X, R) is a bitransformation group by Theorem 2.2.3.  → X be a finite-sheeted covering map, let .(X, R) be Proposition 2.2.5 Let .π : X  R) be its lifted flow. If .π( a flow, and let .(X, x ) = x, then the following hold:

2.2 Lifting Flows

(a) (b) (c) (d) (e)

47

π(O( x )− ) = O(π( x ))− + − + .π(O ( x ) ) = O (π( x ))− − − − .π(O ( x ) ) = O (π( x ))− .π(ω( x )) = ω(x) .π(α( x )) = α(x) .

Proof We begin with a simple observation that is very useful. Assume the covering has k sheets. Let U be an evenly covered open connected subset of X and let k be the components of .π −1 (U ). If xt is in U and .π −1 (x) = { 1 , . . . , U .U x1 , . . . ,  xk }, i contains exactly one . then each .U xj t. Beginning with part (a), the left-hand side is contained in the right-hand side by continuity of .π . Suppose y is in .O(π( x ))− , and let .π −1 (y) = { y1 , . . . ,  yk }. If − −1 k of 1 , . . . , V .O( x ) ∩ π (y) = φ, then there exist disjoint open neighborhoods .V  . y1 , . . . ,  yk , respectively, such that .O( x ) ∩ Vj = φ for .j = 1, . . . , k. Since .π is  j ) is an open neighborhood of y. There exists an an open function, .V = kj =1 π(V open connected evenly covered neighborhood U of y contained in V and .τ ∈ R such that .xτ ∈ U . It follows from the observation in the previous paragraph that each of the k −1 components of .π −1 (U ) contains one of the k points ), which includes . xτ . k of .π (xτ  But U was constructed so that . x τ was not in . j =1 Vj ⊃ π −1 (U ), a contradiction that completes the proof of part (a). The same argument applies to parts (b) and (c) with .O+ ( x ) and .O− ( x ), respectively, replacing .O( x ). For part (d), the definition of the .ω-limit set and part (b) imply that the left-hand side is contained in the right-hand side. Suppose y is in .ω(x). Then for every .τ ≥ 0, there exists . y ∈ π −1 (y) such that . y is in .O+ ( x τ )− by part (b). Because .π −1 (y) −1 is finite, there exists . y ∈ π (y) such that . y is in .O+ ( x τ )− for arbitrarily large .τ . + − + − Since .O ( x τ ) ⊂ O ( x σ ) when .σ < τ , the point . y is in .ω( x ) to complete the proof of part (d). The proof of part (e) is similar. 

 → X is said to be a normal covering map or sometimes a A covering map .π : X

   regular covering provided that .π∗ 1 (X, x ) is a normal subgroup of . 1 (X, π( x ))   for all . x ∈ X. It turns out, however, that the covering map .π : X → X is normal    x )) is a normal subgroup of . 1 (X, π( x )) for some . x∈X if and only if .π∗ ( 1 (X, (Lemma 11.20 in [42]). The significance of normal covering spaces for lifted flows is the following property:  → X is normal if and only • Fibers of normal coverings: A covering map .π : X  (Proposition 11.29 (b) in [42]). if the orbit . x = π −1 (π( x )) for all . x∈X  → X is a normal covering map with covering group ., then .θ ( If .π : X x) =   onto to X by the uniqueness of quotient π( x ) defines a homeomorphism of .X/  → X is a quotient map. Thus a normal covering map is, for spaces because .π : X  onto the orbit space .X/  . all practical purposes, a natural projection of .X   R) is If, in addition, .(X, R) is a flow with lifted flow .(X, R), then .(, X, a bitransformation group by Theorem 2.2.3. Recall (Proposition 1.1.23) that

48

2 Flows and Covering Spaces

 . Hence for a normal covering, .θ is also ( x , t) → ( x t) defines a flow on .X/  , R) onto .(X, R) and there is the possibility of using an isomorphism of .(X/  R) to study .(X, R). For finite-sheeted coverings, the bitransformation group .(, X, recursive properties can be lifted, but for infinite coverings geometric techniques will be used to study the lifts of recurrent orbits.

.

 → X be a finite-sheeted normal covering map, let Theorem 2.2.6 Let .π : X   .(X, R) be a flow, and let .(X, R) be its lifted flow. The following hold for . x ∈ X:  R) if and only if .π( (a) . x is a periodic point of .(X, x ) is a periodic point of .(X, R).  R) if and only if .π( (b) . x is an almost periodic point of .(X, x ) is an almost periodic point of .(X, R).  R) if and only if .π( (c) . x is a positively {negatively} recurrent point of .(X, x ) is a positively {negatively} recurrent point of .(X, R).  R) if and only if .π( (d) . x is a recurrent point of .(X, x ) is a recurrent point of .(X, R). Proof In each of the four results, the “only if” part of the proof is an easy consequence of the fact that the covering map .π is a homomorphism of the flow  R) onto the flow .(X, R) and will be omitted. Throughout the proof, .x = π( .(X, x ). x τ ) = x and there exists Suppose x is a periodic point of period .τ . Then .π( .T ∈  such that .T  x = x τ . It follows that .T 2 x = T ( x τ ) = (T  x )τ =  x (2τ ) and k by induction .T  x =  x (kτ ). There exists a smallest positive integer m such that mx =  .T  x and so . x = x (mτ ). Thus . x is periodic, but the period can be an integer multiple of .τ .  be an open neighborhood of . Next suppose x is almost periodic and let .U x.  is a component of an evenly Without loss of generality, we can assume that .U covered connected open neighborhood U of x. To prove that there exists a syndetic , we need to select for each .T ∈  an open neighborhood subset A such that . xA ⊂ U  and .σT = 0. For  ι = U .VT of .T  x and .σT ∈ R. For .T =  ι, the identity of ., set .V − .T =  ι, there are two cases—either .T  x is or is not in .O( x) . If .T  x is in .O( x )− , then .O(T  x )− ⊂ O( x )− and O(T 2 x )− = T O(T  x )− ⊂ T O( x )− = O(T  x )− .

.

Letting m, as above, be the smallest positive integer such that .T m x = x , it follows that O( x )− ⊂ O(T m−1 x )− ⊂ · · · ⊂ O(T  x )− ⊂ O( x )−

.

and .O(T  x )− = O( x )− . Therefore, there exist .σT ∈ R and an open neighborhood  and .V .  T σT ⊂ U .VT of .T  x such that .VT ⊂ T U − T of .T  If .T  x is not in .O( x ) , there exists an open neighborhood .V x such that −    T are .VT ⊂ T U and .O( x ) ∩ VT = φ. Set .σT = 0. We can assume the sets .V chosen to be disjoint. Then set

2.2 Lifting Flows

49

V =



.

T ), π(V

T ∈

which is an open neighborhood of x because .π is open and . is finite. Since x is almost periodic, there exists a syndetic set B of .R such that .xB ⊂ V . Set T }. BT = {t ∈ B :  xt ∈ V

.

T . The sets Note that .BT = φ, when .T  x is not in .O( x )− . If .BT = φ, then . x BT ⊂ V  .BT are disjoint because the sets .VT are disjoint and B=



.

BT .

T ∈

Set A=



.

BT + σT

T ∈

. There exists a compact subset L of .R such that .B + L = and observe that . xA ⊂ U R. Setting .K = T ∈ L − σT , it follows that .A + K = R. Therefore, A is syndetic x is almost periodic. and . Suppose x is positively recurrent. Then there exists .T ∈  such that .T  x ∈ ω( x) by part (d) of Proposition 2.2.5. It follows that .ω(T  x ) ⊂ ω( x ) and that .T 2 x ∈ x ∈ ω( x ) for all .k ∈ Z+ . Hence T ω( x ) = ω(T  x ) ⊂ ω( x ). So, by induction, .T k . x = T m x ∈ ω( x ) for a positive integer, and . x is positively recurrent. 

For a more general version of Theorem 2.2.6 and other results about lifting dynamical properties, see [54].  → X is a 2-sheeted covering map, then .π is a normal Proposition 2.2.7 If .π : X covering map. Proof Because the cardinality of .π −1 (x) is 2 for all x in X, setting .T ( x) =  x if  x ) = π( x ) and . x =  x defines a function of .X onto itself such that and only if .π( −1 = T and .π ◦ T = π . If U is an open evenly covered connected set, then .T −1 (U ) has two components, .U 1 ) = U 2 . 1 and .U 2 , and .T (U .π Since X has a basis of open evenly connected sets, the components of .π −1 (U )  because .π is for the open evenly covered connected sets of X form a basis for .X −1 is an open function. Therefore, T is a an open function. It follows that .T = T homeomorphism. 

Although the lift of a periodic point need not be periodic, for normal coverings there is at least the useful covering criterion for the periodicity of points in the base of the covering. The proofs of it and a corollary are left to the reader.

50

2 Flows and Covering Spaces

 → X be a normal covering with covering group ., Proposition 2.2.8 Let .π : X  R) be a flow on X and its lift to .X,  respectively. A point x and let .(X, R) and .(X, in X is periodic if and only if for some {for all} . x in .π −1 (x) there exist .τ > 0 and .T ∈  such that .T  x = xτ .  → X be a normal covering with covering group ., Corollary 2.2.9 Let .π : X  R) be a nontrivial flow on X and  where .X is a nontrivial cover. Let .(X, R) and .(X,  its lift to .X, respectively. A point x in X is not periodic if and only if .O( x )∩O(T  x) = φ for . x ∈ π −1 (x) and .T ∈  \ {ι}. Exercise 2.2.10 Prove Proposition 2.2.8 and Corollary 2.2.9. 1 → X and .π2 : X 2 → X of the same space, a Given two coverings .π1 : X 1 → X 2 such that covering homomorphism from .π1 to .π2 is a continuous map .θ : X .π2 ◦ θ = π1 . A covering homomorphism that is a homeomorphism is a covering isomorphism. It is easy to check that a covering isomorphism from .π1 to .π2 satisfies −1 = π . A covering homomorphism from .π to .π is a covering map ( [42], .π1 ◦ θ 2 1 2 Lemma 12.1).  → X is a universal covering provided that .X  is simply A covering .π : X  x )) is the connected. Every universal covering space is normal because .π∗ ( 1 (X,  trivial subgroup of . 1 (X, π( x )). Universal coverings have the following important properties:  → X is a universal covering and .π1 : X 1 → X is a • Universality: If .π : X →X 1 such that .π = π1 ◦  covering, then there exists a covering . π :X π ([42], Proposition 12.6). 1 → X and .π2 : X 2 → X • Uniqueness of universal coverings: If .π1 : X are two universal coverings, then .π1 and .π2 are isomorphic coverings ([42], Proposition 12.6).  → X is a universal covering, then . is • Universal covering group: If .π : X isomorphic to . 1 (X, x) ([42], Corollary 11.32). • Existence of universal covering: Every connected and locally simply connected space has a universal covering space ([42], Theorem 12.8).  → X is a covering of an n-dimensional manifold, then Proposition 2.2.11 If .π : X  the covering space .X is an n-dimensional manifold. Proof Using a basis of evenly covered open sets of X homeomorphic to .Bn , it is not  is Hausdorff and every point of .X  has an open neighborhood difficult to check that .X homeomorphic to .Bn . Since every manifold has a universal covering space, it suffices to prove that the universal covering space of a manifold is second-countable because the universal covering space is a covering space of every covering space of X (universality property).  → X be a universal covering of the n-dimensional manifold X. Since Let .π : X the fundamental group of a manifold is countable ([42], Theorem 8.11), the covering group . is also countable. Let .Ui , .i ∈ Z+ , be a countable basis of connected

2.3 Compact Connected Surfaces

51

i be a component of .π −1 (Ui ). Then evenly covered open sets of X. For fixed i, let .U −1 i for a unique .T ∈  because every component of .π (Ui ) can be written as .T U i,j of universal covers are normal. Since . is countable, there exists an indexing .U the components of .π −1 (Ui ) with .j ∈ Z+ .  given by The collection of open connected sets .B of .X B=

∞  ∞ 

.

i∈Z+

ij U

i∈Z+

is clearly a countable basis of the universal cover.



2.3 Compact Connected Surfaces A surface is just a 2-dimensional manifold, such as .R2 . The sphere .S2 , the projective space .P2 , and the torus .T2 are examples of connected compact surfaces. These three surfaces are all topologically distinct because their fundamental groups are different. (From their universal coverings, we know that their fundamental groups are the trivial group, .Z2 and .Z2 , respectively.) Also .S2 and .T2 can be embedded in .R3 and hence represented in a figure, but .P2 cannot be embedded in .R3 . If you remove the Arctic and Antarctic open regions from .S2 and project the remaining band into .P2 , you obtain a Möbius band (Figure 2.2), which is a nonorientable surface with one boundary component that can be embedded into .R3 as shown in Figure 2.2. Given a flow on a compact surface, the components of the surface are both compact connected surfaces and invariant sets of the flow. Thus the interesting dynamical properties of flows on compact surfaces all occur in flows on compact connected surfaces. This section is devoted to understanding the classical description and classification of all compact connected surfaces in preparation for studying the dynamical properties of flows on them. Our discussion will also include compact connected surfaces with boundary because flows on them can be extended to flows on compact connected surfaces. A compact manifold with boundary has only a finite number of boundary components Fig. 2.2 The Möbius band is a nonorientable surface with boundary

52

2 Flows and Covering Spaces

and they are necessarily compact. Consequently, the boundary of a compact surface with boundary consists of a finite number of components each homeomorphic to 1 1 .S because every compact 1-dimensional manifold is homeomorphic to .S ([42], Theorem 6.1). Compact connected surfaces and compact connected surfaces with boundary are polyhedrons and have triangulations. A proof of this fact can be found in Riemann Surfaces by Alfors and Sario [2]. Although not needed here, it is worth noting that all surfaces are polyhedrons. For an outline of the proof and an overview of triangulations of manifolds, see the Triangulation Theorems section in Chapter 5 of [42]. Since every compact connected surface has a triangulation (Theorem 5.12, [42]), every compact connected surface can be obtained from polygonal presentations (Proposition 6.4, [42]). There are a number of elementary transformations that can be applied to the polygonal presentation of a surface and always produce a polygonal presentation of a surface homeomorphic to the original one. (See the section Polygonal Presentations of Surfaces beginning on p. 129 in [42].) Using these elementary transformations, it can be shown that every compact connected surface is homeomorphic to one of the following: (a) The sphere .S2 (b) A connected sum .T2 # . . . #T2 (c) A connected sum .P2 # . . . #P2 Proofs can be found in the Classification of Surface Presentations section in Chapter 6 of [42] and in Sections 1.5, 1.6, and 1.7 of [63]. This result does not, however, rule out the possibility that there exists a homeomorphism between two of the compact connected surfaces in the above list. The fundamental group is used to rule out this possibility. For a group G, the commutator subgroup .[G, G] is the smallest normal subgroup of G such that .G/[G, G] is abelian. The group .G/[G, G] is called the abelianization of G and denoted by .Ab(G). It is clearly an invariant of the group. The abelianizations of the fundamental groups of all the possible compact connected surfaces complete the classification of the compact connected surfaces. Specifically, their abelianizations are (a) .Ab( 1 (S2 )) = {1}, the trivial group 2 (b) .Ab( 1 ( T . . #T2 )) = Z2n  # . n 2 (c) .Ab( 1 ( P . . #P2 )) = Zn−1 × Z2  # . n

The determination of these groups is independent of the location or size of the regular Euclidean balls used to construct the connected sum. The integer n, denoting the number of copies of .T2 or .P2 in the connected sum, that appears in these abelianizations is clearly a topological invariant and is called the genus of the compact connected surface. For example, a surface of genus 3 is shown in Figure 2.3. The genus of the sphere .S2 is 0 by definition and consistent

2.3 Compact Connected Surfaces

53

Fig. 2.3 The connected sum .T2 #T2 #T2 is a surface of genus 3

with the usual convention that .Z0 = {1}. The nonorientable compact connected surfaces can also be identified algebraically as those whose abelianizations contain an element of order 2. The others, including .S2 , are all orientable and have torsionfree abelianizations. Notice that no two of these groups are isomorphic, and hence no two of the fundamental groups from which they were derived are isomorphic. Therefore, no two of the compact connected surfaces 2 S2 , T . . #T2 , and P2 # . . . #P2  # .

.

m

n

are homeomorphic. This completes the classification of compact connected surfaces which was originally established by Dehn and Heegard [26] in 1907. In summary, there are two infinite sequences, orientable and nonorientable, of compact connected surfaces indexed by the genus. The indexing begins at zero for the orientable sequence and at one for the nonorientable sequence. Thus every compact connected surface is uniquely determined by its genus and whether or not it is orientable. The genus appears in the standard polygonal presentations of compact connected surfaces as a factor of the number of edges. For example, Figure 2.4 is the standard polygonal presentation of .T2 #T2 shown in Figure 2.1 on p. 38. When a compact connected orientable surface has positive genus, the genus is the number of edges of its standard polygonal presentation divided by 4. With a little more work, the Euler characteristic can also be computed from a standard polygonal presentation in terms of the number of edges. Thus there exist equations connecting the genus and the Euler characteristic. Letting .γ (X) denote the genus of a compact connected surface, we have χ (X) = 2 − 2γ (X)

(2.2)

χ (X) = 2 − γ (X)

(2.3)

.

for X orientable and .

54

2 Flows and Covering Spaces

Fig. 2.4 A polygonal presentation of .T2 #T2

b a

a

d

b

c

c d

for X nonorientable. Hence .χ (X) is also a topological invariant of compact connected surfaces, and it is equivalent to .γ (X) because one can solve for .γ (X) in terms of .χ (X). It follows from Theorem 1.2.11 that all flows on most compact connected surfaces have at least one fixed point because there are only two compact connected surfaces with Euler characteristic equal to zero, namely, .T2 and .P#P. The latter merits further discussion. A convenient consequence of (2.2) and (2.3) is that .χ (X) < 0 if and only if X is orientable with .γ (X) ≥ 2 or X is nonorientable with .γ (X) ≥ 3. Let . be the group of homeomorphisms of .R2 generated by . S(x, y) = (x + 1/2, −y) and .T(0,1) (x, y) = (x, y + 1). Observe that . S 2 = T(1,0) and .G = {T(m,n) : (m, n) ∈ Z2 }, the covering group of the torus, is a subgroup of .. Clearly, . acts properly and freely on .R2 . Then using Proposition 1.1.19 and Corollary 1.1.21, a routine argument shows that .R2 /  is a surface and that the natural projection is a covering map. The natural projection .π maps .[0, 1/2] × [−1/2, 1/2] onto .R2 / . Thus .R2 /  is a compact connected surface called the Klein bottle. The Klein bottle is homeomorphic to .P#P ([42], Lemma 6.15), making it a nonorientable surface of genus 2 and Euler characteristic 0. In other words, the Klein bottle can be thought of as two copies of the Möbius strip in Figure 2.2 glued together along their boundary curves. Like the projective plane, the Klein bottle cannot be embedded in .R3 and is usually pictured with self-intersections. It is easy to verify that . S. It follows that . is not an S ◦ T(m,n) = T(m,−n) ◦  abelian group and that G is a normal subgroup of . of index 2. Moreover, . S is the lift of .S ∈ HT2 defined by .S(z, w) = (eπ i z, w −1 ) = (−z, w). Now it is a straightforward calculation to show that .(e2π ix , e2π iy ) → π(x, y) defines a 2sheeted normal covering of the Klein bottle .R2 /  by the torus .T2 with covering group .{S, S 2 = ι}, which is isomorphic to .Z2 . Proposition 2.3.1 The torus is the only compact connected surface that is a topological group. Proof If X is a topological group and g is an element of X, then .x → xg is a homeomorphism of X onto itself. Since .xg = x implies that .g = e, the identity, it

2.3 Compact Connected Surfaces

55

has no fixed points unless it is the identity map. If X is also a compact connected surface, then X must be the torus or the Klein bottle. But the fundamental group of a topological group is abelian ([42], Problem 8-3), and by the universal covering group property on p. 50, the fundamental group of the Klein bottle is isomorphic to the group of homeomorphisms of .R2 generated by . S(x, y) = (x + 1/2, −y) and 2 .T(0,1) (x, y) = (x, y + 1), which is not abelian. Thus X is homeomorphic to .T , which is a topological group. 

We now know that the universal covering space of the projective plane .P2 is the 2sphere .S2 and the universal covering space of the torus .T2 and the Klein bottle .P#P is .R2 . But more importantly in each case the group of covering transformations has a concrete representation as a group of known functions that are easy to manipulate. For example, the covering transformations of the universal covering space of the torus are just translates by a subgroup of .R2 . The universal covering spaces of the rest of the compact connected surfaces is also .R2 , but that is not the best way think about it. It is better to use .B2 , which is homeomorphic to .R2 , because the universal covering transformations of the orientable surfaces can be represented as a group of complex hyperbolic linear fractional transformations of .B2 onto itself. We will discuss these groups in more detail in Chapter 7. For convenience, a surface with a nonempty boundary will be called a bordered surface. The Möbius band in Figure 2.2 is an example of a bordered surface obtained by removing an open disk from the projective plane .P2 . Because every component of the boundary of a compact connected bordered surface is homeomorphic to .S1 , Theorem 2.1.20 can be applied a finite number of times to a compact connected bordered surface to obtain a fundamental structural theorem that also reduces the study of flows on compact connected bordered surfaces to the study of flows on compact connected surfaces. Theorem 2.3.2 Let X be a compact connected bordered surface such that .∂X has r components, then there exist a compact connected surface .X∗ , an embedding h of X into .X∗ , and regular Euclidean balls .B1 , . . . , Br of .X∗ with the following properties: (a) .Bj− ∩ Bk− = φ for .j = k. (b) For each component C of .∂X, there exists a unique j such that h(C) = ∂Bj− = Bj− \ Bj .

.

(c) .X∗ \ h(X) =

n

j =1 Bj .

Furthermore, if .(X, R) is a flow on X, then there exists a flow .(X∗ , R) such that .h(X) is a closed invariant set of .(X, R) and h is an isomorphism of flows. Proof Assume the boundary of a compact connected bordered surface has r components and proceed by induction on r. For .r = 1, the result is an immediate consequence of Theorem 2.1.20. Assuming the result holds when X has r boundary components, consider a compact connected bordered surface X with .r + 1 boundary

56

2 Flows and Covering Spaces

components. Then apply Theorem 2.1.20 again to obtain a compact connected bordered surface .X with r boundary components. Since .X has r boundary components, the induction assumption applies. It is now straightforward to check that .(X )∗ = X∗ has the required properties. 

Proposition 2.3.3 If X is a compact connected bordered surface, then .X∗ , the genus of .X∗ , and the orientability of .X∗ are topological invariants of X. Proof Suppose f is a homeomorphism of X onto Y . Obviously, the number of boundary components is a topological invariant of X and so X and Y have the same number of boundary components, which will be denoted by .r ∈ Z+ . To start, let .h : X → X∗ and .h : Y → Y ∗ be the embeddings given by Theorem 2.3.2. Then .H = h ◦ f ◦ h−1 is a homeomorphism of .h(X) onto .h (Y ) that will be extended to a homeomorphism of .X∗ on .Y ∗ . Let .Bj and .Bj denote the regular Euclidean balls obtained from Theorem 2.3.2 for .X∗ and .Y ∗ , respectively. Set .Cj = Bj− \ Bj and .Cj = (Bj )− \ Bj for .1 ≤ j ≤ r. There exists a permutation .θ of .{1, . . . , r} such that .H (Cj ) = Cθ(j ) for .1 ≤ j ≤ − − n r. Let .αj and .βj be homeomorphisms of .Bj and .(Bj ) onto .D . Then

σj (x) = βθ(j ) H [αj−1 (x)])

.

defines a homeomorphism of .S1 onto itself. Equation (2.1) on p. 37 can be used to extend .σj to a homeomorphism of .D2 onto itself by setting σˆ j (x) =

.

xσj (x/x) 0

when x > 0 when x = 0.

The homeomorphism H can now be extended to a homeomorphism .Hˆ of .X∗ onto Y ∗ by setting

.

ˆ (x) = .H

H (x)  −1 βθ(j ˆ j [αj (x)] ) σ

when x is in h(X) when x is in Bj−

and leaving the details to the reader. Thus .X∗ is a topological invariant of the bordered surface X. Since the genus and the orientability are topological invariants of the surface .X∗ , they are also 

topological invariants of the bordered surface X. We now have three topological invariants for compact connected bordered surfaces, namely, the number of boundary components, the genus of .X∗ , and whether or not the surface .X∗ is orientable. Because a compact connected bordered surface is a polyhedron, it also has a polygonal presentation. Massey, [63], in Section 1.10, shows how the techniques used for classifying the polygonal presentations of

2.3 Compact Connected Surfaces

57

compact connected surfaces can be modified for compact connected bordered surfaces. As a result, the polygonal presentation of a compact bordered surface of genus .γ with r boundary components and either orientable or nonorientable is equivalent to a standard or normal polygonal presentation. Therefore, these three topological invariants completely classify the compact connected bordered surfaces. In other words, if you remove any r disjoint regular Euclidean balls from say 1 1 .S #S , which can be done in many ways, then the surfaces you obtain are all homeomorphic. From the standard or normal polygonal presentation, it is not difficult to see that ∗ .χ (X) = χ (X ) − r. Consequently, the Euler characteristic of a compact connected bordered surface X is a topological invariant of X. Defining the genus of a compact connected bordered surface X by .γ (X) = γ (X∗ ), then χ (X) = 2 − 2γ (X∗ ) − r = 2 − 2γ (X) − r

.

(2.4)

for .X∗ orientable and χ (X) = 2 − γ (X∗ ) − r = 2 − γ (X) − r

.

(2.5)

for .X∗ nonorientable. The final topic in this section is a covering space relationship between the nonorientable and orientable compact connected surfaces that can be used to study the dynamics of flows on nonorientable compact connected surfaces. We begin with a special case of a general construction (Problem 11–7 in [42]).  is a compact connected Let X and Y be compact connected surfaces. Suppose .X  surface and .π : X → X is a 2-sheeted covering. Let U be a regular Euclidean ball in X such that .U − is contained in an evenly covered open connected set .U of X, and let V be a regular Euclidean ball in Y . Set .X = X \ U and .Y = Y \ V . Then both .X and .Y are compact connected surfaces with one boundary component. So both .C = ∂X and .D = ∂Y are homeomorphic to .S1 , and there exists a surjective homeomorphism .σ : C → D. Using U , V , and .σ , construct the compact connected surface .X#Y (see Proposition 2.1.18 and the discussion preceding it). Let .f : X  Y → X#Y be the quotient map used in the construction of .X#Y on p. 38. Because .U is evenly covered, .π −1 (U ) consists of two disjoint regular Euclidean   = X\(U balls .U1 and .U2 . Set .X 1 ∪U2 ), a compact connected bordered surface with two boundary components .C1 and .C2 . Note that .π |Cj maps .Cj homeomorphically   (Y × {1, 2}) with the coherent topology. It is a compact onto C. Consider .X bordered surface with 3 components, and its boundary components are .C1 , .C2 , .D × {1}, and .D × {2}. Define .σ : C1 ∪ C2 → D × {1, 2} by .σ ( x ) = (σ ◦ π( x ), j ) if and only if . x ∈ Cj . Then .σ is a surjective homeomorphism that can be used to attach  . The same  \ (U1 ∪ U2 ) and produce a space denoted by .Y #X#Y .Y × {1, 2} to .X  is a ideas used to prove Proposition 2.1.18 can be used here to prove that .Y #X#Y compact connected surface.

58

2 Flows and Covering Spaces

 (Y ×{1, 2}) → X#Y defined There is also a natural continuous function .g : X  and .g(y, j ) = f (y) for .(y, j ) in .Y × {1, 2}. Then g by .g( x ) = f (π( x )) for . x in .X  → X#Y . passes to the quotient to obtain .π : Y #X#Y Because 2-sheeted coverings are normal coverings (Proposition 2.2.7), there  → X  for .π such that .S 2 = ι and the exists a covering transformation .S : X  is an S invariant set. covering group . = {S, ι}. Notice that by construction .X Extend S to a homeomorphism of .X  (Y × {1, 2}) onto itself by .S(y, 1) = (y, 2) and .S(y, 2) = (y, 1) and check that S also passes to the quotient to obtain  a homeomorphism T of .Y #X#Y onto itself such that .T 2 = ι. It follows that .π ◦ T = π . Now Proposition 2.2.1 can be used to show that .π is a normal 2sheeted covering of .X#Y . This construction of double coverings will be used to prove the following: Proposition 2.3.4 If X is a nonorientable compact connected surface, then there  → X such that .γ (X)  = γ (X) − 1. exists a 2-sheeted orientable covering .π : X Proof It has already been shown that the projective plane .P2 is double covered by 2 2 .S and the Klein bottle is double covered by .T . Since X is homeomorphic to a 2 2 connected sum .P # . . . #P with n terms, it suffices to use these specific examples to prove that .P2 # . . . #P2 has an orientable double covering depending on whether n is odd or even. Since .S2 double covers .P2 , the construction preceding the statement of this proposition shows that .T2 #S2 #T2 , which is homeomorphic to .T2 #T2 , double covers 2 2 2 2 2 .P #T , which is homeomorphic to .P #P #P ([42], Lemma 6.16). Proceeding by induction, suppose that π : T2 # . . . #T2 → P2 # . . . #P2

.

2n

2n+1

is a 2-sheeted covering, then with .Y = T2 there exists a 2-sheeted covering π : T2 # . . . #T2 → P2 # . . . #P2 #T2 .

.

2(n+1)

2n+1

Because the spaces .

2 2 P . . #P2 #T2 and P . . #P2  # .  # . 2n+1

2(n+1)+1

are homeomorphic, the result holds by induction for n odd. For n even, simply reuse the argument for n odd with .P2 and .S2 replaced by 2 2 2 .P #P , the Klein bottle, and .T .

 Collectively, the results about flows in this chapter make the case that the dynamical behavior of flows on compact surfaces with or without boundary is completely determined by the dynamical behavior of flows on compact connected orientable surfaces.

Chapter 3

A Family of Examples

A suspension flow is similar to a flow on a covering space with .Z as the covering group. In fact, starting with a cascade on a manifold, the construction uses an actual covering space and a covering group. In this case, the suspension flow is on a manifold of dimension 1 more than the original manifold. Thus the suspension flow of a cascade on .S1 is a flow on a surface. Section 3.1 is devoted to suspension flows and their dynamical properties. Section 3.2 contains a brief overview of the dynamics of homeomorphisms of the unit circle onto itself, which is accompanied by several examples. The suspension flow of a cascade on the circle is either a flow on the torus or Klein bottle because suspensions have no fixed points. The degree of a function from the circle to itself provides a convenient way of characterizing these two possibilities. The key tool used in this discussion is the degree of a function from the circle to itself. The analysis and examples of the dynamics of homeomorphisms of the circle with periodic points comes first in Section 3.2 and is relatively straightforward. We have already seen that there are minimal circular cascades, namely, irrational rotations. The heart of Section 3.3 is showing that there is a rich family of circular cascades without periodic points that are not minimal. Any minimal set of such a homeomorphism must be a Cantor set by Corollary 1.1.14. The section closes with some of the dynamical properties of these cascades. Suspension flows have no fixed points. (The fixed points of the cascade become periodic points of period 1.) Since a suspension flow constructed from a cascade of the unit circle is a compact connected surface, it must be either the torus or the Klein bottle by Theorem 1.2.11 because they are the only compact connected surfaces with zero Euler characteristic. The degree of the homeomorphism determines whether a suspension of flow of a circular cascade is the torus or the Klein bottle. Thus suspensions of circular cascades produce a family of flows on both these surfaces, but the most interesting ones come from orientation preserving homeomorphisms without periodic points. In particular, the Denjoy flows on the torus are the suspensions of cascades on the circle with Cantor minimal sets. They © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_3

59

60

3 A Family of Examples

are an important class of examples in the theory of flows on surfaces and lead to other interesting examples.

3.1 Suspension Flows This section begins with a useful general construction of a flow from a cascade that further illustrates the use of bitransformation groups. It begins by considering a homeomorphism .f : X → X of a topological space X onto itself and defining a homeomorphism of .X × R onto itself by setting T (x, s) = (f −1 (x), s + 1).

.

(3.1)

It is easy to see that the continuous left action of .Z on .X×R defined by .(n, (x, s)) → T n (x, s) = (f −n (x), s + n) is proper and free. There is also a natural flow on .X × R defined by .(x, s)t = (x, s + t). Observe that  n    T (x, s) t = (f −n (x), s + n + t) = T n (x, s)t ,

.

making .(Z, X × R, R) a bitransformation group. By Proposition 1.1.23, there exists a flow on .(X × R)/Z defined by ϕ(π(x, s), t) = π(x, s + t)

.

or more simply as .π(x, s)t = π(x, s + t), and the natural projection .π : X × R → (X × R)/Z is a homomorphism of flows. This flow is called the suspension flow of f and will be denoted by .(S(X, f ), R). Note for future use that equation (3.1) implies that suspension flows satisfy π(x, s)t = π(x, s + t) = π(f [s+t] (x), s + t − [s + t]),

.

(3.2)

where .[·] denotes the greatest integer function. The topological space .S(X, f ) is often as reasonable a space as X is: Proposition 3.1.1 Let f be a homeomorphism of a topological space X onto itself. (a) If X is a Hausdorff topological space, then .S(X, f ) is a Hausdorff topological space. (b) If X is a compact topological space, then .S(X, f ) is a compact topological space. (c) If X is a second-countable locally compact metric space, then .S(X, f ) is a second-countable locally compact metric space. (d) If X is a compact metric space, then .S(X, f ) is a compact metric space.

3.1 Suspension Flows

61

(e) If X is an n-dimensional manifold, then .S(X, h) is an .(n + 1)-dimensional manifold. Proof Part (a) is an immediate consequence of Proposition 1.1.19. Since the natural projection .π is continuous, part (b) follows from the observation that .π(X × [0, 1]) = S(X, f ). Turning to part (c), it follows that .S(X, f ) is a second-countable locally compact space because .π is open and onto. It is Hausdorff by part (a). Hence the points of .S(X, f ) are closed sets. It is regular because locally compact Hausdorff spaces are regular ([39], 5.17 Theorem). Now Urysohn’s metrization theorem ([39], 4.16 Theorem) applies and .S(X, f ) is metrizable. Since compact metric spaces are second-countable, part (d) follows from parts (b) and (c). Finally, part (e) follows from Proposition 2.2.1.   Proposition 3.1.2 Given a cascade .(X, f ), the function .ν(x) = π(x, 0) is an embedding of X in .S(X, f ) and the cascade .(X, f ) is isomorphic to the time one map of the flow on .S(X, f ) restricted to .ν(X). Proof Observe that .π is injective on .X × (−ε, ε) when .0 < ε < 1/2. It follows that π restricted to .X × (−ε, ε) is a homeomorphism onto an open subset of .S(X, f ) because .π is an open function. Consequently, .ν is a homeomorphism onto .ν(X). The calculation

.

ν(f (x)) = π(f (x), 0) = π [T −1 (x, 1)] = π(x, 1) = π(x, 0)1 = ν(x)1

.

completes the proof.

 

Corollary 3.1.3 Given a cascade .(X, f ), a point x in X is a periodic point of period p (fixed points of the cascade are periodic points of period 1) if and only if .ν(x) = π(x, 0) is a periodic point of period p in the suspension flow .S(X, f ). Exercise 3.1.4 Prove Corollary 3.1.3. Proposition 3.1.5 Let .(X, f ) and .(Y, g) be cascades. If .θ : (X, f ) → (Y, g) is a homomorphism, then there exists a homomorphism of flows .θS : S(X, f ) → S(Y, g) such that .νY (θ (x)) = θS (νX (x)). Proof Define S to be the homomorphism mapping .Y × R onto itself by .S(y, s) = (g −1 (y), s + 1), and let .π  : Y × R → S(Y, g) be the natural projection. Extend .θ θ (x, s) = (θ (x), s). Then to .X × R by setting . Sn ◦  θ (x, s) = (g −n (θ (x)), s + n) = (θ (f −n (x)), s + n) =  θ ◦ T n (x, s),

.

and π  ( θ (x, s)) = π  (S n ◦  θ (x, s)) = π  ( θ ◦ T n (x, s)).

.

62

3 A Family of Examples

Thus .π  ◦  θ passes to the quotient by setting .θS (π(x, s)) = π  ◦  θ (x, s). Routine calculations show that .θS is a homomorphism and .νY (θ (x)) = θS (νX (x)).   Exercise 3.1.6 In the proof of Proposition 3.1.5, verify that .θS is a homomorphism and .νY (θ (x)) = θS (νX (x)). To avoid confusion with orbits of the flows .(X × R, R) and .(S(X, f ), R) with the orbits of the cascade .(X, f ), let .Of (x) = {f n (x) : n ∈ Z}. The orbit of .(x, s) for the flow on .X × R is just .{x} × R and so .π({x} × R) is a typical orbit of .(S(X, f ), R). If .u = π(x, s), then .O(u) = π({x} × R) and π −1 (O(u)) =



.

T k ({x} × R) =

k∈Z

  {f −k (x)} × R = Of (x) × R.

(3.3)

k∈Z

It follows that .Of (x)×R is a saturated set. If Y is a subset of X such that .f (Y ) = Y , then .Y × R is also a saturated set by equation (3.3) because it contains .Of (x) × R for every x in Y . Proposition 3.1.7 If Y is a subset of X such that .f (Y ) = Y , then π(Y × R)− = π(Y − × R).

.

(3.4)

Proof Clearly, .(Y × R)− = Y − × R, and .T (Y × R) = Y × R because Y is f invariant. It follows that .T (Y − × R) = Y − × R because the closure of an invariant set is an invariant set. Since .Y − is f -invariant, .(Y − × R) is a saturated closed set. Hence .π(Y − × R) is a closed subset of .S(X, f ) ([42], Lemma 3.16) containing − ⊂ π(Y − × R). .π(Y × R). It follows that .π(Y × R) To prove the reverse inclusion, note that the set .π −1 (π(Y × R)− ) contains .Y × R and is closed because .π is continuous. Thus .Y − × R ⊂ π −1 (π(Y × R)− ) and − × R) ⊂ π(Y × R)− . .π(Y   Corollary 3.1.8 If .u = π(x, s), then O(u)− = π(Of (x)− × R),

.

(3.5)

and .u = π(x, s) is a transitive point of the flow .(S(X, f ), R) if and only if x is a transitive point of the cascade .(X, f ). Proof Let .Y = Of (x). Then π(Of (x)− × R) = π(Of (x) × R)− = O(u)−

.

by the proposition and equation (3.3). Since .Of (x)− ×R is a saturated set in .X ×R, the set .π(Of (x)− ×R) = S(X, f ) if and only if .Of (x)− × R = X × R if and only if .Of (x)− = X.  

3.1 Suspension Flows

63

Proposition 3.1.9 If A is a syndetic subset of .Z, then A is a syndetic subset of .R. Proof There exists a finite set C of .Z such that .Z = A + C because a subset of .Z is compact if and only if it is finite. Then .R = Z + [0, 1] = A + C + [0, 1] and .C + [0, 1] is a compact subset of .R.   More generally a left syndetic subset of a closed syndetic subgroup H of a topological group G is a left syndetic subset of G ([31], Remark 2.03). (A subgroup is left syndetic if and only if it is right syndetic.) Theorem 3.1.10 Let .(X, f ) be a cascade on a metric space such that .S(X, f ) is also a metric space. (a) If x is an almost periodic point of the cascade .(X, f ), then .π(x, 0) is an almost periodic point of the suspension flow .(S(X, f ), R). (b) If Y is a compact minimal subset of the cascade .(X, f ), then .π(Y × R) is a compact minimal set of the flow .(S(X, f ), R) . Proof For part (a), let U be an open neighborhood of .π(x, 0). There exist open neighborhoods .V1 and .V2 of x and 0, respectively, such that .V1 × V2 ⊂ π −1 (U ). By hypothesis there exists a syndetic subset A of .Z such that .f n (x) ∈ V1 for all .n ∈ A. Let n be in A. Then π(x, 0)n = π(x, n) = π [T −n (x, n)] = π(f n (x), 0) ∈ π(V1 × V2 ) ⊂ U.

.

By Proposition 3.1.9, A is a syndetic subset of .R and .π(x, 0) is an almost periodic point of .(S(X, f ), R) because .π(x, 0)n is in U for all .n ∈ A. Now suppose that Y is a compact minimal subset of the cascade .(X, f ), and let x be an element of Y . Then x is almost periodic by Theorem 1.1.11 and .u = π(x, 0) is almost periodic by part (a). Since .π(Y × R) = π(Y × [0, 1]), the set .π(Y × R) is compact. Therefore, .O(u)− is a compact minimal subset of the flow .(S(X, f ), R) by Theorem 1.1.11. Finally, O(u)− = π(Of (x)− × R) = π(Y × R)

.

by Corollary 3.1.8 and the minimality of Y .

 

Corollary 3.1.11 If .(X, f ) is a minimal cascade on a compact metric space, then (S(X, f ), R) is a minimal flow on a compact metric space.

.

Proof By Proposition 3.1.1, the space .S(X, f ) is a compact metric space, and the theorem applies.   Proposition 3.1.12 If .(X, f ) is a cascade on a metric space such that .S(X, f ) is also a metric space, then for all x in X ω(π(x, 0)) = π(ω(x) × R)

.

64

3 A Family of Examples

and α(π(x, 0)) = π(α(x) × R).

.

Proof Let y be in .ω(x). So there exists a sequence .mi of positive integers increasing to infinity such that .f mi (x) converges to y. Then π(x, s)mi = π(x, s + mi ) = π ◦ T −mi (x, s + mi ) = π(f mi (x), s)

.

converges to .π(y, s) in .ω(π(x, s)), proving for all s in .R that π(ω(x) × {s}) ⊂ ω(π(x, s)) = ω(π(x, 0)s) = ω(π(x, 0)).

.

Let .v = π(y, s) be in .ω(π(x, 0)). Since .ω-limit sets are invariant, .v  = π(y, 0) is also in .ω(π(x, 0)). So there exists a sequence .ti of positive real numbers increasing to infinity such that .π(x, 0)ti = π(x, ti ) converges to .π(y, 0). Recall that .π restricted to the open set .U = X × (−1/4, 1/4) is a homeomorphism onto an open set containing .π(y, 0). There exists .K > 0 such that .π(x, ti ) ∈ π(U ) when .i > K. Consequently, for .i > K, there exist a sequence of positive integers .mi and a sequence of real numbers .τi ∈ (−1/4, 1/4) such that (a) .ti = mi + τi (b) .T −mi (x, ti ) = (f mi (x), τi ) ∈ U (c) .π(f mi (x), τi ) = π(x, ti ) Obviously, .mi increases to infinity and .(f mi (x), τi ) converges to .(y, 0) in .X × R. Hence y is in .ω(x) and .v  = π(y, 0) is in .π(ω(x) × R). Because .π(ω(x) × R) is invariant, it follows that .v = π(y, s) is also in .π(ω(x) × R). Therefore, .ω(π(x, 0)) ⊂ π(ω(x) × R). The same basic proof works for .α-limit sets.   Corollary 3.1.13 Let .(X, f ) be a cascade on a metric space such that .S(X, f ) is also a metric space. A point x in X is positively recurrent {negatively recurrent} if and only if .π(x, 0) is positively recurrent {negatively recurrent} in .(S(X, f ), R). Exercise 3.1.14 Prove Corollary 3.1.13. As an example, consider .(S1 , f ) where .f (z) = ze2π ia and a is a real number. We saw in Section 1.1 (page 6) that .(S1 , f ) is a minimal transformation group if and only if a is irrational, and in this case .(S(S1 , f ), R) is a minimal flow by Corollary 3.1.11. If a is rational, then .(S1 , f ) and .(S(S1 , f ), R) are periodic flows. Construct the suspension flow .(S(S1 , f ), R) as usual with the flow .(z, s)t = (z, s + t) on .S1 × R and .T (z, s) = (ze−2π ia , s + 1) with .π : S1 × R → S(S1 , f ) denoting the natural projection. We will show that the suspension flow .(S(S1 , f ), R) is isomorphic to the straight line flow on .T2 determined by a and 1. Define 2 homeomorphisms . S and .T of .R2 onto itself by . S(x, y) = (x + 1, y) and .T(x, y) = (x − a, y + 1). The map .(x, y) → (e2π ix , y) is a covering map with covering group the cyclic group generated by . S. It now follows that the group of

3.1 Suspension Flows

65

homeomorphisms generated by . S, T], is an abelian S and .T, which is denoted by .[ group and a universal covering group of .(S(S1 , f ), R). So .S(S1 , f ) is the orbit space of .[ S, T]. The lifted flow of .(S(S1 , f ), R) on .R2 is .(x, y)t = (x, y + t). So that T is not used in this example for 2 different mappings, let .S  (x, y) = (x+1, y) and .T  (x, y) = (x, y+1) be the generators of the universal covering group for .T2 . Define a homeomorphism Q of .R2 onto itself by .Q(x, y) = (x − ay, y). A  = [ simple calculation shows that .Q ◦ S  =  S ◦ Q and .Q ◦ T  = T ◦ Q. Let .G S, T]    −1 and .G = [S , T ]. Then Q and .Q are isomorphisms of the right actions of .G  on .R2 . Therefore, their orbit spaces are homeomorphic and .θ (e2π ix , e2π iy ) = and .G  G(Q(x, y)) is a homeomorphism of .T2 onto .S(S1 , f ). The lift of the straight line flow determined by a and 1 is .(e2π ix , e2π iy )t = 2π (e i(x+at) , e2π i(y+t) ) and its lifted flow is .(x, y)t = (x + at, y + t). Hence,   θ (e2π ix , e2π iy )t = θ (e2π i(x+at) , e2π i(y+t) )

.

 = G(Q(x + at, y + t))  − ay, y + t) = G(x  = G(Q(x, y)t) = θ (e2π ix , e2π iy )t and the flow .(S(S1 , f ), R) is isomorphic to a straight line flow. There is also an alternate approach to suspensions that will be useful in later chapters. Given a homeomorphism f of X onto itself, define an equivalence relation on .X × [0, 1] by .(x, s) ∼f (y, t) if and only if one of the following holds: (a) .(x, s) = (y, t) (b) .s = 1, .t = 0, and .y = f (x) (c) .t = 1, .s = 0, and .x = f (y) Let .π  be the quotient map of .X × [0, 1] onto the quotient space .X/ ∼f . Theorem 3.1.15 If f is a homeomorphism of X onto itself, then   π  (x, s)t = π  f [s+t] (x), s + t − [s + t]

.

(3.6)

defines a flow .(X/ ∼f , R) isomorphic to the suspension flow .(S(X, f ), R). Proof In the proof of Proposition 3.1.1, it was noted that .π(X × [0, 1]) = S(X, f ). Let .πˆ denote the restriction of .π to .X × [0, 1]. It is readily shown that .π(x, ˆ s) = πˆ (y, t) if and only if .(x, s) ∼f (y, t). Since .X × {0} and .X × {1} are closed subsets of .X × [0, 1], it follows that the saturation of a closed set of .X × [0, 1] is also closed because .(x, 1) → (f (x), 0) is a homeomorphism of .X × {1} onto .X × {0}. Thus .πˆ is a closed function and a quotient map. Now the uniqueness of the quotient topology implies that there exists a unique homeomorphism . of .X/ ∼f onto .S(X, f ) such that . ◦π  = πˆ and . −1 ◦ πˆ = π  .

66

3 A Family of Examples

Consequently, . −1 ( [π  (x, s)]t) defines a flow on .X/ ∼f isomorphic to the flow .(S(X, f ), R). Then using equation (3.2), the calculation

−1 ( [π  (x, s)]t) = −1 (πˆ (x, s)t)

.

= −1 (π(x, s)t)    = −1 π f [s+t] (x), s + t − [s + t]    = −1 πˆ f [s+t] (x), s + t − [s + t]   = π  f [s+t] (x), s + t − [s + t]

completes the proof.

 

3.2 Cascades on the Circle Throughout this section, .π : R → S1 will be the usual universal covering map 2π ix with covering group . = {S n : n ∈ Z} where .S(x) = x + 1. Given .π(x) = e distinct points a and b in .S1 , it will be convenient to let .(a, b) and .[a, b] always denote the open and closed counterclockwise intervals of .S1 from a to b. To start, consider a continuous function .f : S1 → S1 . There exists a lift .f : R → R such that .π ◦ f = f ◦ π and .f(0) is in the interval .[0, 1) because .π([0, 1)) = S1 . Then all functions . g : R → R such that .π ◦  g = f ◦ π have the form .S n ◦ f for some .n ∈ Z. Since .π ◦ f◦ S = f ◦ π ◦ S = f ◦ π , there exists a unique integer d such that d  = f◦ S. With a little manipulation, it follows that .f◦ S m = S md ◦ ffor all .S ◦ f 1 1 .m ∈ Z. The integer d is called the degree of f and written .deg(f ). If .g : S → S is another continuous function, then .deg(f ◦ g) = deg(f ) deg(g) follows from ◦ S m = S md ◦ f. Observe that .deg(f ) = 1 if and only if .S ◦ f = f◦ S, that is, .f .f and S commute. (It is an exercise to verify that this definition of degree agrees with the more standard definition found in [42], Problem 8–7, page 191.) If .f (1) = 1, then .f(0) = 0 by the convention that .f(0) is in .[0, 1) and .

deg(f ) = S deg(f ) (0) = S deg(f ) (f(0)) = f(S(0)) = f(1).

In particular, .deg(zn ) = n because .f(x) = nx for .f (z) = zn . Since .(S◦ f)◦S = S deg(f ) ◦(S◦ f), the degree of f does not depend on the choice of the lift .f. If .ρ(z) = e2π ia z, and .a ∈ R is a rotation of .S1 , then .ρ (x) = x + a satisfies .π ◦ ρ  = ρ ◦ π and .ρ ◦ S = S ◦ ρ . Thus .deg(ρ) = 1. By replacing f with .ρ ◦ f , it can be assumed that .f (1) = 1 without changing the degree of f .

3.2 Cascades on the Circle

67

Strengthening our initial hypothesis, we assume that f is a homeomorphism of S1 onto itself. Then .deg(f ) deg(f −1 ) = deg(f ◦ f −1 ) = deg(ι) = 1. Since the degree function is integer-valued, either .deg(f ) = deg(f −1 ) = 1 or .deg(f ) = deg(f −1 ) = −1, dividing the homeomorphisms into 2 types. As an aside, it follows that the homeomorphisms of .S1 onto itself of degree 1 are an index 2 subgroup of the group .HS1 of all homeomorphisms of .S1 onto itself. If, in addition, the homeomorphism f is replaced by the homeomorphism .ρ ◦ f for a suitable rotation, it can be assumed that .f (1) = 1 without changing the degree of f . Clearly, .f −1 (1) = 1. Let .f and . g be the lifts of f and .f −1 such that .f(0) =  g ◦ f = π = π ◦ f◦  g and . g ◦ f =  ι = f◦  g by the unique 0= g (0). Then .π ◦   lifting property applied at 0. Therefore, .f is a homeomorphism of .R onto itself, when f is a homeomorphism of .S1 onto itself when .f (1) = 1. The requirement that .f (1) = 1, however, can be dropped because for any rotation of .S1 the function .ρ ◦ f is a homeomorphism if and only if f is a homeomorphism. So the lift .fof a homeomorphism of .S1 onto itself is either strictly increasing or strictly decreasing, another dichotomy. Moreover, .f is strictly increasing {decreasing} if and only if f preserves {reverses} the counterclockwise ordering of .S1 , which leads to the following result: .

Proposition 3.2.1 A homeomorphism f of .S1 onto itself preserves the counterclockwise order of .S1 if and only if .deg(f ) = 1. Proof Let f be a homeomorphism of .S1 onto itself that preserves the counterclockwise order of .S1 . Then .ρ ◦ f preserves the counterclockwise order of .S1 for every rotation .ρ of .S1 . There exists .ρ such that .h(1) = ρ ◦ f (1) = 1 and a lift . h:R→R such that . h(0) = 0. It follows that .deg(h) = deg(ρ) deg(f ) = deg(f ) = ±1 because f is a homeomorphism. Since h preserves the counterclockwise order of 1 .S , the lift . h must be strictly increasing, in particular, . h(1) >  h(0) = 0. Therefore,  .deg(f ) = deg(h) = h(1) = 1 because .deg(f ) = ±1. Now suppose that .deg(f ) = 1. With h and . h as in the first half of the proof,  .h(0) = 0 < 1 = deg(h) =  h(1) and . h must be strictly increasing because it is a homeomorphism of .R onto itself. Thus h preserves the counterclockwise order of 1 −1 ◦ h. .S and so does .f = ρ   1 Corollary 3.2.2 Let a and  b be distinct points in .S . If f is a homeomorphism 1 of .S onto itself, then .f [a, b] equals .[f (a), f (b)] or .[f (b), f (a)] depending on whether .deg(f ) = 1 or .deg(f ) = −1.

Exercise 3.2.3 Prove Corollary 3.2.2. Corollary 3.2.4 If f is a homeomorphism of .S1 onto itself such that .deg(f ) = −1, then f has a fixed point. Exercise 3.2.5 Prove Corollary 3.2.4 by showing that a lift .f of f must intersect the graph of .ι(x) = x. Since the suspensions of cascades .(S1 , f ) on the circle are flows on surfaces, an understanding of the dynamics of circular cascades will provide an understanding

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3 A Family of Examples

of the dynamics of this family of flows on surfaces by applying the results of Section 3.1. The simplest homeomorphisms of .S1 onto itself are the ones with at least one fixed point, which includes all those of degree .−1. Slightly more complicated are those with a periodic point that is not a fixed point. The treatment of the dynamics of these two classes will be brief. The homeomorphisms without periodic points will be carefully investigated in Section 3.3. Suppose f is a homeomorphism of .S1 onto itself of degree 1 with a fixed point, that is, .f (ζ ) = ζ for some .ζ ∈ S1 . Let .ρ(z) = e2π ia z be a rotation with .0 ≤ a < 1 such that .ρ(1) = ζ . Then .(S1 , ρ −1 ◦ f ◦ ρ) is a cascade isomorphic to .(S1 , f ) such that 1 is a fixed point because .ρ ◦ (ρ −1 ◦ f ◦ ρ) = f ◦ ρ and .ρ is an isomorphism. So without loss of generality, .f (1) = 1. Let .fbe the lift of f such that .f (0) = 0. Then (1) = 1 because .deg(f ) = 1. It follows that .g = f|[0, 1] is a homeomorphism of .f .[0, 1] onto itself. The function .θ = π |[0, 1] maps the closed interval .[0, 1] onto .S1 with only 0 and 1 identified and is a quotient map because .[0, 1] is compact and .S1 is Hausdorff space. Clearly, .θ is a homomorphism of the cascade .([0, 1], g) onto the cascade 1 .(S , f ). Since .θ maps the invariant open interval .(0, 1) homomorphically onto the invariant open set .S1 \ {1}, the dynamics of .(S1 , f ) can be completely determined from the dynamics of .([0, 1], g). The set of fixed points of g, which includes both 0 and 1, will be denoted by F . The set .[0, 1] \ F is the disjoint union of a finite or countable collection of open intervals, and each of these finite open intervals must be an invariant set of .([0, 1], g). Let .(a, b) be one of these open intervals. It is easily verified that .

· · · < f −2 (x) < f −1 (x) < x < f (x) < f 2 (x) < . . .

occurs for all .x ∈ (a, b) or .

· · · > f −2 (x) > f −1 (x) > x > f (x) > f 2 (x) > . . .

occurs for all .x ∈ (a, b). It follows that .ω(x) is a single point in .[a, b]. Consequently, either .ω(x) = {a} or .ω(x) = {b}, and .α(x) = {b} or .α(x) = {a}, respectively. Moreover, .ω(y) = ω(x) and .α(y) = α(x) for all y in .(a, b). There is little else to say about the dynamics of either .([0, 1], g) or .(S1 , f ) except that the fixed point set of f or g can be any closed subset of .[0, 1] containing 0 and 1 or any closed subset of .S1 containing 1, respectively. Next suppose that .deg(f ) = −1. Then Corollary 3.2.4 implies that f has a fixed point. As in the degree 1 case, we can, using a rotation, assume that .f (1) = 1 and .f is the lift such that .f(0) = 0, so .deg(f ) = f(1) = −1. Set .g = S ◦ f|[0, 1]. Then g is again a homeomorphism of .[0, 1] onto itself, but now g interchanges 0 and 1. .ι(x) = x. It follows that f Clearly, the graph of g must intersect the graph of  has at least two fixed points, 1 and .ζ . Then .S1 \ {1, ζ } consists of the two disjoint open counterclockwise intervals .(1, ζ ) and .(ζ, 1) of .S1 . Because .deg(f ) = −1, .f ([1, ζ ]) = [ζ, 1] by Corollary 3.2.2, and f has exactly 2 fixed points.

3.2 Cascades on the Circle

69

Obviously, .deg(g 2 ) = 1 and .g 2 has at least 2 fixed points, placing it in a class of homeomorphisms of .S1 already studied. A fixed point of .g 2 , however, need not be a fixed point of g. It can happen that .g(z) = w and .g(w) = z with .z = w, that is, g can interchange pairs of points other than 0 and 1. So f can have points of period 2. For example, .f (z) = z has as many as possible. If .f 2 (z) = z, then .f 2 (f (z)) = f (z). Letting .F 2 be the set of fixed points of .f 2 , the set .S1 \F 2 is a finite or countable union of open counterclockwise intervals of .S1 . If .(u, v) is one of these intervals, then so is .(f (v), f (u)) and .(u, v) ∪ (f (v), f (u)) is an open f invariant set of .S1 . If z is in .(u, v), then .f 2n (z) converges to either u or v and .f 2n+1 (z) converges to either .f (u) or .f (v) accordingly. Note that u or v could be one of the 2 fixed points of f . Thus .ω(z) can contain one fixed point of f or a periodic orbit of g of period 2. For example, see Figure 3.1. Suppose f is a homeomorphism of .S1 onto itself of degree 1 with a periodic point of period .p > 1. As before we can assume that 1 is the periodic point. The distinct points .1, f (1), f 2 (1), . . . , f p−1 (1) are not necessarily in counterclockwise order. For example, let .f (z) = e2π iq/p z = e2π i3/5 z and see Figure 3.2. But there exists a cyclic permutation (i0 i1 . . . ip−1 )

.

of .0, 1, . . . , p − 1 such that .i0 = 0 and the p distinct points f i0 (1), f i1 (1), . . . , f ip−1 (1)

.

with .f i0 (1) = 1 are in counterclockwise order. Applying f to them preserves the counterclockwise order because .deg(f ) = 1. In Figure 3.2, the permutation is .(0 2 4 1 3).

f 2 (z  )

Fig. 3.1 Example of = −1 with one orbit of period 2

.deg(f )

z



i

f 2 (z) z f −2 (z)

f −2 (z  )

−1

1 f −3 (z)

f −3 (z  ) f −1 (z  )

f −1 (z) f (z  ) 3  f (z) f (z )−i f 3 (z)

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3 A Family of Examples

Fig. 3.2 The .f −orbit of 1 when .q = 3 and .p = 5

e 2 πi/ 5 e4 πi/ 5

1

e6 πi/ 5 e 8 πi/ 5

If   f f ij (1) = f ij  (1),

.

then, looking at Figure 3.2,   f f ij +1 (1) = f ij  +1 (1).

.

  Let .Ij = f ij (1), f ij +1 (1) for .j = 0, 1, . . . , p − 1 mod p. Then it follows from the preceding equations that given j there exists .j  such that .f (Ij ) = Ij  . So p p−1 (I ) is a reordering of the p distinct intervals .f (Ij ) = Ij and .I0 , f (I0 ), . . . , f 0 .I0 , I1 , . . . , Ip−1 . The dynamics of f on the set .S1 \ O(1) is determined by the dynamics of .f p on the closed interval .I0− . Then .f p |I0− is a homeomorphism onto itself with fixed points 1 and .f i1 (1). Let .F = {z ∈ I0− : f p (z) = z}, which can be any closed subset of .I0− . Then .I0− \ F is a finite or countable union of disjoint open intervals, each an invariant set of .f p . Let .(a, b) be such an interval. Then .f np (z) converges to a as np (z) converges to b as .n → −∞ for all .z ∈ (a, b) or vice versa. .n → ∞ and .f The basic dynamical properties of a homeomorphism f of .S1 onto itself of degree 1 with a periodic point of period .p > 1 now follow from the previous paragraph. Obviously, z is in F if and only if z is a periodic point of f of period p in .I0− . If z is a periodic point of f , then its period is p. If z is a point in .S1 that is not a periodic point of f , then .α(z) and .ω(z) are 2 distinct periodic orbits of f (Figure 3.3). We can now summarize the dynamics of the cascades .(S1 , f ) with periodic points (including fixed points) as a theorem.

3.2 Cascades on the Circle

71

Fig. 3.3 .ω(z) = O(1) with = 1 and .p = 3

e 2 πi/ 3 f 4 (z )

.q

z

f (z)

f 3 (z) 1

e 4 πi/ 3 5 f (z)

f 2 (z)

Theorem 3.2.6 The following hold for a homeomorphism f of .S1 onto itself: (a) If .deg(f ) = 1 and .f (ζ ) = ζ for some .ζ ∈ S1 , then for .z ∈ S1 either z is a fixed point or both .α(z) and .ω(z) consist of a single fixed point such that .α(z) = ω(z) unless .ζ is the only fixed point of f and then .α(z) = ω(z). (b) If .deg(f ) = −1, then .z ∈ S1 can be one of the exactly 2 fixed points of f , a periodic point of period 2, a non-periodic point such that .α(z) {.ω(z)} is a fixed point, or a periodic orbit of period 2, and .α(z) never equals .ω(z). (c) If .deg(f ) = 1 and there exists periodic point .ζ ∈ S1 of period .p > 1, then 1 .z ∈ S can be a periodic point of period p or .ω(z) and .α(z) can be periodic orbits of period p with .ω(z) = α(z) when .O(ζ ) is the only periodic orbit. Corollary 3.2.7 If a cascade on the circle has a periodic point, then every minimal subset of it is a periodic orbit of the same period. Since suspension flows have no fixed points, the Euler characteristic of the surface .S(S1 , f ) must be 0 by Theorem 1.2.11. The only two compact connected surfaces with Euler characteristic zero are the torus and the Klein bottle. The strategy will be to construct the universal covering space of .S(S1 , f ). Then the covering group will determine .S(S1 , f ) because it is isomorphic to the fundamental group of .S(S1 , f ), which we know for the torus and Klein bottle. The key to this strategy will be the degree of f . Let T be the homeomorphism of .S1 × R onto itself defined by .T (z, s) = −1 (f (z), s + 1). The cyclic group .[T ] generated by T is isomorphic to .Z and acts on the left. Let .π : S1 × R → S1 × R/[T ] be the natural projection onto the orbit space. Recall that .π is a covering by Proposition 2.2.1. Continuing to let .S(x) = x + 1, define . S : R2 → R2 by  S(x, s) = (S(x), s) = (x + 1, s)

.

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3 A Family of Examples

and . π : R2 → S1 ×R by . π (x, s) = (e2π ix , s). Then . π is the universal covering map 1 of .S × R and the covering group is the infinite cyclic group .[ S]. Let .f : R → R be the lift of f such that .f(0) lies in the interval .[0, 1]. Then T(x, s) = (f−1 (x), s + 1)

.

is a lift of T , that is, . π T = T  π . Let . be the group .[ S, T]. It naturally acts on .R2 and this action will be written on the left using the juxtaposition notation. Proposition 3.2.8 The group . has the following properties: (a) (b) (c) (d)

If .deg(f ) = 1, then . S T = T S. If .deg(f ) = −1, then . S −1 T = T S. The group . is abelian if and only if .deg(f ) = 1. If Q is in ., then there exist unique integers m and n such that .Q =  S m Tn or m n such that .Q = T  S .

Proof From the discussion of degree in the beginning of Section 3.2, we know that S ◦ f = f ◦ S or .S −1 ◦ f = f ◦ S depending on whether .deg(f ) equals 1 or .−1. Parts (a) and (b) are immediate consequences of these facts. And then part (c) follows from parts (a) and (b). It is obvious from parts (a) and (b) that .Q =  S m Tn for some choice of m and n m n p q m−p q−n       in .Z. If .S T = S T , then .S = T . It follows from the definitions of . S  and .T that .q − n = 0 which in turn implies that .m − p = 0.   .

Proposition 3.2.9 Given .(x, s) and .(y, t) in .R2 , .π ◦  π (x, s) = π ◦  π (y, t) if and only if there exists Q in . such that .(y, t) = Q(x, s).     Proof Observe that .π  π (x, s) = π  π (y, t) implies that     π (x, s) =  π Tn (x, s)  π (y, t) = T n 

.

  for some .n ∈ Z. Thus there exists .m ∈ Z such that .(y, t) =  S m Tn (x, s) . Conversely and without loss of generality, let .Q =  S m Tn . Then .

π◦  π (y, t) = π ◦  π ◦ Q(x, s) = π ◦  π ◦ S m Tn (x, s) = π ◦ π ◦ Tn (x, s) = π ◦ T n ◦  π (x, s) = π ◦  π (x, s)

to complete the proof.

 

The homeomorphism . S has very regular horizontal movement with no vertical movement, and .T has very regular vertical movement. These properties were used in the proof of part (d) of Proposition 3.2.8 and will be used to prove the next result. Proposition 3.2.10 The group . acts freely and properly on .R2 . Proof It follows from part (d) of Proposition 3.2.8 that the action of . is free. To prove that this action is proper, it must be shown that for every pair of points

3.2 Cascades on the Circle

73

(x, s) and .(y, t) in .R2 there exist open neighborhoods U and V of .(x, s) and .(y, t), respectively, such that

.

{Q ∈  : U ∩ QV = φ}

.

is a finite subset of .. Let U and V be open 1 by 1 squares in .R2 centered at .(x, s) and .(y, t), respectively. The rectangles .T m S n V (the order here is deliberate) are disjoint 1 by 1 squares with edges parallel to the axes. Furthermore, every element of . can be written in this form by part (d) of Proposition 3.2.8. It follows that at most 4 of these rectangles can intersect U . (First, picture .∪n∈Z S n V and then n .T (∪n∈Z S V ).) Therefore, .{Q ∈  : U ∩ QV = φ} is finite and the action is proper.   Theorem 3.2.11 If .(S1 , f ) is a cascade on the circle, then the compact connected surface .S(S1 , f ) is the torus or Klein bottle depending on whether .deg(f ) is 1 or .−1. Proof Since the action of . on .R2 is both proper and free, the natural projection  2 2 .π : R → R /  is a covering map. In fact, it is a universal cover with covering group . (Proposition 2.2.1). Since .π and . π are open functions, .π ◦  π is also an open function and a quotient map. It follows from Proposition 3.2.9 that .π ◦  π and  .π make the same identifications. Therefore there exists a unique homeomorphism 2 1  .θ : R /  → S(S , f ) such that .θ ◦ π = π ◦  π by the uniqueness of the quotient topology ([42], Corollary 3.32). It follows that .π ◦  π is the universal covering of 1 .S(S , f ) and its covering group is .. The universal covering group is isomorphic to the fundamental group of the space it covers (page 50). So . is isomorphic to the fundamental group of .S(S1 , f ). Whether or not the fundamental group is abelian distinguishes the torus from the Klein bottle. Therefore, .S(S1 , f ) is homeomorphic to the torus or Klein bottle depending on whether .deg(f ) = 1 or .deg(f ) = −1 by part (c) of Proposition 3.2.8.   To close this section, we summarize the dynamics of the suspensions of cascades (S1 , f ) with periodic points (including fixed points) as a theorem which follows from Theorem 3.2.6.

.

Theorem 3.2.12 The following hold for the suspension of a homeomorphism f of S1 onto itself:

.

(a) If .deg(f ) = 1 and .f (ζ ) = ζ for some .ζ ∈ S1 , then for .w ∈ S(S1 , f ) either w is a periodic point of period 1 or both .α(w) and .ω(w) consist of a single periodic orbit such that .α(w) = ω(w) unless .ζ is the only fixed point of f . (b) If .deg(f ) = −1, then .w ∈ S(S1 , f ) can be one of the 2 periodic points of period 1, a periodic point of period 2, a non-periodic point such that .α(w) {.ω(w)} is a periodic orbit of period 1 or 2 and .α(w) never equals .ω(w). (c) If .deg(f ) = 1 and there exists a periodic point .ζ ∈ S1 of period .p > 1, then .w ∈ S(S1 , f ) can be a periodic point of period p or .ω(w) and .α(w) are periodic orbits of period p with .ω(w) = α(w) when .O(ζ ) is the only periodic orbit of .S1 .

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3 A Family of Examples

3.3 Denjoy Flows on the Torus The most interesting class of homeomorphisms of .S1 onto itself are those without periodic points. They are all of degree 1 because those of degree .−1 have 2 fixed points which are periodic points. Of course, the ones without periodic points include the irrational rotations which are minimal cascades, and hence their suspensions are minimal (Corollary 3.1.11). It was also shown in Section 3.1 that the suspension of the rotation .ϕ(z) = e2π ia z is isomorphic to the straight line flow on .T2 determined by a and 1. Corollary 1.1.14 holds out the possibility of the existence of a cascade .(S1 , f ) with a minimal Cantor set C and hence of a flow on the torus with a nowhere dense minimal set that is not a periodic orbit. The main result of this section is proving the existence of cascades on .S1 with Cantor minimal sets. Given a cascade .(X, f ) on a compact metric space, two distinct points x and y in X are said to be positively asymptotic or negatively asymptotic depending on whether .

  lim d f n (x), f n (y) = 0 or

n→∞

  lim d f n (x), f n (y) = 0

n→−∞

where d is a metric for X. The assumption that X is compact makes the definition independent of the choice of the metric and hence invariant under isomorphisms of dynamical systems. The points x and y are said to be doubly asymptotic provided that they are both positively and negatively asymptotic. A Cantor set C in the circle can be written in the form C=S \ 1

.

∞ 

(ai , bi ),

i=1

where ∞  .

(ai , bi )

i=1

is an open dense set of .S1 and .[ai , bi ]∩[aj , bj ] = φ when .i = j . The open intervals 1 .(ai , bi ) are the components of .S \ C and are called the complementary intervals of C. The inaccessible and accessible points of C are defined, respectively, by I = S1 \

∞ 

.

i=1

[ai , bi ] and A = C \ I =

∞ 

{ai , bi }.

i=1

The notation set forth in this paragraph will be used throughout this section.

3.3 Denjoy Flows on the Torus

75

Proposition 3.3.1 Let C be a minimal Cantor set of a cascade .(S1 , f ). If .(a, b) is a complementary interval of C, then every pair of distinct points z and w in .[a, b] is doubly asymptotic and .ω(z) = C = α(z). Proof First observe that f cannot have any periodic points by Corollary 3.2.7 and n so .deg(f ) = 1. Corollary 3.2.2 now implies that .f n ([a, b]) = [f n (a),  f (b)] for 1 n n all .n ∈ Z. Since .(a, b) is a component of .S \ C, so is . f (a), f (b) because C   is invariant. Thus either .(a, b) ∩ f n (a), f n (b) = φ or .(a, b) = f n (a), f n (b) . In the latter case, .a = f n (a), which is impossible when .n = 0 because f has no periodic points. Because a Cantor set has no isolated points, the closed intervals n n .[a, b] and .[f (a), f (b)] cannot intersect at an endpoint. Therefore, the intervals n n .[f (a), f (b)] are all disjoint and their arc length goes to zero as .|n| goes to infinity. It follows that .|f m (z) − f m (w)| goes to zero as .|m| goes to infinity and z and w are doubly asymptotic. Since z and a are now doubly asymptotic, .ω(z) = ω(a) and .α(z) = α(a). The sets .ω(a) and .α(a) are clearly closed invariant subsets of C, and hence .ω(z) = ω(a) = C = α(a) = α(z) because C is minimal.   From Proposition 3.3.1, we know that if a cascade on the circle has a Cantor minimal subset, the dynamics of the cascade outside the Cantor set is not very interesting. In particular, the cascade has at most one Cantor minimal set. Proving the existence of a cascade on the circle having a Cantor minimal set, however, begins by identifying the points that must be doubly asymptotic for a given Cantor set in a systematic way that can then be used to build the cascade on the circle for which this specific Cantor set is minimal. There is a useful equivalence relation on .S1 associated with a Cantor set C. It is defined by .z ∼C w provided that .z = w or .z, w ∈ [ai , bi ] for some i. Here a Cantor function is by definition a continuous function .κ : S1 → S1 for which there exists a Cantor set C in .S1 such that .κ(z) = κ(w) if and only if .z ∼C w. This is, of course, motivated by the Cantor ternary function of real analysis. A Cantor function, .κ : S1 → S1 , is a closed function because .S1 is compact and Hausdorff and hence a quotient map. It follows from the definition of a Cantor function that .κ −1 (κ(z)) = {z} if and only if .z ∈ I . If F is a closed subset of .S1 , then .κ(F ∩I ) = κ(F )∩κ(I ) and .κ|I is a closed function. Thus .κ|I is a homeomorphism of I onto .κ(I ). Uniform continuity will play a key role in two critical arguments using Cantor functions. Specifically, if E is a dense subset of a compact metric space and .f : E → X is uniformly continuous on E, then f extends to a continuous function of X to X. Although this is a special case of Theorem 26 in Chapter 6 of [39], it can readily be proved directly. The following result can be found in [52]. Proposition 3.3.2 If C is a Cantor set in .S1 and D is a countable dense subset of 1 1 1 .S , then there exists a Cantor function .κ : S → S such that .κ(A) = D. ∞ Proof The Cantorfunction .κ will be defined first   on . i=1 [ai , bi ]. Letting .D = {d1 , d2 , . . . }, set .κ [a1 , b1 ] = d1 and .κ [a2 , b2 ] = d2 . For notational uniformity,

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3 A Family of Examples

let .k1 = 1 and .k2 = 2. Next let .k3 be the smallest integer not in .{k1 , k2 } such that .{dk1 , dk2 , dk3 } and  .[a1 , b1 ], [a2 , b2 ], [a3 , b3 ] have the same1 counterclockwise ordering. Then set .κ [a3 , b3 ] = dk3 . Because D is dense in .S , this process can be continued inductively to define .κ on . ∞ i=1 [ai , bi ]. Every point in D will be the image of some .[ai , bi ] because A, the set of accessible points, is dense in C. By construction, .κ maps distinct complementary intervals to distinct points and preserves their counterclockwise order. Since .κ is now defined on A, to extend .κ to a continuous function √ on C it suffices to show that .κ is uniformly continuous on A. Given .0 < ε < 2, there exists N such that .|z −w| < ε when z and w are in the closure of the same component of .S 1 \  {d1 , . . . , dN }. Let .(aij , bij ) be the complementary interval such that .κ [aij , bij ] = dj . There exists .δ > 0 such that z and w are in the same component of E = S1 \

.

N 

(aij , bij )

j =1

when .|z − w| < δ. If z and w are in A and satisfy .|z − w| < δ, then they are in the same component of E and there exist distinct integers m and n in .{1, . . . , N } such that z and w are in .(bim , ain ) and .(aij , bij ) is not contained in .(bim , ain ) for all .j ∈ {1, . . . , N}. Since .κ preserves the cyclic order of the complementary intervals, the points .κ(z) and 1 .κ(w) are in the closure of the same component of .S \ {d1 , . . . , dN }. Therefore, .|κ(z) − κ(w)| < ε when .|z − w| < δ, proving that .κ is uniformly continuous on A. So .κ has a continuous extension to .S1 which will also be denoted by .κ. Clearly, 1 .κ(C) = S . .(ai , bi ) and .(aj , bj ), it follows that intervals  Given distinct  complementary      .κ (bi , aj ) = κ(bi ), κ(aj ) and .κ (bj , ai ) = κ(bj ), κ(ai ) . If z is in I and w is in .S1 , then there exist distinct complementary intervals .(ai , bi ) and .(aj , bj ) such that z is in .(bi , aj ) and w is in .(bj , ai ). Consequently, .κ(w) = κ(z) and −1 (κ(z)) = {z} when .z ∈ I . Therefore, .κ is a Cantor function such that .κ(A) = D. .κ   Corollary 3.3.3 If .C is a Cantor set in .S1 , then .S1 /∼C is homeomorphic to .S1 . Proof Since .S1 is separable, Cantor functions exist for every Cantor set C contained S1 . Then .S1 and .S1 / ∼C are homeomorphic because Cantor functions are quotient maps.  

.

The existence of cascades on the circle with Cantor minimal sets was first proved by Denjoy [28]. To prove the Denjoy theorem, we will make use of a general result (Theorem 3.3.4) on constructing invariant Cantor sets and then apply it to an irrational rotation of the circle. Theorem 3.3.4 is in turn a special case of Theorem 1.1 in [52], where it is used to classify the cascades .(S1 , f ) having Cantor minimal sets. The proof of Theorem 3.3.4 uses a uniform continuity argument similar to the one in the proof of Proposition 3.3.2.

3.3 Denjoy Flows on the Torus

77

Theorem 3.3.4 Let .κ : S1 → S1 be a Cantor function for a Cantor set C in .S1 . If g is a homeomorphism of .S1 onto itself such that .g(κ(I )) = κ(I ), then there exists a homeomorphism f of .S1 onto itself such that .f (C) = C and .κ ◦ f = g ◦ κ. Proof To start, let .f (z) = κ −1 ◦ g ◦ κ(z) for .z ∈ I . Since it has already been pointed out that .κ|I is a homeomorphism of I onto .κ(I ), the function f must be a homeomorphism of I onto itself. Using arguments similar to those used to prove Proposition 3.3.2, it will be shown that the homeomorphism of I onto itself given by .f = κ −1 ◦ g ◦ κ√ is uniformly continuous on I . Given .0 < ε < 2, there exists a finite set of distinct complementary intervals .(aij , bij ), .j = 1, . . . , N such that .|z − w| < ε when z and w are in the same component of S1 \

.

N 

[aij , bij ].

j =1

Set ⎛ F2 = κ ⎝

.

N 

⎞ [aij , bij ]⎠ ⊂ S1 \ κ(I ) = κ(A)

j =1

and F1 = g −1 (F2 ).

.

Then .F1 ⊂ S1 \ κ(I ) = κ(A) because .g ◦ κ(I ) = κ(I ). Also note that .F2 is finite and hence so is .F1 . Let z and w be in I . Because .κ −1 (F1 ) is √ the closure of a finite collection of complementary intervals, there exists .0 < δ < 2 such that when .|z − w| < δ they are in the same component of .S1 \ κ −1 (F1 ). It follows that .κ(z) and .κ(w) are in the same component of .S1 \ F1 when .|z − w| < δ. Consequently, .z = g ◦ κ(z) and  1 .w = g ◦ κ(w) are in the same component of .S \ F2 when .|z − w| < δ. N 1 The components of .S \ j =1 [aij , bij ] are N open saturated intervals of .S1 , so their images under .κ must be the components of .S1 \ F2 . So if .|z − w| < δ and −1 (z ) and .κ −1 (w  ) are in the same component of .S1 \ N [a , b ] .z, w ∈ I , then .κ ij j =1 ij and |f (z) − f (w)| = |κ −1 (z ) − κ −1 (w  )| < ε

.

when .|z − w| < δ, completing the proof that f is uniformly continuous on I . Therefore, f has a continuous extension to .I − = C, and .f (C) = C because .f (I ) = I . Similarly, .κ ◦ f (z) = g ◦ κ(z) holds for all .z ∈ C because it holds for .z ∈ I by the definition of f on I .

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3 A Family of Examples

The proof thus far applies to .g −1 as well as to g, so there exists a continuous function .fˆ such that .κ ◦ fˆ(z) = g −1 ◦κ(z) for all .z ∈ C. It follows that .κ ◦ fˆ ◦f (z) = κ(z) for all .z ∈ C and .fˆ ◦ f (z) = z for all .z ∈ I . Thus .fˆ = f −1 on C. To extend f to .S1 , first observe that .g◦κ(I ) = κ(I ) implies that .g◦κ(A) = κ(A). Given a complementary  interval  .(ai , bi ) of C, there exists a complementary interval .(aij , bij ) such that .f {ai , bi } = {aij , bij } because .κ ◦ f = g ◦ κ and f is a homeomorphism of C onto itself. Now let .fi be a homeomorphism of .[ai , bi ] onto .[aij , bij ] that extends f and set  f (z) =

.

f (z)

if z ∈ C

fi (z) if z ∈ (ai , bi ).

To show that f is continuous, let .zn be a sequence in .S1 converging to z and show that .f (zn ) converges to .f (z). When z is in a complementary interval, this is obviously true. Likewise when .zn is in C for all n. Suppose .zn falls into neither case. Define a sequence .wn by  wn =

.

ai

if zn is in (ai , bi ) for some i

zn

if zn is in C.

Clearly .wn is in C for all n. Observe that .wn also converges to z and .f (zn ) = f (wn ) for all n. Thus .

lim f (zn ) = lim f (wn ) = f (z),

n→∞

n→∞

completing the argument that f is continuous. Finally, .fi−1 can be used to extend .fˆ to the inverse of f on .S1 .   Using Theorem 3.3.4, the next result establishes the existence of circular cascades with minimal Cantor sets first proved by Denjoy [28]. Actually, he proved more by constructing examples with a continuous first derivative (see Theorem 7.2.3 in [23]). Theorem 3.3.5 (Denjoy) Let .(S1 , g) be a rotation .g(z) = e2π iα z with .α irrational. If D is a countable invariant set for .(S1 , g) and C is a Cantor subset of .S1 with accessible points A, then there exist a cascade .(S1 , f ) and a homomorphism 1 1 1 .κ : (S , f ) → (S , g) such that C is the only minimal set of .(S , f ), the homomorphism .κ is a Cantor function, and .κ(A) = D. Furthermore, .κ(z) = κ(w) if and only if z and w are doubly asymptotic or .z = w. Proof Recall that .(S1 , g) is a minimal cascade because every orbit is dense. In particular, D is a countable dense set of .S1 . So by Proposition 3.3.2, there exists a Cantor function .κ for C such that .κ(A) = D. Then Theorem 3.3.4 applies because 1 .D = κ(A) is g invariant by hypothesis. So there exists a homeomorphism f of .S

3.3 Denjoy Flows on the Torus

79

onto itself such that .f (C) = C and .κ ◦ f = g ◦ κ, that is, .κ is a homomorphism of (S1 , f ) onto .(S1 , g). Because .(S1 , g) is a minimal cascade, every point of .S1 is almost periodic by Theorem 1.1.11. Let z be in .I = C \ A, and let U be an open neighborhood of z. Then there exist w and .w  in I such that .z ∈ (w, w  ) ⊂ U . Note that .(w, w  ) is an open saturated set of .S1 , that is,

.

  κ −1 κ[(w, w  )] = (w, w  ).

.

Since .κ is clearly a quotient map, .κ[(w, w  )] is an open neighborhood of .κ(z) and there exists a syndetic subset Q of .Z such that g n (κ(z)) = κ(f n (z)) ∈ κ[(w, w  )]

.

for all .n ∈ Q. It follows that .f n (z) ∈ (w, w  ) for all .n ∈ Q because .(w, w  ) is saturated. Therefore, z is an almost periodic point of .(S1 , f ). Since .D = κ(A) is a countable dense invariant set of .(K, g), its complement 1 .S \ κ(A) = κ(I ) is an uncountable dense invariant set. So given .z ∈ I , it follows   that .O(κ(z)) = κ O(z) is dense in .κ(I ), which implies that .O(z) is dense in I because .κ|I is a homeomorphism. Therefore, .C = I − ⊂ O(z)− ⊂ C, and C is a minimal set by Theorem 1.1.11. It follows from Proposition 3.3.1 that C is the only minimal set of .(S1 , f ) and that z and w are doubly asymptotic when .κ(z) = κ(w). Conversely, if z and w are doubly asymptotic, then using the uniform continuity of .κ, it follows that .κ(z) and .κ(w) are doubly asymptotic. But g is an isometry and has no pairs of distinct points that are doubly asymptotic. So .κ(z) = κ(w).   The hypotheses of Theorem 3.3.5 require only a Cantor set C in .S1 , a positive irrational number a, and a countable invariant set D of the rotation .g(z) = e2π ia z to construct a cascade on the circle with a Cantor minimal set. Such a cascade will be called a Denjoy cascade. The choice of the irrational number a and the set D does make a difference, however, in the isomorphism class of the cascade [52] and the cardinality of the isomorphism classes of the Denjoy cascades is infinite. For example, D can contain a finite or countable number of g-orbits, that is, there exists a finite or countable set . of .S1 such that for distinct points .ζ and .ζ  in . the sets .O(ζ ) and .O(ζ  ) are disjoint and D=



.

O(ζ ).

ζ ∈

Then for each .ζ in ., there exists a unique complementary interval .(aζ , bζ ) contained in .κ −1 (ζ ), S1 \ C =



.

ζ ∈ n∈Z

  and the sets .f n (aζ , bζ ) are all disjoint.

  f n (aζ , bζ ) ,

(3.7)

80

3 A Family of Examples

With the proof of Theorem 3.3.5, we have seen that all three types of minimal sets possible for a homeomorphism of .S1 onto itself (Corollary 1.1.14) do occur. Were we to continue the study of cascades on the circle, the next step would be to prove that for a lift .f of a degree 1 homeomorphism f of .S1 the limit fn (x) − x n→∞ n

ρ(f) = lim

.

exists and is independent of x and .f. In fact, .ρ(f ) = π(ρ(f)) is a topological invariant of f called the rotation number. For example, if .f (z) = e2π ia z, then 2π ia . A full treatment of the rotation number and its properties can be .ρ(f ) = e found in Section 7.1 of [23]. There are two additional results about cascades on the circle worth mentioning. First, every minimal cascade on the circle is isomorphic to an irrational rotation. A proof using the classical rotation number of Poincaré appears in [23] (Theorem 7.1.9), and one based on more general results can be found in [49]. Second, every cascade on the circle with a Cantor minimal set is isomorphic to a Denjoy cascade (see [52] or use Theorem 7.1.9 in [23]). Up to isomorphism, there are no circular cascades with Cantor minimal sets that cannot be constructed using Theorem 3.3.5. Given a Denjoy cascade .(S1 , f ) with minimal Cantor set C, its suspension flow 1 1 .(S(S , f ), R) is called a Denjoy flow. The space .S(S , f ) is always a torus, that is, 2 homeomorphic to .T because .deg(f ) = 1. Using the notation of Section 3.1, .CS = π(C × R) is a minimal subset of .(S(S1 , f ), R) by Theorem 3.1.10. If .w = π(z, t) is not in .CS , then .α(w) = ω(w) = CS (Propositions 1.2.4 and 3.1.12). Given a complementary interval .(a, b) of C, the covering map .π is injective on the open set .(a, b) × R because .(a, b) ∩ (f n (a), f n (b)) = φ for all .n ∈ Z. Since .π 1 is open, .π |(a, b) × R is a homeomorphism onto an open  subset of .S(S , f ). Using equation (3.7), it follows that for each .ζ ∈  the set .π (aζ , bζ )×R is a component of .S(S1 , f ) \ CS homeomorphic to .(aζ , bζ ) × R. Exercise 3.3.6 Prove that the double asymptoticity of a Denjoy cascade .(S1 , f ) persists in its suspension. That is, given a complementary interval .(a, b) of the Cantor  minimal set C of .(S1 , f ) and a metric d for .S(S1 , f ), prove that .lim|t|→∞ d π(z, t), π(w, t) = 0 for all z and w in .(a, b). Part of the conclusion of Theorem 3.3.5 is the existence of a Cantor function κ that is a homomorphism of the Denjoy cascade .(S1 , f ) onto an irrational rotation .(S1 , g). By Proposition 3.1.5, there exists a canonical homomorphism 1 1 1 .κS : S(S , f ) → S(S , g). It follows from the discussion on page 64 that .S(S , g) is isomorphic to a minimal straight line flow. A minimal set of a flow on a compact surface is said to be nontrivial minimal set provided it is not a periodic orbit (which includes fixed points by definition of periodic) and it is a closed nowhere dense set in the compact surface. The minimal sets of Denjoy flows are examples of nontrivial minimal sets as will be shown in the next paragraph. .

3.3 Denjoy Flows on the Torus

81

Given a Denjoy cascade .(S1 , f ) with minimal Cantor set C and its suspension flow .(S(S1 , f ), R) with minimal set .CS = π(C × R), we know that .(S1 , f ) has no periodic points, and hence .(S(S1 , f ), R) has no periodic points by Corollary 3.1.3. Similarly, since C is a closed nowhere dense subset of .S1 and since a vertical line of 1 1 .S × R is nowhere dense, it follows that .C × R is a nowhere dense subset of .S × R. Finally, .CS = π(C × R) is closed because it is a minimal set of flow on a compact metric space and cannot have interior because .C × R has no interior. Combining the suspension flows on the torus and Klein bottle constructed in this chapter with Beck’s theorem (page 22), one can exhibit a wide range of qualitative behaviors of flows on the torus. For example, by adding a fixed point to the minimal set of a Denjoy flow, one obtains a recurrent orbit closure that is not a minimal set and is closed nowhere dense. This general approach can also be used to construct flows on surfaces of higher genus. Theorem 3.3.7 If X is a compact connected orientable surface, then there exists a flow .(X, R) with at least .γ (X) or .[(γ (X) − 1)/2] nontrivial minimal sets depending on whether X is orientable or nonorientable. Proof A Denjoy flow, of course, is an example of a flow on a compact connected orientable surface of genus 1 with 1 nontrivial minimal set. So we proceed by induction in the orientable case. Suppose .(Xm , R, ϕm ) is a flow on a compact connected orientable surface of genus .m > 1 with m nontrivial minimal sets. Let .(X1 , R, ϕ1 ) be a Denjoy flow. Since nontrivial minimal sets are closed nowhere dense sets, there exist regular Euclidean balls .Bm and .B1 in .Xm and .X1 , respectively, such that the intersections − and .B − with the nontrivial minimal sets are empty. of .Bm 1 Apply Theorem 1.3.3 to obtain new flows .(Xm , R, ψm ) and .(X1 , R, ψ1 ) such that .∂Bi is contained in the set of fixed points of .ψi for .i ∈ {1, m}. Note that a nontrivial minimal set M of .(Xm , R, ϕm ) or .(X1 , R, ϕ1 ) is a nontrivial minimal set of .(Xm , R, ψm ) or .(X1 , R, ψ1 ) because the orbits of M are not changed by the application of Theorem 1.3.3. Then .ψi |(Xi \ Bi ) is a flow on a bordered surface with a single boundary component consisting of fixed points for .i ∈ {1, m}, and we can form the compact connected surface .Xm #X1 . Clearly, .γ (Xm #X1 ) = m + 1, and the flows .ψm |(Xm \ Bm ) and .ψ1 |(Xi \ Bi ) define a flow on .Xm #X1 with .m + 1 nontrivial minimal sets, to complete the orientable case. Notice that in the nonorientable case the statement provides no information about 2 2 2 .P and .P #P because .[(γ (X) − 1)/2] = 0 in those cases. These two cases will be resolved in Sections 4.3 and 5.3. So we will assume that .γ (X) ≥ 3 when X is not orientable. It is shown in Lemma 6.16 of [42] that .T2 #P2 = P2 #P2 #P2 . It follows from this basic connection between .T2 and .P2 that

82

3 A Family of Examples

.

2 T . . #T2 #P2 = P2 # . . . #P2 = X2m+1  # . m copies

2m+1 copies

and .

2 T . . #T2 #P2 #P2 = P2 # . . . #P2 = X2m+2  # . m copies

2m+2 copies

for .m ≥ 1. Then .X2m+1 and .X2m+2 are compact connected nonorientable surfaces such that .γ (X2m+1 ) = 2m + 1 and .γ (X2m+2 ) = 2m + 2. These two sequences of surfaces include all compact connected nonorientable surfaces of genus at least 3 for .m ≥ 1. Now the orientable case can be applied to construct a flow .ϕm with m nontrivial minimal sets on .

2 T . . #T2 .  # . m copies

It follows that there exist flows on the surfaces .

2 2 T . . #T2 #P2 and T . . #T2 #P2 #P2  # .  # . m copies

m copies

with m nontrivial minimal sets by adding trivial flows on .P2 and .P2 #P2 .

 

We will return to counting nontrivial minimal sets and more generally recurrent orbit closures in Section 9.3.

Chapter 4

Local Sections

The two most important mathematical tools in the study of flows on surfaces are local sections and the Jordan separation theorem. Typically they are used together to limit or control to some extent where orbits can and cannot go. Starting with a well-placed local section, this approach can lead to results about the dynamical behavior of orbits on a surface. The classical Poincaré–Bendixson theorem for planar autonomous differential equations is the prototype result. In this context, the local section is a small line segment perpendicular to the vector field at a specific noncritical point of the vector field. Section 4.3 will revisit these results in the context of flows on the 2-sphere. The local section will no longer be a line segment but an arc. The main result of the chapter is proving in Section 4.2 that there is always a local section that is an arc at a moving point of a flow on a surface. This was first proved by Whitney [79] in 1933. Six years later, Bebutoff [17] proved the general existence of local sections at moving points of flows on metric spaces, but his local sections were just closed subsets of the metric space with no other specified topological structure. The approach here will be to define local sections (also known as local cross sections) for flows on metric spaces, prove Bebutoff’s theorem, and discuss general properties of local sections including the Poincaré return function in Section 4.1. The proof of Whitney’s theorem in Section 4.2 will make use of Bebutoff’s theorem, Peano continua, and the Jordan separation theorem. The second section concludes with several important applications of Whitney’s theorem to flows on surfaces.

4.1 Bebutoff’s Theorem Given a flow .(X, R) on a metric space X, a closed subset .λ of X is by definition a local section of length .2α provided the function .h : λ × [−α, α] → X defined by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_4

83

84

4 Local Sections

z

λ

y

x Fig. 4.1 A local section for a flow in .R3

s 1 F

−α

α

t

B

Fig. 4.2 A homeomorphism of .[0, 1] × [−α, α] onto B

h(x, t) = xt is a homeomorphism of .λ × [−α, α] onto .B = λ[−α, α] = {xt : x ∈ λ and − α ≤ t ≤ α} and that

.

λi ≡ B o ∩ λ = φ.

.

(4.1)

Define .λ = (λi )− . Local sections are also known as local cross sections. When .λ is a local section of length .2α, the set .B = λ[−α, α] is called a flow box. Figure 4.1 shows a local section for a flow in .R3 . The local section is a closed disk and an orbit flows through every point of it from the left to the right. The flow box is not shown but is a tube with .λ cutting it into two pieces. And .λi consists of all the points in .λ that are not on the rim of the disk. Figure 4.2 shows the flow box of a local section .λ that is an arc on the torus.

4.1 Bebutoff’s Theorem

85

Note that .y ∈ λ is in .λi if and only if the flow box B is a neighborhood of y. The opposite of a fixed point is by definition a moving point, that is, x is a moving point of a flow provided that .xt = x for some .t ∈ R. Obviously, every point in a local section is a moving point. Without the requirement that the flow box .B = λ[−α, α] is a neighborhood of at least one moving point in .λ, the concept of a local section would be of little value. It is frequently important to know that a particular moving point y of X is in i i .λ for some local section .λ. When .y ∈ λ for a local section .λ, we say that .λ is a local section at the point y or simply a local section at y. As you would expect, a local section is usually a local section at many different points and the context dictates which ones are important. The main goal of the section is to prove that given a moving point y of a flow on a metric space, there exists a local section at y, but first some elementary but important properties of local sections that follow primarily from the definition need to be included here. Proposition 4.1.1 If .λ is a local section of length .2α for a flow .(X, R) on a metric space X at a point y with flow box B, then .λ has the following properties: (a) If x is in B, then there exist a unique .w ∈ λ and a unique .t ∈ [−α, α] such that .x = wt. (b) If .xσ and .xτ are distinct elements of .λ, then .{(xσ )s : −α ≤ s ≤ α} and .{(xτ )t : −α ≤ t ≤ α} are disjoint pieces of the orbit of x contained in B. (c) If .xσ and .xτ are distinct elements of .λ, then .|σ − τ | > 2α. (d) Given .x ∈ X, the set .{t ∈ R : xt ∈ B} is the disjoint union of closed intervals of length .2α or empty. (e) The set .{t ∈ R : xt ∈ λ} is a finite or countable discrete subset of .R. Proof Parts (a) and (b) follow from the requirement that h is a homeomorphism of λ × [−α, α] onto .B = λ[−α, α]. Then (c) follows from (b), (d) from (c), and (e) from (d). 

.

The following general fact has a useful corollary about local sections and will be used to prove the existence of local sections: Proposition 4.1.2 Let .(X, G) be a continuous (right) group action of a metric group G on a metric space X. If E is a closed subset of X and C is a compact subset of G, then EC is a closed set of X. Proof Because X and G are metric, it suffices to use sequences in the proof. If xn gn is a sequence in EC converging to z in X, then without loss of generality it can be assumed that .gn converges to .g ∈ C because C is compact. It follows that .xn = (xn gn )gn−1 converges to .x = zg −1 , which is in .E − = E. Then .z = limn→∞ xn gn = xg is an element of EC, proving that EC is closed. 

.

Corollary 4.1.3 If .λ is a local section of a flow .(X, R) at a point y on a metric space X, then the flow box .B = λ[−α, α] is a closed set of X.

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4 Local Sections

The concept of a local section comes from differential equations and goes back at least to 1900 and the work of Poincaré and Bendixson. The following general existence theorem for local sections is a version of Bebutoff’s theorem from Nemytskii and Stepanov [67] and is proved using the less cumbersome approach of Hajek [33]. Theorem 4.1.4 (Bebutoff) If y is a moving point of a flow .(X, R) on a metric space X, then there exists a local section at y. Proof Let d be a metric for X. There exists .σ > 0 such that .y = yt when .0 < t ≤ σ . In fact, when y is not periodic, any .σ > 0 works. The function .g(x) = d(x, y)/d(y, yσ ) is a continuous real-valued function on X. Set 

σ

G(x) =

g(xs) ds.

.

0

Then .G(x) is a continuous real-valued function on X by Proposition 1.3.2 and 

σ

G(xt) =

.



σ

g((xt)s) ds =

0

 g(x(t + s)) ds =

0

σ +t

g(xu) du t

for .x ∈ X and .t ∈ R. Hence for fixed x, the function .t → G(xt) is a differentiable real-valued function on .R and G (xt) =

.

dG(xt) = g(x(t + σ )) − g(xt), dt

which is a continuous real-valued function on .X × R such that G (y) = g(yσ ) − g(y) = 1.

.

Using the continuity of .G , there exist an open neighborhood .U1 of y and α > 0 such that .G (xt) > 0 for all .(x, t) ∈ U1 × (−4α, 4α). Consequently, .G(y(−α)) < G(y) < G(yα), and there exists an open neighborhood of .U2 of y such that .G(x(−α)) < G(y) < G(xα) for all .x ∈ U2 . Because metric spaces are regular, there exists an open neighborhood of U of y such that .U − ⊂ U1 ∩ U2 . Set .

λ = {x : G(x) = G(y)} ∩ U − [−α, α]

.

and B = λ[−α, α].

.

By Proposition 4.1.2, .U − [−α, α] is closed. Now .λ is a closed set of X because {x : G(x) = G(y)} is obviously closed.

.

4.1 Bebutoff’s Theorem

We will use two lemmas to complete the proof.

87



Lemma 4.1.5 The sets U , .λ, and B have the following properties: (a) .U − ⊂ B and B is a neighborhood of y. (b) If .x ∈ λ, .xs ∈ λ, and .|s| < 2α, then .s = 0. (c) Given .x ∈ B, there exist unique .z ∈ λ and .t ∈ [−α, α] such that .x = zt. Proof For part (a), let x be in .U − . So .G(x(−α)) < G(y) < G(xα), and there exists .t ∈ (−α, α) such that .G(xt) = G(y). Thus xt is in .λ and x is in B. Next suppose that .x ∈ λ, .xs ∈ λ, and .|s| < 2α. By definition, .λ ⊂ U − [−α, α] and .x = zτ for some .z ∈ U − and .|τ | ≤ α. Clearly, .s + τ is in .[−3α, 3α]. By hypothesis, x and xs are .λ, and hence G(zτ ) = G(x) = G(y) = G(xs) = G(z(s + τ )).

.

Since .U − ⊂ U1 , it follows that .G (zt) > 0 on .[−3α, 3α] and .G(zt) is increasing on .[−3α, 3α]. Therefore, .G(zτ ) = G(z(s + τ )) if and only if .s = 0, and the proof of part (b) is complete. Given x in B, there exist .z ∈ λ and .t ∈ [−α, α] such that .x = zt. Suppose there exist .z1 and .z2 in .λ and .τ1 and .τ2 in .[−α, α] such that .z1 τ1 = x = z2 τ2 for some x in B. Then .z1 = z2 (τ2 −τ1 ) and .|τ2 −τ1 | ≤ 2α. Part (b) now implies that .τ2 −τ1 = 0 and then .z1 = z2 .  It follows from part (c) of Lemma 4.1.5 that there exist unique functions .μ : B → λ and .ν : B → [−α, α] such that .x = μ(x)ν(x). Lemma 4.1.6 The functions .μ and .ν are continuous on B and .λ = μ(U − ). Proof Let .xn be a sequence in B converging to x in B. If .ν is continuous, then .ν(xn ) converges to .ν(x) and .μ(xn ) = xn (−ν(xn )) converges to .x(−ν(x)) = μ(x). Thus the proof is reduced to showing that .ν is continuous. If .ν is not continuous at .x ∈ B, using the compactness of .[−α, α], there exists a sequence .xn converging to x such that .ν(xn ) converges to .τ = ν(x). Then .xn (−ν(xn )) converges to .x(−τ ) which is an element of .λ because .λ is a closed set of X. Since .τ is clearly in .[−α, α], the uniqueness of .ν(x) implies that .ν(x) = τ , a contradiction. Therefore, .ν is a continuous function. Part (a) of Lemma 4.1.5 implies that .μ(U − ) ⊂ λ. Since .λ ⊂ U − and .μ(λ) = λ by the definitions of .λ and .μ, it follows that .λ = μ(λ) ⊂ μ(U − ) ⊂ λ.  We can now complete the proof of the theorem. It follows from Lemma 4.1.5 part (c) that the continuous function .h(x, t) = xt is a bijective function from .λ×[−α, α] to B. Moreover, .h−1 : B → λ × [−α, α] is given by .h−1 (z) = (μ(z), ν(z)) and is continuous by Lemma 4.1.6. Finally, .B = λ[−α, α] is a neighborhood of y by part (a) of Lemma 4.1.5. The functions .μ and .ν are not particular to the construction used to prove Theorem 4.1.4. Let .λ be a local section of length .2α for a flow .(X, R) on a metric space, and for .(x, t) ∈ λ × [−α, α] let .h(x, t) = xt. Define .μ and .ν by

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μ = π1 ◦ h−1 and ν = π2 ◦ h−1 ,

.

(4.2)

where .π1 and .π2 are the projections of .λ × [−α, α] onto .λ and .[−α, α], respectively. Clearly, .μ and .ν are continuous functions such that .x = μ(x)ν(x) for all .x ∈ B = λ[−α, α]. These functions have additional nice properties that will be used in the next section. Proposition 4.1.7 If .λ is a local section of length .2α for a flow .(X, R) on a metric space, then the continuous functions .μ and .ν defined by equation (4.2) are open functions and the function .μ is a closed function. Proof Since .h−1 is a homeomorphism and thus an open and closed function, it suffices to show that .π1 and .π2 are open functions and that .π1 is a closed function. It follows from the definition of the product topology that the coordinate projections are open functions. Because .[−α, α] is compact, it follows that .π1 is a closed function.  We have already seen one example of a local section. Let .(X, h) be a cascade on a second-countable locally compact metric space X. By Proposition 3.1.1, .(S(X, h), R) is a flow on a metric space. It follows from the proof of Proposition 3.1.2 that the embedded copy of X in .S(X, h) given by .x → π(x, 0) is a local section of length .1/2. It is actually an example of a global section by definition because every orbit crosses it. Having an understanding of the subsets of a flow box that are open subsets of the space will prove useful. In particular, we will be able to construct open subsets of the space using our knowledge of the topology of the local section. Proposition 4.1.8 Let .λ be a local section for a flow .(X, R) of length .2α on a metric space X and, as usual, let .B = λ[−α, α] be the flow box. (a) A point .x ∈ B is in .B o if and only if there exist an open set .ϒ of .λ and an open interval .(β, γ ) contained in .(−α, α) such that .ϒ(β, γ ) is an open neighborhood of x in X. (b) If x is in .B o , then .μ(x) is in .λi and .μ(x)(−α, α) is contained in .B o . (c) .B o = λi (−α, α) and .B o is homeomorphic to .λi × (−α, α). (d) If .ϒ is an open set of .λi and .(β, γ ) is an open interval in .(−α, α), then o .ϒ(β, γ ) ⊂ B is an open set of X homeomorphic to .ϒ × (β, γ ).  i − (e) .λ ≡ (λ ) is a local section such that .(λ )i = λi . Proof For part (a), first note that .B o , which is an open set of X and contains x, is also an open set of B with its relative topology. In addition, .y(α + ε) and .y(−α − ε) are not in B for y in .λ and small .ε by part (d) of Proposition 4.1.1. Hence −1 (B o ) is an open neighborhood of .h−1 (x) in .λ × [−α, α] that does not intersect .h .λ × {−α} ∪ λ × {α}. Using the definition of the product topology, there exist an open set .ϒ of .λ and an open interval .(β, γ ) contained in .(−α, α) such that .h−1 (x) ∈ ϒ × (β, γ ) ⊂ h−1 (B o ). Because h is a homeomorphism, .h(ϒ × (β, γ )) = ϒ(β, γ ) is an open subset of B contained in .B o . So .ϒ(β, γ ) is also open in .B o with its

4.1 Bebutoff’s Theorem

89

relative topology. Then .ϒ(β, γ ) is an open neighborhood of x in X because .B o is open in X ([42], Proposition 3.4 (e) and( f)). The converse is a trivial consequence of the definition of the interior. For part (b), let x be in .B o . By part (a), there exist an open set .ϒ of .λ and an open interval .(β, γ ) contained in .(−α, α) such that .ϒ(β, γ ) is an open neighborhood of x in X. For .−α −β < t < α −γ , the set .ϒ(β, γ )t = ϒ(β +t, γ +t) is an open subset of X contained in B. Thus these sets are contained in .B o and cover .μ(x)(−α, α). Then .μ(x)(−α, α) is contained in .B o and .μ(x) is in .λi . Part (c) is an immediate consequence of part (b). If .ϒ is an open subset of .λi and .(β, γ ) is an open interval contained in .(−α, α), then .ϒ(β, γ ) is a subset of .B o because .B o = λi (−α, α) by part (c). Also .ϒ(β, γ ) is open in X because .B o is homeomorphic to .λi × (−α, α) by (c). Since .λ∗ ⊂ λ and the open set .B o = λi (−α, α) of X is contained in .λ (−α, α), the set .λ is a local section and .λi ⊂ (λ )i . Finally, .λ ⊂ λ implies that  o (λ )i (−α, α) = λ (−α, α) ⊂ B o = λi (−α, α),

.

which then implies that .(λ )i ⊂ λi to complete the proof of part (e).



Corollary 4.1.9 If y is a moving point of a flow .(X, R) on a metric space X, then there exists a countable neighborhood base at y consisting of the flow boxes of local sections at y. Proof Given a moving point y of .(X, R), let .λ be a local section at y of length .2α. Without loss of generality, .λ = (λi )− by part (e), and .λi is an open subset of .λ with the relative topology. For large n, the set .{x ∈ λ : d(x, y) ≤ 1/n} ⊂ λi . For such n, set .λn = {x ∈ λ : d(x, y) ≤ 1/n}− ⊂ λ Then .Vn = λn [−α/n, α/n] is a countable neighborhood base of flow boxes at y.  Just the existence of a local section is a tool for studying the dynamical behavior of returning points by introducing the Poincaré return function. It is particularly useful when the local section is homeomorphic to a familiar space. Theorem 4.1.10 (Poincaré) Let .λ be a local section at y for a flow .(X, R) on a metric space X such that .{t > 0 : yt ∈ λ} = φ, and let .τ = inf{t > 0 : yt ∈ λ}. If i i .yτ is in .λ , then there exists an open neighborhood .ϒ of y in .λ such that .f (x) = inf{t > 0 : xt ∈ λ} is a continuous real-valued function on .ϒ and .g(x) = xf (x) is a homeomorphism of .ϒ into .λi such that .g(y) = yτ . Proof As usual .2α denotes the length of .λ; .B = λ[−α, α], its flow box, and d, a metric for X. Clearly, .f (x) > 2α whenever .{t > 0 : xt ∈ λ} =  φ. There exists .η > 0 such that (a) .{x : d(x, yτ ) < η} ⊂ λ(−α/2, α/2) (b) .{x : d(x, yτ ) < η} ∩ λ ⊂ λi (c) .η < d(λ, y [α/2, τ − α/2])

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By the continuity of the flow, there exists .δ > 0 such that .d(xt, yt) < η for 0 ≤ t ≤ τ , when .d(x, y) < δ (Proposition 1.3.1). Because .λ is a local section at y, we can also assume that .{x : d(x, y) < δ} ∩ λ ⊂ λi . Consider .x ∈ λ such that .d(x, y) < δ. Then, on the one hand, .d(xt, yt) < η for t in the closed interval .[α/2, τ − α/2] implies .x[α/2, τ − α/2] ∩ λ = φ. On the other hand, .xτ is in .λ(−α/2, α/2) and xt crosses .λ exactly once in the interval .[τ − α/2, τ + α/2], say at .xσ . Therefore, .σ = f (x) and .f (x) satisfies .τ − α/2 ≤ f (x) ≤ τ + α/2 on i i .ϒ = {x : d(x, y) < δ} ∩ λ , which is an open subset of .λ . Suppose .xn is a sequence in .ϒ converging to .x ∈ ϒ. To prove that f is continuous at x, it suffices to show that every convergent subsequence of .f (xn ) converges to .f (x) because .f (xn ) is in the compact set .[τ − α/2, τ + α/2] for all n. So, without loss of generality, it can be assumed that .f (xn ) converges to .s ∈ [τ − α/2, τ + α/2]. Then .xn f (xn ) converges to xs and xs is in .λ because .xn f (xn ) is in .λ for all n. Thus .f (x) = s because .f (x) is the unique number in .[τ − α/2, τ + α/2] such that .xf (x) is in .λ, and f is continuous on .ϒ. The function .g(x) = xf (x) is obviously a continuous function from .ϒ to .λ. Suppose .xf (x) = wf (w) and .f (x) ≤ f (w). Then .x = w(f (w) − f (x)) and .0 ≤ f (w)−f (x) ≤ α which is impossible unless .f (w)−f (x) = 0. Then .f (x) = f (w) implies .xf (x) = wf (x) and .x = w. Thus g is injective on .ϒ. The last step in the proof is to show that g is a closed function on .ϒ. Let C be a closed subset of .ϒ. If w is in the closure of .g(C), then .w ∈ λ and there exists a sequence .xn in C such that .g(xn ) = xn f (xn ) converges to w. Without loss of generality, we can assume that .f (xn ) converges to .s ∈ [τ − α/2, τ + α/2]. Then .xn = [xn f (xn ) − f (xn )] converges to .w(−s). It follows that .g(xn ) converges to .g(w(−s)). In a metric space, the limit of a sequence is unique. Therefore, .w = g(w(−s)) is in .g(C) and g is a closed function.  .

The function .f (x) = inf{t > 0 : xt ∈ λ} is called the first return time and is often extended to .λ by setting .f (x) = ∞ when .{t > 0 : xt ∈ λ} = φ. The function .g(x) = xf (x) is called the Poincaré return function and often contains useful information about the dynamics of a flow. Proposition 4.1.10 has an analogue for autonomous differential equations (Theorem 6.4, [58]). Similar to Proposition 4.1.10, there is also something locally intrinsic about the topology of .λi that is worth stating for insight. The proof is left to the reader. Proposition 4.1.11 Let .(X, R) be a flow on a metric space X, and let .λ1 and .λ2 be local sections. If .xt1 and .xt2 are in .λi1 and .λi2 , respectively, then there exist open neighborhoods .ϒ1 and .ϒ2 of .xt1 and .xt2 in .λi1 and .λi2 , respectively, such that .ϒ1 and .ϒ2 are homeomorphic. Exercise 4.1.12 Prove Proposition 4.1.11.

4.2 Whitney’s Theorem

91

4.2 Whitney’s Theorem Building on the existence of local sections at moving points (Theorem 4.1.4) for flows on metric spaces, we study the topological properties of local sections for flows on manifolds culminating with the theorem that at a moving point of a flow on a surface there exists a local section that is homeomorphic to the unit interval. This is Whitney’s theorem, which is essential for the rest of this book. This is also the first section requiring the Jordan separation theorem, which is the other essential tool for the study of flows on surfaces. Because of the importance of the Jordan separation theorem and related results, we review these fundamental topological theorems from a broad perspective. The general theorem for .Rn in this circle of ideas is as follows: Jordan–Brouwer separation theorem An .(n − 1)-sphere embedded in .Sn separates .Sn into two components of which it is their common boundary. A proof of the Jordan–Brouwer separation theorem can be found in Spanier [72]. The first version of the Jordan–Brouwer separation theorem for .S2 was due to Jordan: Jordan separation theorem An embedded circle J in the 2-sphere .S2 separates 2 − = U ∪ J and .V − = V ∪ J , that .S into two components U and V such that .U is, J is the common boundary of both U and V . An obvious corollary is that both U and V are open sets of .S2 . There is also an equivalent version of the Jordan separation theorem for the complex plane .C. If J is an embedded circle in .C, then it is also an embedded circle of .C∞ , the one-point compactification of .C. Since .C∞ is homeomorphic to .S2 with the north pole .(0, 0, 1) playing the role of .∞, it follows from the Jordan separation theorem that .C∞ \ J consists of two connected sets U and V such that .U − = U ∪ J and − = V ∪ J . Obviously, .U − and .V − are compact and either U or V contains .V .∞, say V . Consequently, U and .W = V \ {∞} are disjoint sets of .C such that − = U ∪ J , and .W − = W ∪ J are in .C. Furthermore, U .C \ J = U ∪ W , .U is bounded because .U − = U ∪ J is a compact subset of .C and W is unbounded because .W = C\U − . It is not hard to show that W is connected, and U is obviously connected. Putting it together, we have the following: Jordan Separation Theorem for .C An embedded circle J in the complex plane − = U ∪ J and .C separates .C into two components U and W such that .U − .W = W ∪ J , that is, J is the common boundary of both U and W . Furthermore, one component, called the interior of J , denoted .JI , is bounded and the other component, called the exterior of J , denoted .JE , is unbounded. Obviously, the Jordan separation theorem for .C is equivalent to the Jordan separation theorem for .R2 , and it is sufficient to refer to the Jordan separation theorem. If J is an embedded circle in .S2 or .C, then J is compact and closed, implying that .S2 \ J or .C \ J is an open subset of .S2 or .C and a manifold. Thus U and V

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are path-connected because the path components of a manifold coincide with its components. (In fact, they are arcwise connected by Theorem 3–16 in [37].) A path in .S2 starting at a point in U and ending at a point in V must intersect J because a path is a connected set. This leads to a useful test for when two points are in the same component: if .f : [0, 1] → S2 or .f : [0, 1] → C is a continuous function such that .f ([0, 1]) ∩ J = φ, then .f (0) and .f (1) are both in U or both in V . This will be referred to as the same component test. The Jordan separation theorem for .C has also been generalized in a different direction that is false for higher dimensions. Schoenflies theorem If J is an embedded circle in .C, then there exists a homeomorphism h of .C onto itself such that .h(J ) = S1 . For a proof of the Schoenflies theorem, see [24]. The Alexander horned sphere shows that the Schoenflies theorem is not true for an embedding of .S2 in .R3 without additional hypotheses. For a picture of the Alexander horned sphere and further references, see [37]. There are already two immediate consequences of the Schoenflies theorem worth noting as propositions. Proposition 4.2.1 If J is an embedded circle in .C with interior .JI and exterior .JE , then the following hold: (a) The open set .JI is a regular Euclidean ball. (b) .JI− = JI ∪ J is a compact connected bordered surface homeomorphic to .D2 . (c) .JE− = JE ∪ J is a noncompact bordered surface homeomorphic to .{z ∈ C : 1 ≤ |z| < 2}. Using the one-point compactification .S2 of .C, it follows the following proposition: Proposition 4.2.2 If J is an embedded circle in .S2 , then the two components of 2 .S \ J are regular Euclidean balls and their complements are compact connected bordered surfaces homeomorphic to .D2 . For future use, we prove the following proposition: Proposition 4.2.3 Let J and .J be disjoint embedded circles in .C. Then exactly one of the following holds: (a) .J ∪ JI ⊂ JI . (b) .J ∪ JI ⊂ JI . (c) .(J ∪ JI ) ∩ (J ∪ JI ) = φ. Proof Either .J ⊂ JI or .J ⊂ JE . Suppose .J ⊂ JI . It can be assumed that .J = S1 and .JI = B2 by the Schoenflies theorem. Since .B2 is homeomorphic to .C, the Jordan curve theorem implies that the components of .B2 \ J consist of two open sets U and V such that .U − = U ∪ J is a compact subset of .B2 and .V − = V ∪ J is not compact. Hence .U = JI and (a) holds.

4.2 Whitney’s Theorem

93

The second case .J ⊂ JE has two subcases: either .J ⊂ JI or .J ⊂ JE . The same argument used to prove that .J ⊂ JI implies property (a) holds can be used to show that .J ⊂ JI implies (b). Suppose .J ⊂ JE . Taking the complement yields .JI ∪ JE ⊃ J ∪ JI . The left side is the disjoint union of two open connected sets, and the right side is a compact connected set. Thus .J ∪ JI ⊂ JE because .J ⊂ JE , proving (c).  Whitney [79], in his general investigation of regular families of curves, first discovered that on a surface one could always find a local section at a moving point that was homeomorphic to .[0, 1]. Whitney was interested in the existence of local cross sections and defining flows for regular families of curves. The disadvantage to Whitney’s proof, however, is that it is embedded in a deep abstract study of families of curves on separable metric spaces. To give a direct proof of Whitney’s theorem, we will first show that for a flow on a manifold every moving point has a local section that is a Peano continuum. Then, using the Jordan separation theorem, we will obtain the stronger result that at every moving point of a flow on a surface there exists a local section homeomorphic to .[0, 1]. This direct approach to Whitney’s theorem was developed by Helen Colston [25] in her University of Maryland master’s thesis with the author as her advisor. There is also a direct proof of Whitney’s theorem using dendrites given by Hajek in [33]. There may be others. A Peano continuum is a compact, connected, and locally connected metric space. For example, .Dn is a Peano continuum. A particularly important Peano continuum is the closed unit interval .[0, 1], which is homeomorphic to .D1 . Notice that if any point other than 0 or 1 is removed from the closed unit interval the result is a disconnected metric space. Thus 0 and 1 can be distinguished topologically from the other points of .[0, 1]. An arc is by definition a homeomorphic image of .[0, 1]. If Y is an arc, then it also contains exactly two points, called endpoints, that can be removed from Y without disconnecting the space. And any homeomorphism of .[0, 1] onto Y maps 0 and 1 onto these endpoints. More generally, a space X is arcwise connected provided that for every pair of distinct points x and y in X, there exists an arc Y in X with endpoints x and y. A key theorem that we need is that a Peano continuum is arcwise connected ([37], Theorem 3–15). Let y be a moving point of a flow .(X, R) on a metric space X. Recall that the proof of Theorem 4.1.4 began with the construction of two open neighborhoods, .U1 and .U2 , of y and .α > 0 depending on the metric and the flow and then required only that U be an open neighborhood of y such that .U − ⊂ U1 ∩ U2 allowing flexibility in the selection of U . We shall now take advantage of that flexibility to obtain local sections of manifolds that are Peano continua. Theorem 4.2.4 If y is a moving point of a flow .(X, R) on a topological manifold X, then there exists a local section at y that is a Peano continuum and hence arcwise connected.

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Proof Since manifolds are metric spaces (Proposition 2.1.2), Theorem 4.1.4 can be applied. Let .U1 and .U2 be open neighborhoods of y constructed as in the proof of Theorem 4.1.4. Because X is a manifold, there exists a Euclidean ball .U of X contained in .U1 ∩U2 and a homeomorphism f of .U onto .Enr (0) such that .f (y) = 0. Clearly, we can assume that .r > 1. Then .U = f −1 (Bn ) is a regular Euclidean ball at y such that .U − ⊂ U1 ∩ U2 . Thus by Theorem 4.1.4 and Lemma 4.1.6, there exists a local section .λ at y such that .μ(U − ) = λ. Since .U − is homeomorphic to .Dn , the local section .λ is compact and connected. Because .Dn is also locally connected and .μ a closed function (Proposition 4.1.7), .λ is locally connected by Lemma 3-21 of [37]. Therefore, .λ is a Peano continuum and arcwise connected.  Let X be a locally compact Hausdorff space that is not compact, and let .∞ denote an abstract point not in X. Set .X∞ = X ∪ {∞}. Then the collection of subsets U of .X∞ such that either U is an open subset of X or .X∞ \ U is a compact subset of X defines a compact Hausdorff topology on .X∞ ([39], 5.21), and .X∞ with this topology is called the one-point compactification. It follows that if C is a compact subset of X, then .U = X∞ \ C is an open neighborhood of the added point, .∞. The one-point compactification is the unique smallest compactification of a locally compact Hausdorff space that is not compact. The one-point compactification of a locally compact second-countable metric space that is not compact is a compact metric space ([59], Theorem 6.4.8). In particular, the one-point compactification of a manifold that is not compact is a compact metric space. For some topological spaces, there are well-known concrete representations of the one-point compactification. For example, as noted in Section 2.1 on page 32, the one-point compactification of n n .R is .S . It is convenient to adapt the usual notation for closed and open intervals to arcs. Given an arc .λ, if x and y are points in .λ, then the segment of .λ joining x and y in .λ is also an arc and will be denoted by .[x, y]λ . Similarly, .(x, y)λ will denote the segment of .λ joining the points x and y in .λ, without the endpoints x and y. Clearly, .(x, y)λ is homeomorphic to the open interval .(0, 1). Similarly, it is convenient to introduce notation for a connected segment of an orbit similar to the notation for sub-arcs of a local section. Although most of the time we have suppressed the function .ϕ : X ×R → X that defines a flow on a topological space X and written simply .(X, R) and xt instead of .(X, R, ϕ) and .ϕ(x, t), there are times when .ϕ is useful. For .x ∈ X and .σ < τ , let .(xσ, xτ )ϕ = {xt : σ < t < τ } and .[xσ, xτ ]ϕ = {xt : σ ≤ t ≤ τ }. Proposition 4.2.5 Let .(X, R) be a flow on a surface, and let .λ be an arc in X with endpoints p and q. If there exists a positive real number .α such that .h(x, t) = xt is an injective function on .λ × [−α, α], then .λ is a local section of length .2α such that i  .λ = λ \ {p, q} and .λ = λ. Proof Since .λ × [−α, α] is compact and X is Hausdorff, h is a homeomorphism of λ × [−α, α] onto its image .B = λ[−α, α]. So to prove that .λ is a local section, it suffices to prove that .λi = λ \ {p, q}.

.

4.2 Whitney’s Theorem

95

Let y be an element of .λ \ {p, q} and U be a Euclidean ball of X at y. There exist v and w in .λ such that y ∈ (v, w)λ ⊂ [v, w]λ ⊂ U.

.

Because .[v, w]λ is compact, there exists .δ > 0 such that .[v, w]λ [−δ, δ] ⊂ U and δ < α. Observe that .(v, w)λ (−δ, δ) is homeomorphic to the open set .(0, 1)2 of .R2 . Since U is homeomorphic to .R2 , invariance of domain implies that .(v, w)λ (−δ, δ) is an open set of U and thus an open set of X. Therefore, .λ \ {p, q} ⊂ λi . Similarly, invariance of domain can be used to show that p and q are not in .λi . Clearly, .λ = λ. 

.

Theorem 4.2.6 (Whitney) If y is a moving point of a flow .(X, R) on a surface X, then there exists a local section at y that is an arc and y is not an endpoint of it. Proof As in the proof of Theorem 4.2.4, construct a local section .λ at y that is a Peano continuum using a regular Euclidean ball U such that .U − ⊂ U1 ∩ U2 . Since .λ is arcwise connected, it suffices by Proposition 4.2.5 to prove that there exists an arc in .λ through y but not beginning or ending at y. Set σ = inf{t < 0 : ys ∈ U for t ≤ s ≤ 0}

.

and τ = sup{t > 0 : ys ∈ U for 0 ≤ s ≤ t}.

.

Using part (a) of Lemma 4.1.5 and part (d) of Proposition 4.1.1, it follows that .

−α ≤σ 2α. But .|s − s| ≤ |s | + |s| ≤ α + ν(v) < α + α/k < 2α. The same argument works when .ν(v) < 0. Therefore, .μ(v) is in V when v is in .V ∩ U and, likewise, .μ(w) is in W when w is in .W ∩ U . Fix v and w in .U ∩ V and .U ∩ W , respectively. Obviously every Euclidean ball in .Rn is arcwise connected, and Euclidean balls of a manifold, such as .U , are arcwise connected. So there exists an arc D in .U with endpoints v and w. Let .E = μ(D). It is a compact connected metric space because D is compact and connected, and it is locally connected by Lemma 3-21 of [37] because .μ is a closed function. Therefore, E is a Peano continuum and arcwise connected. Note that .E ⊂ U ∩ λ because .D ⊂ U and it was shown earlier in the proof that .μ(U ) ⊂ U ∩ λ. Therefore, there exists an arc .λ ⊂ E ⊂ U ∩ λ with endpoints .μ(v) ∈ V and .μ(w) ∈ W . The Jordan separation theorem now implies that φ = λ ∩ C ⊂ λ ∩ C = {y}

.

to complete the proof.



Whitney’s theorem naturally raises the question of whether flows on ndimensional manifolds always have local sections .λ at moving points such that i n−1 . In subsequent papers ([78] and [80]), Whitney proved .λ is homeomorphic to .B that this was true for .n = 3. Twenty years later a counterexample for .n = 4 emerged from a theorem about .R4 . In 1957, Bing [20] constructed a quotient space of .R3 that is often called Bing’s dogbone space. We will denote Bing’s equivalence relation by .∼db . The equivalence classes are all either points or tame arcs, but .R3 / ∼db is not homeomorphic to .R3 . In fact, .R3 / ∼db is not a manifold, so at least one point does not have an open neighborhood homeomorphic to .B3 . Bing [21] also proved that .(R3 / ∼db ) × R is homeomorphic to .R4 . This is where flows and local sections enter the picture. We can always define a flow on .X × R by .((x, s), t) → (x, s + t). When .X = R3 / ∼db , this produces a flow on the 4-dimensional manifold .R4 with .(R 3 / ∼db ) × {0} as a global section at every point of the form .(x, 0) that every orbit crosses exactly once. Now we can invoke Proposition 4.1.11. If the dogbone space is not a manifold at y, then no local section .λ at .(y, 0) can have .λi homeomorphic to .B3 , because otherwise .R3 / ∼db would be a manifold at y.

4.2 Whitney’s Theorem

97

There is, however, an important situation in which the conjecture holds, namely, locally Lipschitz autonomous differential equations. Consider .x˙ = f(x) where .f :  → Rd is locally Lipschitz on an open subset . of .Rd . When the solutions are all defined on .R, such a system defines a flow because the solutions are continuous in initial conditions (see [58], Theorem 1.23). This includes, for example, all the linear constant coefficient systems .x˙ = Ax, where A is a .d × d real matrix and . = Rd . But even when the solutions are not all defined on .R, at a noncritical point there exists a local section .λ homeomorphic to .Dd−1 with .λi homeomorphic to .Bd−1 ([58], Theorem 6.1). These local sections are usually constructed by taking a small disk perpendicular to the vector field at the given point as illustrated in Figure 4.1. The conjecture also holds for differentiable flows on differentiable manifolds by applying the result for differential equations locally. Standing Assumption 1 A local section .λ of a flow on a surface will always be homeomorphic to .[0, 1]. The next theorem illustrates how the structure of a flow box can be used on a surface as a result of Whitney’s theorem (Theorem 4.2.6). Theorem 4.2.7 If M is a nontrivial minimal subset of a flow .(X, R) on a compact connected surface, then M is not locally connected at every point in M. Proof Assume that M is a nontrivial minimal set, that is, M is not a periodic orbit and is nowhere dense in X. Let y be any point in M. Then, as in the proof of Whitney’s theorem, y is a moving point, there exists a local section .λ at y of length .2α, and . = λ ∩ M is a closed Cantor set contained in .λ. By Whitney’s theorem, there exists a homomorphism mapping the flow box .λ[−α, α] onto .B = [−1, 1] × [−α, α]; y onto .(0, 0); .λ onto .[−1, 1], and . onto C, a Cantor subset of .[−1, 1] because .λ is both an arc and a local section. A Cantor set is totally disconnected, that is, its components are the points in it. Consequently, the components of .C × [−α, α] are simply the collection of disjoint lines .{x} × [−α, α] for .x ∈ C. If U is an open connected neighborhood of .(0, 0) in the relative topology of .C × [−α, α], then there exists an open set V of B such that .U = V ∩ (C × [−α, α]). Clearly, V must intersect infinitely many components of  C. Therefore, U has infinitely many components. The rest of this section uses the Poincaré return function from Section 4.1 and Whitney’s theorem. Let .λ be a local section for a flow .(X, R) on a surface X. So there exists a homeomorphism .θ of .[0, 1] onto .λ mapping endpoints to endpoints. It imposes an order on .λ by defining .x < y provided that .θ −1 (x) < θ −1 (y). There are exactly two orderings of .λ that can be defined this way and they are the exact opposites of each other. A homeomorphism g of .[u, v]λ to .λ is either order preserving or order reversing independent of which order has been imposed on .λ. Our convention in this section will be to write .[u, v]λ when u is less than v in the order on .λ and .[v, u]λ when v is less than u. In this context, a function .g : [u, v]λ → λ will be called a Poincaré return function provided that it satisfies the following conditions:

98

4 Local Sections

(a) .[u, v]λ ⊂ λi . (b) The return function .f (x) = inf{t > 0 : yt ∈ λ} is a continuous real-valued function on .[u, v]. (c) The Poincaré function .g(x) = xf (x) is a homeomorphism of .[u, v]λ onto .[uf (u), vf (v)]λ or .[vf (v), uf (u)]λ . Given y in .λi such that .{t > 0 : yt ∈ λ} = φ, Theorem 4.1.10 guarantees the existence of a Poincaré return function on some .[u, v]λ such that .y ∈ (u, v)λ . Theorem 4.2.8 Let .λ be a local section of a flow .(X, R) on a surface X, and let g be a Poincaré return function that is a homeomorphism of .[u, v]λ into .λ. If X is orientable, then g is order preserving. Proof Let .2α be the length of .λ. Let .θ : [0, 1] → λ be the homeomorphism used to order .λ, and use .s → θ (s)t to order .λt consistently for .−α ≤ t ≤ α. Then .h(s, t) = θ (s)t is a homeomorphism of .[0, 1] × [−α, α] onto .B = λ[−α, α], the flow box of .λ, preserving the order .[0, 1] × {t} for .−α ≤ t ≤ α. Recall that the first return function is .f (x) = inf{t > 0 : yt ∈ λ} and the Poincaré return function is .g(x) = xf (x). Observe that g is order preserving {reversing} on .[u, v]λ if and only if g is order preserving {reversing} restricted to .[u , v ]λ contained in .[u, v]λ because g is a homeomorphism. To take advantage of this remark, let y be in .(u, v)λ and set .τ = f (y). By continuity, there exists .[u , v ]λ contained in .[u, v]λ such that .τ −α/4 < f (x) ≤ τ +α/4 for all .x ∈ [u , v ]λ . It follows that .|f (x)−f (x )| ≤ α/2 for any .x, x ∈ [u , v ]λ . For convenience, replace the original .[u, v]λ by .[u , v ]λ , without loss of generality. Let .a = θ −1 (u) and .b = θ −1 (v) and recall that .f (x) > 2α for .x ∈ [u, v]λ . Similarly, let .c = θ −1 (g(u)) and .d = θ −1 (g(v)) for later use. Define h : [a, b] × [0, 1] → {xt : x ∈ [u, v]λ and α/2 ≤ t ≤ f (x) − α/2} = B

.

by   h (s, t) = θ (s) α/2 − αt + f (θ (s))t .

.

It is easily seen that .h is continuous and its image is .B . Hence, .h is a homeomorphism if it is injective. Suppose .h (s, t) = h (s , t ) or, equivalently,     θ (s) α/2 − αt + f (θ (s))t = θ (s ) α/2 − αt + f (θ (s ))t .

.

Letting .θ (s) = x and .θ (s ) = x , the above simplifies to   x (f (x) − α)t − (f (x ) − α)t = x .

.

4.2 Whitney’s Theorem

99

Then, without loss of generality, .0 ≤ (f (x) − α)t − (f (x ) − α)t = β, and either .β = 0 or .β is a positive return time because both x and .x are in .λ. If .β = 0, then it quickly follows that .x = x and .t = t . Suppose .β > 0. Then .β ≥ f (x) > 2α and .f (x ) > 2α. So .(f (x ) − α)t ≥ 0. Hence .0 < t ≤ 1 and we have the following contradiction: β ≤ f (x) − α < τ + α/4 − α = τ − 3α/4 < τ − α/4 < f (x) ≤ β.

.

Consequently, .β = 0, the function .h is injective, and .h is a homeomorphism. Set .B/2 = λ[−α/2, α/2] and notice that .B/2 ∩ B consists of 2 arcs .{x(α/2) : x ∈ [u, v]λ } and .{x(f (x)−α/2) : x ∈ [u, v]λ }. These arcs were already consistently ordered with the .θ ordering on .λ. Thus .B is a strip of the surface to the opposite sides of .B/2. Next triangulate .[0, 1] × [−α/2, α/2] starting with the vertices at .(0, ±α/2), .(1, ±α/2), .(a, α/2), .(b, α/2), .(c, −α/2), .(d, −α/2), and .(1/2, 0) as shown in Figure 4.3. Using the earlier definitions of .a, b, c, d, add the obvious edges and triangles when g is order reversing as shown in Figure 4.3. Then triangulate .(a, b) × [0, 1] with vertices at the corners and one in the center .((a + b)/2, 1/2) and the obvious edges and triangles (see Figure 4.4). These Euclidean simplicial complexes will be denoted by .1 and .2 , respectively. Clearly .1  2 with the coherent topology is a bordered surface and another simplicial complex. Also  H (x) =

.

h(x)

when x ∈ 1

h (x)

when x ∈ 2

is a continuous map of .1  2 onto .B/2 ∪ B and a quotient map. Since h and .h are homeomorphisms onto .B/2 and .B , the equivalence relation .∼H on .1  2 determined by H can only have equivalence classes consisting of 1 or 2 points.

(0, α2 )

(a, α2 )

(b, α2 )

(1, α2 )

( 12 , 0)

(0, − α2 ) (c, − α2 )

(d, − α2 )

Fig. 4.3 .1 for an order reversing Poincaré return function

(1, − α2 )

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4 Local Sections

Fig. 4.4 The simplicial complex .2

(a, 1)

(a, 0)

(b, 1)

(b, 0)

Moreover, an equivalence class containing 2 points must be the inverse image of a point in .B/2 ∩ B . Observe that .h(s, α/2) = θ (s)(α/2) = h (s, 0) and .(s, α/2) ∼H (s, 0) for all .s ∈ [a, b]. It follows that .∼H identifies the edge .(a, 0), (b, 0) of .2 with the edge .(a, α/2), (b, α/2) of .1 . The same kind of relationship exists between the edge .(a, 1), (b, 1) of .2 and the edge .(c, −α/2), (d, −α/2) of .1 because   h (s, 1) = g(θ (s))(−α/2) = h θ −1 (g(θ (s))), −α/2 ,

.

which implies that .(s, 1) ∼H (θ −1 (g(θ (s))), −α/2) for all .s ∈ [a, b]. Thus .∼H identifies the edge .(a, 1), (b, 1) of .2 with the edge .(c, −α/2), (d, −α/2). Of course, we could now construct a simplicial complex .3 homeomorphic to .1  2 / ∼H and to .H (1  2 ) in X by the uniqueness of quotient spaces. Moreover, this simplicial complex would be a bordered surface because the two pairs of edges being identified are boundary edges. But this is not necessary because we can also extract the information we need directly from .1  2 . The Euler characteristic of .1  2 is obviously 2. Counting a pair of identified vertices {edges} as a single vertex {edge}, we obtain the Euler characteristic of .3 . Specifically, the number of vertices decreases by 4 and the number of edges decreases by 2 and hence the Euler characteristic decreases by 2. Therefore, the Euler characteristic of .3 is 0. Carefully counting the boundary curves of .1  2 without using the identified edges shows that .3 must have 1 or 2 boundary curve(s) depending on whether g is order reversing or order preserving. The only solution of equations (2.4) and (2.5) with Euler characteristic 0 and 1 boundary curve is the projective plane .P2 with 1 hole, that is, the Möbius band. (When g is order preserving, .3 is a sphere with 2 holes or a cylinder.) Therefore, if g is order reversing, then the surface X contains a Möbius band and is not orientable, a contradiction. 

4.2 Whitney’s Theorem

101

Recall that an orbit can cross a local section .λ more than once, but it must do so at discrete times (Proposition 4.1.7), and hence it is possible to talk about consecutive crossings of a local section by an orbit. Specifically, .xσ and .xτ with .σ < τ are consecutive crossings of a local section .λ by the orbit of x provided that both .xσ and .xτ are points in .λ and (xσ, xτ )ϕ ∩ λ = φ.

.

Corollary 4.2.9 Let .λ be a local section of a flow .(X, R) on an orientable surface X, and let x and .xτ be distinct consecutive crossings of .λ with .τ > 0 such that x and .xτ are in .λi . If x is in .ω(y) and y is not periodic, then there exists a sequence .tn  ∞ such that .ytn is in .(x, xτ )λ and converges to x or .xτ . Proof By the theorem, there exists an order preserving Poincaré return function g : [u, v]λ → λ such that x is in .(u, v)λ and .g(x) = xτ . Without loss of generality, .[u, v]λ ∩ g([u, v]λ ) = φ. Because x is in .ω(y), there exists a sequence .tn of real numbers increasing to infinity such that .ytn ∈ [u, v]λ and .ytn converges to x. Because y is not periodic, it can be assumed that either .ytn ∈ (u, x)λ for all n or .ytn ∈ (x, v)λ for all n. In the latter case, there is nothing further to prove. If .ytn ∈ (u, x)λ for all n, then .g(ytn ) = yf (ytn ) converges to .xτ because g is  continuous. Since g is order preserving, .g((u, x)) ⊂ (x, xτ ). .

A sequence .an in a local section .λ is monotonic in a local section provided .an is in .[an−1 , an+1 ] for all .n > 1 and strictly monotonic in a local section provided .an is in .(an−1 , an+1 ) for all .n > 1. Given a Poincaré return function .g : [u, v]λ → λ, if .g(y) = y, then y is periodic with period .f (y). Conversely, if y is a periodic point of period .ρ in .[u, v]λ such that .O(y) ∩ λ = {y}, then .g(y) = y and .f (y) = ρ. Theorem 4.2.10 Let y be a periodic moving point of a flow .(X, R) on an orientable surface X. If y is in .ω(x), then .ω(x) = O(y), and if x is not in .O(y), then the set .{w ∈ X : ω(w) = O(y)} is a nonempty open subset of X. Proof The proof is trivial if x is in .O(y), so we assume that x is not in .O(y). By Theorems 4.1.4 and 4.2.6, there exists a local section .λ of length .2α such that .λ is an arc and y is in .λi . Because .O(y) ∩ λ is finite, it can be assumed that .O(y) ∩ λ = {y}. By Theorem 4.1.10, there exists a Poincaré return function g defined on .[u, v]λ such that y is in .(u, v)λ . Then .y ∈ ω(x) implies that there exists a sequence .xτn of consecutive crossings of .λ, that is, .τn < τn+1 and .{xt : τn < t < τn+1 } ∩ λ = φ for all n. And there exists a subsequence of .xτn that converges to y. Note that if .xτn is in .[u, v]λ , then .g(xτn ) = xτn+1 . If .xτn is in .[u, v]λ , then both .xτn and .xτn+1 are in the same component of .λ \ {y} because g is order preserving (Theorem 4.2.8) and .g(y) = y. There must exist .m such that .xτm in .[u, y)λ {.(y, v]λ } and .xτm +1 is in .(xτm , y)λ {.(y, xτm )λ }; otherwise no subsequence converges to y. Because .g(y) = y and g is order preserving, .xτm +2 is in .(xτm +1 , y)λ {.(y, xτm +1 )λ } and, by induction, .xτm +k+1 is in .(xτm +k , y)λ {.xτm +k+1 is in

102

4 Local Sections

(y, xτm +k )λ }. Thus, in either case, .xτm +k is a strictly monotonic sequence. Therefore, .xτn converges to y because a subsequence of .xτn converges to y. Let .ρ denote the period of y. By the continuity of the first return time function .f (w) = inf{t > 0 : wt ∈ λ}, there exists .η > 0 such that .|f (w) − ρ| < α/2 when w is in .λ and .d(w, y) < η. Consequently, .w ∈ λ and .d(w, y) < η implies that .g(w) = w(ρ + s) with .|s| < α/2. For any .ε > 0, there exists a positive .δ < η such that .d(wt, yt) < ε when .0 ≤ t ≤ ρ + α (Proposition 1.3.1), .w ∈ λ, and .d(w, y) < δ. And there exists .mε ∈ Z such that .d(xτn , y) < δ < η for all .n ≥ mε . If .t > τmε , then there exists an integer .n ≥ mε such that xt is in .[xτn , xτn+1 )ϕ and .d(xt, yt) < ε for all .t ∈ [τn , τn + ρ + α]. Since .δ < η, it follows that .xτn+1 = g(xτn ) = xτn (ρ + s) where .|s| < α/2. Therefore, .d(xt, yt) < ε for .τn < t < τn+1 , when .n ≥ mε , and hence .O+ (xτmε ) ⊂ {w ∈ X : d(w, O(y))} < ε. It follows that .ω(x) ⊂ O(y). To finish the proof, go back to .m when .xτn became monotonic in .λ and let .u and .v be points in .λ such that .u is in .(xτm , xτm +1 )λ and .v is in .(xτm +1 , xτm +2 )λ . Because g is order preserving, every point w in .[u , v ] has the property that .g n (w) converges to y. Since .g n (w) = wσn and .σn  ∞, the point y is in .ω(w) and then the first part of the proof implies that .ω(w) = O(y) for all .w ∈ (u , v )λ . It follows that .ω(w) = O(y) for all w in .V = (u , v )λ (−α, α). Finally, there exists an open neighborhood U of x such that .U τm +1 ⊂ V .  .

Corollary 4.2.11 Let .(X, R) be a flow on a connected orientable surface X. If x is a positively recurrent point that is not periodic, then every periodic point in .ω(x) is a fixed point. The previous corollary can be extended to flows on connected nonorientable surfaces that have a finite sheeted orientable covering using Proposition 2.2.5 and Theorem 2.2.6. Theorems 4.2.8 and 4.2.10 are definitely theorems about orientable surfaces as the next example shows. Consider the cascade on .[−1, 1] defined by .h(x) = −x 3 . The fixed point set is .{0}, and there is one periodic orbit, namely, .{−1, 1}. For every other point, .ω(x) = {0} and .α(x) = {−1, 1}. Now form the suspension flow .(S([−1, 1], h), R). Then .S([−1, 1], h) is clearly a compact bordered surface with Euler characteristic zero because a suspension flow has no fixed points. It also has only 1 boundary component and it is the periodic orbit of period 2 coming from the periodic orbit .{−1, 1} of h. As in the proof of Theorem 4.2.8, .S([−1, 1], h) must be a Möbius band. Letting .ν : [−1, 1] → S([−1, 1], h) be the function .ν(x) = π(x, 0), we know from Proposition 3.1.2 that .ν is a homeomorphism and satisfies .ν(h(x)) = ν(x)1. So .(ν([−1, 1]), ϕ1 ) is a copy of .([−1, 1], h) sitting in .S([−1, 1], h). It is also a global section of the suspension flow with a constant first return time of 1. So 3 .g(ν(x)) = ν(x)1 is a Poincaré return function. Then .ν(x)1 = ν(−x ) is clearly not order preserving, but .ν(0)1 = 0, in fact, .ν([0, 1])1 = ν([−1, 0]). Suppose x is in .(0, 1). Then .hn (x) oscillates between .(0, 1) and .(−1, 0) while n + .|h (x)| decreases to 0 as n goes to infinity. Consequently, .O (ν(x)) spirals toward + .ν(0) from both sides of .ν([−1, 1]). We cannot, however, say that .O (ν(x)) spirals

4.3 Classical Applications

103

toward .O(ν(0)) from both sides of .O(ν(0)) because .O(ν(0)) has only one side in the sense that .S([−1, 1], h) \ O(ν(0)) is connected.

4.3 Classical Applications The main theorems in this section are classical results for autonomous differential equations on planar domains that have been recast for flows on .S2 . The underlying idea of their proofs comes from classical Poincaré-Bendixson theory. The qualitative behavior of flows on .S2 and .P2 is much simpler than that of flows on the compact connected surfaces of higher genus. Nevertheless, flows on .S2 are the starting point for the study of flows on compact connected surfaces. Furthermore, the proofs in this section are important because their techniques will be used to investigate lifted flows on the universal covering spaces of compact connected orientable surfaces, namely, .R2 and .B2 . As a result of Whitney’s theorem, we have at our disposal local sections at moving points that are arcs. Consequently, from Proposition 4.1.11, we know that i .λ of any local section for a surface flow will be a 1-dimensional manifold. From this perspective, the standing assumption that a local section .λ of a flow on a surface is homeomorphic to .[0, 1] is very practical. Recall that .xσ and .xτ with .σ < τ are consecutive crossings of a local section .λ by the orbit of x provided that both .xσ and .xτ are points in .λ and (xσ, xτ )ϕ ∩ λ = φ.

.

When .xσ and .xτ are consecutive crossings, it is easy to see that J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ

(4.3)

.

is an embedded circle as shown in Figure 4.5. Of course, if x is periodic, then it is possible that .xσ = xτ and in this case the set .(xσ, xτ )λ is empty and .O(x) is an embedded circle (Theorem 1.2.1). When the Fig. 4.5 The curve = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ is an embedded circle in .S2

xτ

.J

J

xσ

104

4 Local Sections

consecutive crossings are distinct points, the embedded circle J has properties that are essential in the proofs of the main theorems in this section. Proposition 4.3.1 Let .λ be a local section of length .2α for a flow .(S2 , R) on the 2-sphere. If .xσ and .xτ with .σ < τ are distinct consecutive crossings of .λ, then the embedded circle J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ

.

has the property that the sets P = {qt : 0 < t < α and q ∈ (xσ, xτ )λ }

(4.4)

N = {qt : −α < t < 0 and q ∈ (xσ, xτ )λ }

(4.5)

.

and .

are in different components of .S2 \ J . Proof First, the sets P and N are not empty because .xσ = xτ by hypothesis. Let U and V be the two components of .S2 \ J . Since .λi is obtained by removing the endpoints of .λ (Proposition 4.2.5), .(xσ, xτ )λ is an open subset of .λi . Then part (d) of Proposition 4.1.8 applies to the open connected set .(xσ, xτ )λ ×(0, α) of .λ×[−α, α] to show that P is an open connected subset of .S2 such that .P ∩ J = φ. Because U and V are open, P must be contained in either U or V . Likewise for N . (See Figure 4.6.) Fig. 4.6 Orbits can only cross .(xσ, xτ )λ in one direction

λ xτ N

P xσ

4.3 Classical Applications

105

If they both are contained in, say U , then using part (d) of Proposition 4.1.8 again, the set P ∪ N ∪ (x, y)λ = (xσ, xτ )λ (−α, α)

.

is an open set the set of .S2 that does not intersect V , contradicting that .(xσ, xτ )λ is  contained in the closure of V by the Jordan separation theorem. One way of viewing the previous proposition is that .(xσ, xτ )λ is a gate between the two components of .S2 \ J that orbits can only cross in one direction. (See Figure 4.6.) This idea will be made precise in Proposition 4.3.4. For simplicity, whichever of the two components of .S2 \ J contains P = {qt : 0 < t < α and q ∈ (xσ, xτ )λ }

.

will be called the positive side of J and the other one the negative side and denoted by .JP and .JN , respectively. Then .JP and .JN are disjoint open path connected sets such that .JP− = JP ∪ J and .JN− = JN ∪ J . Corollary 4.3.2 Let .λ be a local section of length .2α for a flow .(S2 , R) on the 2-sphere. If .xσ and .xτ with .σ < τ are distinct consecutive crossings of .λ, then .{x(σ − t) : 0 < t ≤ α} ⊂ JN and .{x(τ + t) : 0 < t ≤ α} ⊂ JP , where J is given by equation (4.3). Proof Note that the sets .{x(σ − t) : 0 < t ≤ α} and .{x(τ + t) : 0 < t ≤ α} are connected sets that do not intersect J and hence lie in either .JN or .JP . If q is in .(xσ, xτ )λ , then .[xσ, q]λ (−α/2) and .[q, xτ ]λ (α/2) are arcs that do not intersect J and connect a point in .{x(σ − t) : 0 < t ≤ α} to a point in .JN and connect a point in .{x(τ + t) : 0 < t ≤ α} to a point in .JP . Now apply the same component test (page 92).  Proposition 4.3.3 If .xσ and .xτ with .σ < τ are distinct consecutive crossings of a local section .λ of length .2α for a flow .(S2 , R), then x is not a periodic point. Proof Suppose x is periodic. Let J be given by equation (4.3). By Corollary 4.3.2, {x(σ − t) : 0 < t ≤ α} ⊂ JN and .{x(τ + t) : 0 < t ≤ α} ⊂ JP . Since x is assumed to be periodic, xt must lie in .JN for some .t > τ + α. Set .s = inf{t > τ + α : xt ∈ JN }. Clearly, .s > τ + α and .xs ∈ J because .JP and .JN are open sets. Then .xs ∈ / (xτ, xσ )λ because .x(s − δ) would be in .JN for .0 < δ < α, contradicting the choice of s. (See Figure 4.7.) The only other possibility is that .xs = q for some .q ∈ (xσ, xτ )ϕ (see Figure 4.8). Let .λ be a local section at q of length .2α such that .λ (−α , α ) ∩ λ(−α, α) = φ. Then xt must enter .B = λ [−α , α ] at .x(s − α ) because .xs = q is in .(λ )i . Thus there exists .w ∈ λ ∩ JP− such that .xs = w(−α ). If .w = q, then xs is not in J , contradicting the choice of s. If .w = q, then .w(−α ) = x(s − α ) is in J , again contradicting the choice of s.  .

Proposition 4.3.3 does not hold for surfaces of higher genus.

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Fig. 4.7 .xs ∈ (xσ, xτ )λ would imply that .x(s − δ) ∈ JN

Fig. 4.8 .xs = q for some ∈ (xτ, xσ )φ

xτ

.q

xσ JN

λ q = xs

Proposition 4.3.4 If .xσ and .xτ with .σ < τ are distinct consecutive crossings of a local section .λ of length .2α for a flow .(S2 , R), then J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ

.

has the following properties: (a) If .q ∈ JP , then .O+ (q) ⊂ JP , that is, .JP is positively invariant. (b) If .q ∈ JN , then .O− (q) ⊂ JN , that is, .JN is negatively invariant. Proof The proofs of the two parts are essentially the same. To prove the first part, suppose q is an element of .JP such that .O+ (q) ∩ (JN ∪ J ) = φ and set .s = inf{t > 0 : qt ∈ JN ∪ J }. Clearly, .qs ∈ J because .JP and .JN are open sets. So either .qs = xs for some .s such that .σ < s ≤ τ or qs is in .[xσ, xτ )λ . In the first case, .O(qs) = O(x) and .q(s − ε) is in J for small .ε, contradicting .s = inf{t > 0 : qt ∈ JN ∪ J }. In the second case, .q(s − t) ⊂ JN for .0 < t < α, again contradicting that .s = inf{t > 0 : qt ∈ JN ∪ J }.  Corollary 4.3.5 If .xσ and .xτ with .σ < τ are distinct consecutive crossings of a local section .λ of length .2α for a flow .(S2 , R), then the points in .(xσ, xτ )λ are not in .ω(y) ∪ α(y) for all .y ∈ S2 . Proof Given p in .(xσ, xτ )λ , the set .U = (xσ, xτ )λ (−α, α) is an open neighborhood of p. If p is in .ω(y) ∪ α(y), then for .n ∈ Z+ there exists .tn such that .|tn | > n and .ytn is in U . But it follows from the proposition that .O(y) ∩ U is at most 1 set of the form .y(s − α, s + α).  Theorem 4.3.6 If x is a positively or negatively recurrent point of a flow .(S2 , R), then x is either a fixed point or a periodic point.

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107

Proof Suppose x is positively recurrent, that is, .x ∈ ω(x). It suffices to assume that x is neither a fixed point nor a periodic point and derive a contradiction. Recall from Proposition 1.2.4 that .ω(x) = ω(xt) for all .t ∈ R and that .ω(x) is a closed invariant set of the flow. Hence .O(x) ⊂ ω(x) ⊂ O+ (x)− . Since it is assumed that x is a moving point, there exists a local section .λ of length .2α at x. Because .x ∈ ω(x), there exists a next crossing .xτ so that x and .xτ are consecutive crossings of .λ. Note that .x = xτ because we are assuming that x is not periodic. Now apply Corollary 4.3.2 and Proposition 4.3.4 to J = [x, xτ ]ϕ ∪ [x, xτ ]λ

.

and conclude that xt is in .JP for all .t ≥ τ + α. It follows that ω(x) = ω(x(τ + α)) ⊂ JP ∪ J.

.

Corollary 4.3.2, however, also implies that .x(−α) is in .JN and hence ω(x) ∩ JN ⊃ O(x) ∩ JN = φ,

.

contradicting .ω(x) ⊂ JP ∪ J .

 2 .(P , R),

Corollary 4.3.7 If x is a positively {negatively} recurrent point of a flow on the projective space .P2 , then x is either a fixed point or a periodic point. Proof Let .π : S2 → P2 be the double covering of .P2 by .S2 (page 44), let .(S2 , R) be the lifted flow (Theorem 2.2.3) of the given one, and let . x be in .π −1 (x). Then . x is a positively recurrent point of the lifted flow by Theorem 2.2.6, and by the theorem . x is periodic or fixed. It follows that x is a periodic or a fixed point because .π is a homomorphism of flows.  Corollary 4.3.8 If x is a positively {negatively} recurrent point of a flow .(X, R) on a bordered compact connected orientable surface X of genus 0 or a bordered compact connected nonorientable surface of genus 1, then x is either a fixed point or a periodic point. Proof Apply Theorem 4.3.6 or Corollary 4.3.7 to the flow .(X∗ , R) given by Theorem 2.3.2.  Theorem 4.3.9 Let w be a moving point in .ω(x) for a flow .(S2 , R), and let .λ be a local section of length .2α at w. If .xτn is a sequence of distinct consecutive crossings of .λ with .τn > 0, then .xτn is a strictly monotonic sequence of .λ converging to w. Proof Because w is in .λ ∩ ω(x), the set of crossings times .τn of .λ by .O+ (x) is an infinite increasing sequence converging to infinity and .xτn contains a subsequence converging to w. Consider the first two consecutive crossings .xτ1 and .xτ2 of .λ, and let .J = [xτ1 , xτ2 ]ϕ ∪ [xτ2 , xτ1 ]λ as usual. Then x is neither periodic nor positively recurrent by Proposition 4.3.3 and Theorem 4.3.6. Thus w is not in .[xτ1 , xτ2 ]λ by

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4 Local Sections

Corollary 4.3.5. Hence w must be in .[e− , xτ1 )λ or .(xτ2 , e+ ] where .e− and .e+ are the endpoints of .λ. Using Corollary 4.3.2 and its method of proof, it is an exercise to show that − + + .[e , xτ1 )λ and .(xτ2 , e ] are in .JN and .JP , respectively. Since .O (xτ2 ) is contained + in .JP , the points .xτn are in .(xτ2 , e ]λ for all .n > 2 and w is in .(xτ2 , e+ )λ because i .xτn contains a subsequence converging to .w ∈ λ . (Although .xτ1 could be the − + endpoint .e , the point .xτ2 does not equal .e because .w ∈ λi .) It suffices to prove by induction for .n ≥ 2 that .xτk is in .(xτn , e+ ]λ for all .k > n and that .xτn is in the interval .(xτn−1 , w)λ . Clearly the previous 2 paragraphs prove the initial induction step for .n = 2 because .xτ2 ∈ (xτ1 , w). Assume that .xτk is in .(xτn , e+ ]λ for all .k > n and that .xτn is in the interval = .(xτn−1 , w)λ . Consider the consecutive crossings .xτn and .xτn+1 , and let .J [xτn , xτn+1 ]ϕ ∪ [xτn+1 , xτn ]λ ⊂ JP . Then w cannot be in .[xτn , xτn+1 ]λ by Corollary 4.3.5. Because .xτn+1 is in .(xτn , e+ ]λ , it follows that .(xτn+1 , e+ ]λ is contained in .JP by again using Corollary 4.3.2 and its method of proof. The induction assumption implies that .xτk is in .(xτn+1 , e+ ]λ for .k ≥ n + 2 because .xτn and .xτn+1 are consecutive crossings. Since w is in .ω(x), the sequence .xτk such that .k ≥ n + 2 contains a subsequence converging to w. It follows that w is in .(xτn+1 , e+ )λ and .xτn+1 is in .(xτn , w)λ .  Exercise 4.3.10 Verify that .[e− , xτ1 )λ and .(xτ2 , e+ ] are in .JN and .JP , respectively, in the proof of Theorem 4.3.2. Theorem 4.3.11 Let x, y, and z be moving points of a flow .(S2 , R). If .y ∈ ω(x) and .z ∈ ω(y) ∪ α(y), then y is a periodic point. Proof If x or y is periodic, the result is trivial. Assuming neither x nor y is periodic, let .λ be a local section of length .2α at z. Because .z ∈ α(y) ∪ ω(y), there exist two consecutive crossings .yσ and .yτ of .λ with .σ < τ < 0 or .0 < σ < τ depending on whether .z ∈ α(y) or .z ∈ ω(y). Let J be given by equation (4.3) with y replacing x. Since .y(τ +α) and .y(σ −α) are in the open sets .JP and .JN by Corollary 4.3.2, there exist .s > 0 such that .xs ∈ JP and .s > s such that .xs ∈ JN because .y ∈ ω(x). It follows from Proposition 4.3.4 that .O+ (xs) ⊂ JP , contradicting .xs ∈ JN with .s > s. (See Figure 4.9.)  Corollary 4.3.12 Let x, y, and z be moving points of a flow .(P2 , R). If .y ∈ ω(x) and .z ∈ ω(y) ∪ α(y), then y is a periodic point. Proof Let .π : S2 → P2 be the double covering of .P2 by .S2 , let .(S2 , R) be the lifted flow, and let . x be in .π −1 (x). By Proposition 2.2.5, there exists . y in .ω( x ) such that .π( y ) = y and  .z in .ω( y ) ∪ α( y ) such that .π( z) = z. Clearly, . x , . y , and  .z are moving points. Now the theorem implies . y is periodic and so is y.  Corollary 4.3.13 Given a flow .(S2 , R) {.(P2 , R)} and .x ∈ S2 {.x ∈ P2 }, if there are no fixed points in .ω(x), then .ω(x) is a periodic orbit.

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109

Fig. 4.9 .O+ (xs) cannot include .xs because .JP is positively invariant

JP z

yτ yσ y (σ −

α)

xs α) y (τ +



xs

JN

Proof Since .S2 {.P2 } is compact, there exist .y ∈ ω(x) and .z ∈ ω(y). It follows from the hypothesis that x, y, and z are moving points because .ω(y) ⊂ ω(x). Now Theorem 4.3.11 or Corollary 4.3.12 implies that y is periodic. Theorem 4.2.10 implies that .ω(x) = O(y) for .S2 . For the nonorientable space 2 2 .P , construct points . x , . y , and  .z in .S as in the proof of Corollary 4.3.12. Then . y is periodic, .ω( x ) = O( y ), and Proposition 2.2.5 implies that .ω(x) = O(y).  Similarly, if .y ∈ ω(x) is not a periodic point, then its limit points cannot be moving points, and we have the following: Corollary 4.3.14 Let x and y be moving points of a flow .(S2 , R) {.(P2 , R)}. If .y ∈ ω(x) and y is not a periodic point, then .ω(y) and .α(y) are nonempty subsets of F , the set of fixed points of the flow. If, in addition, F is totally disconnected, that is, the only connected sets in F are points, then .ω(y) and .α(y) consist of a single fixed point. Of course, Theorem 4.3.11 and its corollaries can be extended to bordered surfaces by applying Theorem 2.3.2, but statements of these routine extensions will no longer be included. Using the following proposition, we can also obtain classical results about flows on .R2 such as recurrent implies periodic or fixed. Proposition 4.3.15 If X is a locally compact second countable metric space that is not compact, then a flow .(X, R) extends to a flow .(X∞ , R) on the one-point compactification of X.

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4 Local Sections

Proof Setting .∞ t = ∞ for all .t ∈ R extends the group action of .R to .X∞ and reduces the proof to showing that this action is continuous at .(∞, t). The open neighborhoods of .∞ are sets of the form .X∞ \ C where C is a compact subset of X (see p. 94 for more information about .X∞ ). Let .xn be a sequence in X converging to .∞ in .X∞ , and let .tn be a sequence in .R converging to .t ∈ R. It must be shown that .xn tn converges to .∞. If this is not the case, there exist a compact subset C of X and a subsequence .xni such that .xni tni is in C. Because C is compact, we can assume without loss of generality that .xni tni converges to y in C. It follows that .xni = xni (tni − tni ) = xni tni (−tni ) converges to .y(−t) ∈ X, contradicting the assumption that .xn converges to .∞.  Proposition 4.3.15 can be extended to the continuous action of a topological group G on a locally compact Hausdorff space that is not compact by simply replacing the sequence argument with a net argument. Theorem 4.3.16 (Poincaré Bendixson) If x is a positively or negatively recurrent point of the flow .(R2 , R), then x is either a fixed point or a periodic point. Exercise 4.3.17 Use Proposition 4.3.15 and Theorem 4.3.6 to prove rem 4.3.16.

Theo-

Theorem 4.3.18 Let x, y, and z be moving points of a flow .(R2 , R). If .y ∈ ω(x) and .z ∈ ω(y) ∪ α(y), then y is a periodic point. Exercise 4.3.19 Prove Theorem 4.3.18. Corollaries 4.3.13 and 4.3.14 do not hold, however, as stated for .R2 because they depend on the compactness of .S2 . For example, on .R2 , it is possible to have .y ∈ ω(x) be a moving point such that .O(y) is not bounded and .ω(y) = φ = α(y) ([67], Part II, Fig. 30). With the additional hypothesis that .O+ (x) is bounded, these corollaries hold. Alternatively, results about flows on the plane can be used to prove corresponding results about flows on the sphere. Flows on the plane will be important in the following chapters because the universal covering space of every compact connected surface except .S2 and .P2 is homeomorphic to .R2 or .C. In the plane, the components of the complement of an embedded circle are topologically very different because one is bounded and the other is unbounded. In particular, when .xσ and .xτ are distinct consecutive crossings of a local section .λ of a planar flow, one component of the complement of .J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ is bounded and hence has a compact closure. It will be called a Bendixson sack because it traps either positive or negative orbits in a compact subset of the plane.

Chapter 5

Flows on the Torus

The torus is unique among the compact connected surfaces. It is the only compact connected surface that is a topological group and the only one that has an infinite abelian fundamental group. It is also the only compact connected orientable surface whose universal covering space is closely linked to Euclidean geometry and provides a natural geometric tool to use for studying lifted orbits. The torus and the Klein bottle are the only two compact connected surfaces with Euler characteristic equal to 0. Thus they are the only two compact connected surfaces on which flows without fixed points can and do occur. Examples of minimal flows on the torus were constructed in Section 1.1. On the Klein bottle, however, it will be shown that every positively or negatively recurrent point is a periodic point or a fixed point, making the torus the only compact connected surface that is the minimal set of a flow. In the early 1930s, André Weil ([76] and [77]) had the idea that covering spaces would provide an effective means of studying flows on compact connected surfaces. The second paper reports on his lecture at the 1935 Moscow Topology Conference and includes the announcement of a striking geometric theorem (without proof) about curves on the torus that applies to the lifts of orbits of a flow on the torus. Weil’s theorem is the centerpiece of Section 5.1. Control curves, which consist of pieces of orbits and local sections, replace the simple closed curves used to prove the classical theorems in Section 4.3 and play a similar essential role in using lifted orbits to study flows on compact connected surfaces. Again the Jordan separation theorem will be a critical partner in many proofs. Section 5.2 is devoted to the definitions and basic properties of control curves. In Section 5.3, control curves are used to obtain basic geometric properties of lifted orbits. These geometric properties are then used to prove structural results about flows on the torus and Klein bottle. For example, two positively recurrent but not periodic points of a flow on the torus have identical orbit closures. The chapter

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_5

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ends with a generalization of Kneser’s classical result that a flow on the Klein bottle without fixed points has a periodic orbit.

5.1 Weil’s Theorem At this point, the complex plane, denoted by .C, begins playing an increasingly important role in our study of flows on compact connected orientable surfaces, becoming necessary when the genus is greater than one. Additively, .C is the topological group .R2 with a continuous multiplication that makes it a field. Moreover, .C \ {0} is also a topological group under complex multiplication because .z → 1/z is continuous. In this context, the usual covering of the torus .π : C = R2 → T2 is written π(z) = π(x + iy) = (e2π ix , e2π iy ).

.

Clearly, .π maps .{x + iy : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1} = [0, 1]2 onto .T2 . More generally, given .ζ ∈ C, the map .π maps any set of the form .[0, 1]2 + ζ onto .T2 . Obviously, every disk of radius 1 in .C contains a set of the form .[0, 1]2 + ζ . From the perspective of .C, the covering group . of the torus is just the left action of the Gaussian integers .Z[i] = {m + ni : m, n ∈ Z} on .C by addition, that is, .T ∈  if and only if there exists .m + in such that m and n are integers and .T z = z + (m + in), which is a complex linear fractional transformation. Recall that linear fractional transformations extend to homeomorphisms of .S2 , the one-point compactification of .C. In this case, .∞ is the fixed point of T . Throughout this chapter, . will denote this particular covering group. For .T ∈ , note that .T z − z = T 0 ∈ Z[i] for all .z ∈ C and that .T → T 0 is an isomorphism of . onto .Z[i]. The axis of .T ∈  \ {ι} is the line .LT through the origin such that .T LT = LT , that is, the line .{sT 0 : s ∈ R}. Alternatively, .LT is the T invariant great circle through .∞ on .S2 . All the lines parallel to .LT are also T invariant and with .∞ are circles but not a great circle on .S2 . A line in the plane is said to be rational provided its slope is rational or infinite (to include lines perpendicular to the x axis). Rational lines are closely tied to .Z[i] and .. In our context, it is more convenient to parameterize a line in the complex form .g(s) = sa + b, where .s ∈ R and .a, b ∈ C with .a = 0. Given .σ ∈ R, if .σ = 0, then .h(s) = sσ a + b is another way to parameterize the same line. In this form, a line is a rational line if and only if .σ a has rational real and imaginary parts for some .σ ∈ R or equivalently .σ a ∈ Z[i] for some nonzero real number .σ . Thus for a rational line, we can always assume that a is in .Z[i]. Clearly, .LT is a rational line for .T ∈  \ {ι}. Given a nonzero complex number a, the set .{σ a : σ ∈ R} is a closed additive subgroup of .C. Shifting the point of view slightly, the line .g(s) = sa + b is rational for all .b ∈ C if and only if the subgroup .Ga = {σ a : σ ∈ R} ∩ Z[i] is not the trivial subgroup of .C. Note that .Gsa = Ga for all nonzero real numbers s. The next

5.1 Weil’s Theorem

113

proposition describes the structure of the group .Ga . The proof is an application of elementary number theory and is left to the reader. Proposition 5.1.1 If a is a nonzero complex number, then the following hold: (a) If .m + in ∈ Z[i] is in .Ga and d is a common divisor of m and n, then .

m n 1 + i = (m + in) d d d

is in .Ga . (b) If .Ga is not the trivial subgroup of .C, then there exists .m + in in .Ga such that .m and n are relatively prime. (c) If .m + in is in .Ga and .m and n are relatively prime, then the cyclic subgroup of .Ga generated by .m + in equals .Ga . (d) If .Ga is not the trivial subgroup of .C, then .Ga is a maximal cyclic group of .Z[i]. Exercise 5.1.2 Prove Proposition 5.1.1. Given distinct complex numbers z and w, let .{z, w} and .s{z, w} denote the line and the line segment determined by z and w. Given T in ., the cyclic subgroup generated by T is the subgroup of . defined by .[T ] = {T n : n ∈ Z}. Proposition 5.1.3 The covering group . of .T2 has the following properties with respect to lines in .C: (a) If .L = {ζ, T ζ } for some .ζ ∈ C and .T ∈  \ {ι}, then .T L = L. (b) Given a line L in .C parameterized by .g(s) = sa + b and .T ∈  \ {ι}, it follows that .T L = L if and only if T 0 is in .Ga . (c) If L is a rational line in .C, then .{T ∈  : T L = L} is a maximal cyclic subgroup of .. (d) If T is in . \ {ι}, then T is in a maximal cyclic subgroup of .. (e) Given .T ∈  \ {ι} with .T 0 = m + in, .[T ] is a maximal cyclic group of . if and only if m and n are relatively prime. (f) A line L in .C is rational if and only if .T L = L for some .T ∈  \ {ι}. Proof Since .T ζ − ζ = T 0, the line L is parameterized by .g(s) = sT 0 + ζ because g(0) = ζ and .g(1) = T ζ . Then .T g(s) = sT 0+ζ +T 0 = g(s +1) and .T g(s −1) = (s − 1)T 0 + ζ + T 0 = g(s). This proves part (a). For part (b), if .T L = L, then .T g(s) = sa+b+T 0 = s a+b and .T 0 = (s −s)a ∈ Ga because T 0 is in .Z[i]. Conversely, .T 0 ∈ Ga implies that .g (s) = sT 0 + b also parameterizes L because .T 0 = σ a. Since .g (0) = b and .g (1) = b + T 0 = T b, it follows that .L = {b, T b} and part (a) implies that .T L = L. Because L is rational, it can be parameterized by .g(s) = as + b such that a is in .Z[i], and hence, .Ga is a nontrivial subgroup of .Z[i]. By Proposition 5.1.1, .Ga is a maximal cyclic subgroup of .Z[i]. Using the isomorphism .T → T 0 of . onto .Z[i], it follows from part (b) that .{T ∈  : T L = L} is isomorphic to .Ga and, hence, a maximal cyclic subgroup of .. .

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For part (d), the axis .LT of T is both rational and T invariant. Then T is in {S ∈  : SLT = LT }, which is a maximal cyclic subgroup of .Z[i] by part (c). Part (e) follows from Proposition 5.1.1 using the isomorphism .T → T 0. Part (f) starts with L rational, so it can be parameterized by .g(s) = sa + b with .a ∈ Z[i]. Hence, .a = T 0 ∈ Ga for .T z = z + a, and T is in . \ {ι} because .a = 0. Thus .T L = L by part (b). Conversely, if .T L = L, then .T 0 ∈ Ga by part (b), and .g(s) = sT 0 + b parameterizes L, proving that L is rational. 

.

Since it will be more convenient to use the complex plane to represent the Euclidean plane, both . x and z are natural notations for points in the universal covering space of the torus. We will use . x or . y when .π( x ) = x or .π( y ) = y is playing an important role in the proof. When we simply want a point in the universal covering space, we will use just z or .ζ . This convention will carry over to compact connected surfaces with genus greater than 1 because their universal covering spaces are most conveniently described using .B2 = {z ∈ C : |z| < 1}. Theorem 5.1.4 (Weil) Let .γ : [0, ∞) → T2 be an injective curve on the torus and let . γ : [0, ∞) → C be a lift of .γ . If .

lim | γ (t)| = ∞,

t→∞

then .

(t) γ t→∞ | γ (t)| lim

exists and is in .S1 . We first prove a lemma in preparation for the proof of Weil’s theorem. : Lemma 5.1.5 Let .β : [a, b] → T2 be an injective curve on the torus, and let .β    [a, b] → C be a lift of .β. Suppose .J = β([a, b]) ∪ s{β (a), β (b)} is an embedded (a), β (b)} ∩ β  (a, b) = φ. If .π(z) = β(a) or .π(z) = β(b), circle such that .{β then z is not in the interior of J . Proof Let z be a point in .C such that .π(z) = β(a). So there exists .T ∈  such (a). Set .L = {β (a), β (b)} for convenience. The proof begins by that .z = T β establishing several elementary facts, supposing that z is in the interior of J , denoted by .JI . (See Figure 5.1.) (a) or .β (b), First, consider the 2 open rays .Ra and .Rb of L starting at .β (a), β (b)}. By construction and hypothesis, respectively, and not containing .s{β they do not intersect J . Since they are connected and unbounded, they are contained in the exterior .JE of J . Therefore, .L ⊂ JE ∪ J and .L ∩ JI = φ. Second, T L either equals L or is a line parallel to L because T is a translation. Because .JI ∩ L = φ and .z ∈ JI ∩ T L, it follows that T L lies in one of the two open half planes determined by L, call it H . So .T L = L and .T H ⊂ H .

5.1 Weil’s Theorem

115

Fig. 5.1 Suppose z is in the interior of J

 β(b)

 z = T β(a) JI β(a) L Fig. 5.2 Four consecutive points on .L

β(τ )  β(t)  ) T β(τ L ∩ T L

 β(b)

L ∩ L

L

β(a)

−

L

−

Third, step  it now  follows from the second   that .JI is contained   in .H = H ∪ L  (a, b) ⊂ H . Therefore, .T β  (a, b) ⊂ T H and .T β  (a, b) ∩ L = φ. and .β ([a, b]) ∩ β ([a, b]) = φ because .β is a injective curve. Therefore, Fourth, .T β  ([a, b]) is contained in .JI . .z = T β (a) ∈ JI implies that the connected set .T β The final step is to use the fact that T is a Euclidean isometry to obtain a contradiction. The Euclidean metric on .C can be written as .d(z, w) = |z − w|, and the distance from a point to the line L, written .d(w, L), is a continuous function. So (t), L) assumes its maximum at some .τ ∈ [a, b], that is, .d(β (τ ), L) ≥ d(β (t), L) d(β

.

for all .t ∈ [a, b]. (τ ) in .JI . Let .L be the line through .T β (τ ) perpendicular Consider the point .T β (τ ) divides .L into two rays; let R be the ray that does not to L. The point .T β intersect L and lies in H . Since R must eventually be in .JE , there exists .t ∈ (a, b) (t) lies in R. It follows from the four preliminary steps that .β (t), .T β (τ ), such that .β .L ∩ T L, and .L ∩ L are consecutive points in .L (Figure 5.2). Therefore, (t), L) = d(β (t), L ∩L) > d(T β (τ ), L ∩T L) = d(T β (τ ), T L) = d(β (τ ), L), d(β

.

which contradicts the choice of .τ .



116

5 Flows on the Torus

Proof of Weil’s Theorem Observe that .

(t) γ γ (t) − ζ  = lim t→∞ | γ (t)| t→∞ | γ (t) − ζ | lim

for any .ζ because .limt→∞ | γ (t)| = ∞. Using the topological group structure of T2 , the curve .t →  γ (t) − ζ is the lift of .t → γ (t)π(ζ )−1 that is injective because multiplication by .π(ζ )−1 is a homeomorphism of .T2 onto itself. Therefore, without loss of generality, we can assume that . γ (0) = 0. Of course, . γ is an injective curve because .γ is an injective curve. Set

.

t0 = inf{s > 0 : | γ (t)| > 2 for all t > s},

.

noting that .| γ (t0 )| = 2. Let .Rt be the unique ray starting at . γ (0) = 0 and passing through . γ (t). The first key step is to prove that .Rt1 ∪ Rt2 cannot be a straight line for .t2 > t1 > t0 . Suppose it is. Set s1 = sup{t :  γ (t) ∈ Rt1 and t1 ≤ t < t2 }

.

and s2 = inf{t :  γ (t) ∈ Rt2 and s1 < t ≤ t2 }.

.

Then .Rsi = Rti for .i = 1, 2 and .Rs1 ∪ Rs2 = Rt1 ∪ Rt2 , which is assumed to be a straight line. By construction,   (Rs1 ∪ Rs2 ) ∩  γ (s1 , s2 ) = φ

.

and .t0 < s1 < s2 . Therefore,   J = γ [s1 , s2 ] ∪ s{ γ (s1 ),  γ (s2 )}

.

is an embedded circle whose interior contains either the half of the disc .{z ∈ C : |z| < 2} above or below the line .Rs1 ∪ Rs2 because .t0 < s1 < s2 . (See Figure 5.3.) In particular, a translate of the unit square, .[0, 1] × [0, 1], is contained in .JI because √ . 2 < 2. Hence, there is a point z in .JI such that .π(z) = γ (s1 ), contradicting the conclusion of Lemma 5.1.5 and proving that .Rt1 ∪ Rt2 cannot be a straight line for .t2 > t1 > t0 . Consequently, the connected set  .

(t) γ : t ≥ t0 | γ (t)|



5.1 Weil’s Theorem

Rt2

117

γ (t0 )

2 γ (s2 )

γ (0)

γ (t1 )

γ (s1 )

Rt1

  Fig. 5.3 The interior of . γ [s1 , s2 ] ∪ s{ γ (s1 ),  γ (s2 )} contains a half-disc of radius 2

in an arc of .S1 of at most .π radians. It follows that there exists a line L through 0 with an open half plane H such that . γ (t) ∈ L ∪ H for all .t > t0 . This can be refined further. The second key step is to show that there exists a ray R in L starting at .0 =  γ (0) and .τ ≥ 0 such that . γ (τ ) ∈ R and .{ γ (t) : t > τ } ⊂ R ∪ H . Set .η = sup{t ≥ 0 :  γ (t) ∈ L}. When .η = 0, set .τ = 0, and let R be either of the two rays in L starting at 0. When .0 < η < ∞, set .τ = η, and let R be the ray starting at 0 and passing through . γ (τ ). Then .{ γ (t) : t > τ } ⊂ H . When .η = ∞, set .τ = inf{t > t0 :  γ (t) ∈ L}, and let R be the ray starting at 0 and passing through . γ (τ ). Then .{ γ (t) : t > τ } ⊂ R ∪ H by the first step. Observe, noting the closure symbol in the upper right, that  Dn =

.

− (t) γ :t ≥n | γ (t)|

is a decreasing sequence of compact connected sets of .S1 . Set D=

∞ 

.

Dn =

n=1

− ∞   (t) γ :t ≥n . | γ (t)|

n=1

Because .S1 is compact, the finite intersection property applies to the sequence .Dn , and the set .D is a nonempty compact connected proper subset of .S1 . Clearly, .

lim

t→∞

γ (t)  | γ (t)|

exists if and only if .D is a point. We argue by contradiction again. Let .ζ and .ξ be two distinct points in .D. Thus .S1 \ {ζ, ξ } consists of two open intervals, .U1 and .U2 . Because the connected set

118

5 Flows on the Torus

 En =

.

 (t) γ :t ≥n | γ (t)|

must intersect every open neighborhood of both .ζ and .ξ , either .U1 or .U2 is contained in .En . Since .En is a decreasing sequence of sets, .U1 or .U2 is contained in .En for all + .n ∈ Z . Assume .U1 is contained all of them. Hence, .U1 ⊂ H ∩ D. Let .L∗ be a rational line through 0 intersecting .U1 at a point .z0 , and let .R ∗ be the ray of .L∗ starting at 0 and passing through .z0 . Pick .z1 and .z2 in .U1 that are separated by .z0 in the interval .U1 . Next, let .R1 and .R2 be rays starting at 0 and passing through .z1 and .z2 , respectively. It is clear from the construction that R, .R ∗ , .R1 , and .R2 are distinct rays contained in non-parallel lines. The third key step in the proof is showing that given any two lines .L1 and .L2 parallel to .R ∗ with .R ∗ between them, there exists .N ∈ Z+ such that .{ γ (t) : t > n} intersects both .L1 and .L2 , when .n ≥ N. From Euclidean geometry, it is clear that there exists .r > 0 such that z is not between .L1 and .L2 when .z ∈ R1 ∪ R2 and .|z| > r. Because .| γ (t)| → ∞ by hypothesis, there exists a positive integer N such that .| γ (t)| > r when .t > N. Since {z0 , z1 , z2 } ⊂ U1 ⊂ { γ (t)/| γ (t)| : t ≥ n},

.

the set .{ γ (t) : t > n} intersects both .L1 and .L2 when .n ≥ N (see Figure 5.4). The choice of .L1 and .L2 will be made later. The fourth key step of the proof is the construction of an embedded circle that will lead to a contradiction similar to the one in the proof of Lemma 5.1.5. Figure 5.5 is provided to help keep track of the many details. Recall that .{ γ (t) : t ≥ τ } ⊂ R ∪ H and . γ (τ ) ∈ R. Let .L be the line passing through . γ (τ ) and parallel to the rational line .L∗ , and let .R be the ray of .L starting

Fig. 5.4 For .t > N , the curve . γ (t) must intersect the lines .L1 and .L2

5.1 Weil’s Theorem

119

L1 L2 γ (τ2 ) R∗

R

γ (τ5 ) = γ (τ3 )

γ (τ4 ) γ (τ1 )

T w1 w1

z0 q1

γ (0)

T γ (τ ) D q2 w2 γ (τ )

T w2

TL L

Fig. 5.5 .J = s{w1 ,  γ (τ4 )} ∪  γ ([τ4 , τ5 ]) ∪ s{ γ (τ5 ), w2 } ∪ s{w2 , w1 } is an embedded circle

at . γ (τ ) and contained in H . Then .L is rational and there exists .T ∈  \ {ι} such that −1 R ⊂ R , .T L = L by Proposition 5.1.3. It follows that either .T R ⊂ R or .T and without loss of generality, it can be assumed that .T R ⊂ R . It follows that .T  γ (τ ) ∈ R ∩ H . Let .ρ be a positive real number such that ρ > | γ (τ )| + |T  γ (τ ) −  γ (τ )|,

.

and set −

D = {z ∈ C : |z −  γ (τ )| < ρ} ∩ H .

.

Then .0 =  γ (0) and .T  γ (τ ) are in D. Moreover, .L ∩ {z : |z −  γ (τ )| = ρ} = {q1 , q2 }, and we can assume .q1 ∈ / R and .q2 ∈ R because .ρ > | γ (τ )|. Then set τ1 = sup{t :  γ (t) ∈ D},

.

so that . γ (t) is not in D for .t > τ1 , noting that .τ1 > τ . Let .L1 be a line parallel to .L such that: (a) .L1 ∩ { γ (t) : τ ≤ t ≤ τ1 } = φ. (b) .L1 ∩ D = φ. (c) .|w1 − q1 | + 2ρ = |w1 − q2 |, where .w1 = L1 ∩ L. Note that .w1 does not lie in R or between .q1 and .q2 . It follows from the third key step that .{t :  γ (t) ∈ L1 } = φ, using any .L2 such that .R ∗ is between .L1 and .L2 . Set .τ2 = inf{t :  γ (t) ∈ L1 }, and note that .τ2 > τ1 .

120

5 Flows on the Torus

Now select .L2 parallel to .R such that: (a) .L2 ∩ { γ (t) : τ ≤ t ≤ τ2 } = φ. (b) .L2 ∩ D = φ. (c) .|w2 − q2 | + 2ρ = |w2 − q1 |, where .w2 = L2 ∩ L. Then .w2 is in R, and D is between .L1 and .L2 . Thus .R ∗ is between .L1 and .L2 . Let .τ3 = inf{t > τ2 :  γ (t) ∈ L2 } and .A = {t ∈ [τ2 , τ3 ] :  γ (t) ∈ L1 }. Clearly, A is a nonempty compact subset of .R. So there exists .τ4 ∈ A such that | γ (τ4 ) − w1 | = inf {| γ (t) − w1 | : t ∈ A} .

.

Let .B = {t ∈ [τ4 , τ3 ] :  γ (t) ∈ s{q2 , w2 }} and set  τ5 =

inf B

if B = φ

τ3

if B = φ

.

.

To complete the construction, set  J =

.

s{w1 ,  γ (τ4 )} ∪  γ ([τ4 , τ5 ]) ∪ s{ γ (τ5 ), w1 }

if B = φ

γ (τ4 )} ∪  γ ([τ4 , τ5 ]) ∪ s{ γ (τ5 ), w2 } ∪ s{w2 , w1 } if B = φ. s{w1 , 

In each case, J is an embedded circle with D in its interior and .JI ∩ L2 = φ. The last step of the proof is to derive a contradiction from the properties of J . First .T  γ (s) =  γ (t) for all .s ≥ 0 and .t ≥ 0 because .γ is injective. On the one hand, T was chosen so that .T L = L , .T L = L, and .T  γ (τ ) =  γ (τ ). On the other hand, J was constructed so that .T  γ (τ ) is in the interior of J and .T  γ (t) is in .T R ∪ T H for .t ≥ τ . Because .limt→∞ |T  γ (t)| = ∞, the curve .T  γ must intersect J for some .t > τ . Set .τ6 = inf{t > τ : T  γ (t) ∈ J }. It follows from the previous paragraph that either .T  γ (τ6 ) is in .s{T w1 ,  γ (τ4 )} or .T  γ (τ6 ) is in .s{T w2 ,  γ (τ3 )} when .B = φ. Suppose .B = φ. Then .τ6 ≤ τ3 because .T L2 = L2 , and hence, .T  γ (τ3 ) must be in .L2 ⊂ J ∪ JE . If .τ6 = τ3 and .T  γ (τ6 ) is in .s{T w2 ,  γ (τ3 )}, then the points .w2 , .T w2 , .T  γ (τ6 ) = T  γ (τ3 ), and . γ (τ3 ) are consecutive points on .L2 . Consequently, |w2 −  γ (τ3 )| = |T w2 − T  γ (τ3 )| < |w2 −  γ (τ3 )|,

.

which is impossible. Thus .τ6 < τ3 . Next .T L2 = L2 implies that τ3 = inf{t > τ :  γ (t) ∈ L2 } = inf{t > τ : T  γ (t) ∈ L2 }.

.

Therefore, .T  γ (τ6 ) must be in .s{T w1 ,  γ (τ4 )} because .τ6 < τ3 . Note that .τ6 must now be in A because .T L1 = L1 implies that . γ (τ6 ) is in .L1 . It follows that .w1 , .T w1 ,

5.1 Weil’s Theorem

121

T γ (τ6 ), . γ (τ4 ), and . γ (τ6 ) are consecutive points on .L1 or the first 4 are consecutive and .τ4 = τ6 . Then the contradiction

.

|w1 −  γ (τ6 )| = |T w1 − T  γ (τ6 )| < |w1 −  γ (τ4 )| ≤ |w1 −  γ (τ6 )|

.



completes the proof. flow .(T2 , R)

Corollary 5.1.6 Let x be a non-periodic point of a let .(C, R) be the lifted flow of .(T2 , R). If .π( x ) = x and .

on the torus, and

lim | x t| = ∞,

t→∞

then .

lim

t→∞

 xt | x t|

exists and is in .S1 . Given a flow .(T2 , R) and x in .T2 , for some . x in .π −1 (x) the orbit of . x in the lifted flow .(C, R) satisfies .limt→∞ | x t| = ∞ if and only if .limt→∞ |T  x t| = ∞ for all .T ∈  because . is a group of Euclidean isometries. Then it follows from the first sentence in the proof of Weil’s theorem that .

lim

t→∞

T xt  xt = lim t→∞ |T  x t| | x t|

for all .T ∈  when .limt→∞ | x t| = ∞. This leads to an important definition. A point w in .S1 is said to be a rational point of .S1 provided that the line .g(s) = sw is a rational line, that is, .{sw : s ∈ R} ∩ Z[i] = {0}. When .limt→∞ | x t| = ∞, the limit of the semi-orbit .O+ ( x ) is rational provided that .

lim

t→∞

 xt | x t|

is a rational point of .S1 . Of course, this is equivalent to the line determined by 0 and .limt→∞  x t/| x t| being a rational line. It follows from the discussion that the limit of the semi-orbit + x ) is rational for some . .O ( x ∈ π −1 (x) if and only if the limit of the semi+ orbit .O ( x ) is rational is for all . x ∈ π −1 (x) because .π −1 (x) is a . orbit. When .limt→∞ | x t| = ∞, the limit of the semi-orbit .O+ ( x ) is irrational provided the limit of the semi-orbit .O+ ( x ) is not rational. When the limit of the semi-orbit .O+ ( x ) is rational, there exists .σ > 0 such that .σ limt→∞  x t/| x t| = m + ni is in .Z[i]. Without loss of generality, we can assume that m and n are relatively prime. Then letting .T z = z + m + ni uniquely determines T because .σ > 0. It follows from part(e) of Proposition 5.1.3 that .[T ] is

122

5 Flows on the Torus

a maximal cyclic subgroup of .. The axis of T is clearly the line determined by 0 and .limt→∞  x t/| x t|. A routine calculation shows that .

m + in  xt T k x = = lim , t→∞ | x t| |m + in| k→∞ |T k x| lim

and T will be referred to as the asymptotic covering map of .O+ ( x ). The proof of Weil’s theorem presented here follows the proof in [47] that is more complete than the shorter proof in [51]. Anosov ([4] page 455) points out that the proof of Lemma 2.2 in [51] has a gap and provides a new proof of the lemma. Here Lemma 5.1.5 is the equivalent of Lemma 2.2, but the proof is very different. Anosov’s proof of Weil’s theorem for closed surfaces with Euler characteristic at most zero (.χ ≤ 0) appears in [3], and he revisits the harder case when .χ = 0 in Section 9 of [4] to address criticisms of his proof. We will address the .χ < 0 case in Section 7.3 using hyperbolic geometry and Fuchsian groups. Weil’s lecture and paper from the 1935 Moscow Topology Conference was largely forgotten for twenty five years. Although Weil did many other things of note, he never published his proof of this theorem or returned to its subject. Also any results obtained by his coworkers mentioned in the lecture are completely lost to the mathematical world. World War II more than likely played a role in these outcomes. Anosov confirms these facts on p.18 of [3].

5.2 Control Curves An injective curve .f : R → C is said to be the type of a line provided that there exists a line L in .C and a constant .D > 0 satisfying: (a) .|f (s)| → ∞ as .|s| → ∞. (b) .d(f (s), L) = inf{|f (s) − z| : z ∈ L} < D for all .s ∈ R. (c) For every .z ∈ L, there exists .s ∈ R such that .|f (s) − z| < D. Alternatively, an injective curve .f : R → C such that .d(f (s), L) = inf{|f (s) − z| : z ∈ L} < D is the type of a line if and only if there exist 2 parallel lines such that .f (R) is between them and intersects every line perpendicular to them. The idea of being of the type of a line goes back at least as far 1932 in [36]. If .f : R → C is the type of a line L parameterized by .g(s) = sa + b as in Section 5.1, then the Weil directions are .limt→∞ f (t)/|f (t)| = a/|a| and .limt→−∞ f (t)/|f (t)| = −a/|a|. So L is a rational line if and only if the Weil directions are rational, connecting Theorem 5.1.4 with the type of a line idea. Proposition 5.2.1 Let .f : R → C be an injective curve that is the type of the line L, and let .L1 and .L2 be lines parallel to L such that L and .f (R) are contained in the region between .L1 and .L2 . If .γ : [0, 1] → C is an injective curve such that .γ (0) ∈ L1 and .γ (1) ∈ L2 , then .γ ([0, 1]) ∩ f (R) = φ.

5.2 Control Curves

123

L1 γ(0) V1 V2

L2 γ(1)

Fig. 5.6 The curve .f (R) must intersect .γ ((0, 1))

Proof Without loss of generality, we can assume that .γ (t) lies between .L1 and .L2 when .0 < t < 1 with .γ (0) in .L1 and .γ (1) in .L2 . Because .[0, 1] is compact and .C is Hausdorff, .γ is a homeomorphism. Thus .γ ([0, 1]) is a compact set (or a closed and bounded set) of .C. The lines .L1 and .L2 with .(0, 0, 1) as north pole form a simple closed curve J in 2 .S with 1 component U containing .f (R). By definition, .lim|s|→∞ f (s) = (0, 0, 1). Furthermore, .γ ([0, 1]) divides the component containing .f (R) into 2 disjoint open connected sets .V1 and .V2 bounded by simple closed curves. (See Figure 5.6.) Because .γ ([0, 1]) is bounded, .f (R) must intersect both .V1 and .V2 , contradicting 

the continuity of .f (R) unless .f (R) ∩ γ ([0, 1]) = φ. Recall that a loop or closed curve in a topological space X is a path or curve f : [0, 1] → X with the same initial and terminal points, that is, .f (0) = f (1). The function .ξ : [0, 1] → S1 defined by .ξ(s) = e2π is is a particularly nice loop in .S1 , and its path equivalence class is a generator of the fundamental group .1 (S1 , 1). Clearly, .ξ is surjective and a closed function because .[0, 1] is compact and .S1 is Hausdorff. Thus .ξ is a quotient map. If .f : [0, 1] → X is a loop on the topological space X, then f passes to the quotient, and there exists a continuous function .f o : S1 → X such that .f = f o ◦ ξ . Following [42], .f o will be called the circle representative of f . Conversely, if .g : S1 → X is a continuous function, then .f = g ◦ ξ is a loop in X such that .f o = g. Simple closed curves on surfaces will be used extensively, and it is necessary to understand their relationship with embedded circles. A simple closed curve on a surface X is a loop .f : [0, 1] → X such that f is injective on the interval .[0, 1). It is easily seen that .f : [0, 1] → X is a simple closed curve if and only if the circle representative .f o of f is an embedding of .S1 in X. An embedded circle is a convenient way of describing the image of a simple closed curve. A simple closed  → X is a universal curve, however, is more useful in lifting arguments. If .π : X covering space, then a simple closed curve .f : [0, 1] → X always has a lift .f :  but the circle representative may not have a lift. [0, 1] → X,

.

124

5 Flows on the Torus

The basic result about the circle representative is that the loop .f : [0, 1] → X is null-homotopic (path-homotopic to the constant loop at .f (0) = f (1)) if and only if the circle representative .f o extends to a continuous function on .D2 ([42], Proposition 7.13).  : [0, 1] → Consider a loop .β : [0, 1] → T2 that is not null-homotopic, and let .β 2 C be a lift of .β to the universal covering .R . There exists a unique .T ∈ , the u by (0) = β (1). Define a universal lift .β covering group of .T2 , such that .T β u (s) = T [s] β (s − [s]). β

.

(5.1)

u (s) is given by The image of .β u (R) = β



.

   [0, 1) . T nβ

(5.2)

n∈Z

  Continuing in this context, .J = β [0, 1] = βo (S1 ) is an embedded circle and = β u (t)| → ∞ as .|t| → ∞, and u (R) is a component of .π −1 (J ). Clearly, .|β .J ∪ {∞} = J∞ is an embedded circle in .S2 through the north pole, that is, .(0, 0, 1). .J Moreover, .π |J is a universal covering of J with covering group .{T k |J : k ∈ Z}.  : [0, 1] → C of a simple closed curve .β : [0, 1] → T2 , every lift Given a lift .β  and .S β (1) = ST S −1 S β (0). So the universal lift of .S ◦ β  of .β has the form .S ◦ β has the form )u (t) = (ST S −1 )[t] S β (t − [t]) = S β u (t) (S ◦ β

(5.3)

)u (R) = S β (R). (S ◦ β

(5.4)

.

and .

For the torus, .ST S −1 = T because . is abelian, but when the genus is greater than 1, the conjugates .ST S −1 will be important. Proposition 5.2.2 When .β is a loop that is not null-homotopic, .β is a simple closed u has the following properties: curve if and only if a universal lift .β u is an injective curve. (a) .β u (R) ∩ β u (R) = φ or S is in .[T ]. (b) For all S in ., either .S β  be a lift of .β with .β (1) = T β (0) Proof Assuming .β is a simple closed curve, let .β  for some .T ∈  \ {ι} and let .βu be the universal lift given by equation (5.1). Suppose u (σ ) = β (σ − [σ ]) = T [τ ] β (τ − [τ ]). Then u (τ ), that is, .T [σ ] β .β     (σ − [σ ]) = π T [τ ] β (τ − [τ ]) π T [σ ] β

.

β(σ − [σ ]) = β(τ − [τ ]) σ − [σ ] = τ − [τ ]

5.2 Control Curves

125

because the range of .s → s − [s] is .[0, 1). Setting .c = σ − [σ ] = τ − [τ ], the two (c) = T [τ ] β (c) and .[σ ] = [τ ]  and .T [τ ] ◦ β  of .β agree at c. Thus .T [σ ] β lifts .T [σ ] ◦ β  because . acts freely on .C. It follows that .σ = τ and .βu is injective. u (R) ∩ β u (R) = φ for some .S ∈ . Then there To prove property (b), suppose .S β (σ − [σ ]) = T [τ ] β (τ − [τ ]).   exists .σ and .τ such that .S βu (σ ) = βu (τ ) or .ST [σ ] β [σ ] [τ ] As in the first part of the proof, .σ − [σ ] = τ − [τ ] and .S ◦ T = T because the action of . on .C is free. It follows that .S = T [τ ]−[σ ] and S is in .[T ]. u has properties (a) and (b), and .β(σ ) = β(τ ) for some .0 ≤ σ < Now suppose .β (τ ) = β (σ ) and .S β u (R) ∩ β u (R) = φ. τ < 1. Then there exists .S ∈  such that .S β n n   By condition (b), .S = T for some .n ∈ Z and .T β (τ ) = β (σ ), contradicting (a). 

Proposition 5.2.3 If .β : [0, 1] → T2 is simple closed curve that is not null : [0, 1] → C is a lift of .β, and T is the covering transformation homotopic, .β u of .β is the type of the T invariant (0) = β (1), then the universal lift .β such that .T β (0), β (1)} = {β (0), T β (0)}. rational line .L = {β Exercise 5.2.4 Prove Proposition 5.2.3. Proposition 5.2.5 If .β : [0, 1] → T2 is a simple closed curve that is not null : [0, 1] → C is a lift of .β, and T is the covering transformation such homotopic, .β (0) = β (1), then .J = β u (R) has the following properties: that .T β (a) .T J = J. (b) There exist open connected sets U and V of .C such that .C \ J = U ∪ V , − − .U = U ∪ J, and .V = V ∪ J. (0), β (1)} = {β (0), T β (0)} such (c) If .L1 and .L2 are lines parallel to .L = {β  that .J lies between them, then .L1 ⊂ U and .L2 ⊂ V or vice versa. (d) .T U = U and .T V = V . Proof Part (a) is obvious from equation (5.2). Part (b) follows from the Jordan separation theorem because .J∞ is an embedded circle in .S2 . Turning to part (c), .T L = L by part (a) of Proposition 5.1.3, implying that .T Li = Li for .i = 1, 2 because they are parallel to L. Since .L1 and .L2 are connected and do not intersect .J, they lie entirely in U or V . Since U and V are connected open sets of .S2 , which is a Peano space, it follows that U and V are arcwise connected ([37], Theorem 3-16). If .L1 and .L2 are in U {V }, then there exists an arc .γ : [0, 1] → C such that .γ (0) ∈ L1 and .γ (1) ∈ L2 that does not intersect .J, contradicting Propositions 5.2.1 and 5.2.3 to complete part (c). For part (d), let .L1 and .L2 be parallel lines to L with .J between them. By part (a), .C = T U ∪ T V ∪ J, and by connectivity, either .T U = U and .T V = V or .T U = V and .T V = U . By part (c), we can assume .L1 ⊂ U and .L2 ⊂ V . Then .L1 = T L1 ⊂ U ∩ T U and .T U = U . Likewise .T V = V . 

A covering transformation T not equal to the identity is said to be a primitive element of . provided that .T = S j for some .S ∈  implies that .|j | = 1. In other words, primitive covering transformations have no roots in the group of covering transformations. Equivalently, T is primitive if and only if .[T ], the cyclic

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5 Flows on the Torus

subgroup generated by T , is a maximal cyclic subgroup of .. Letting .L = {0, T 0}, Proposition 5.1.3 implies that .T L = L, the subgroup .{S ∈  : SL = L} is a maximal cyclic subgroup of ., and L is a rational line. Thus T is a primitive covering transformation if and only if there exists a rational line L such that .[T ] = {S ∈  : SL = L}. So the primitive elements of . are the generators of the subgroups .{S ∈  : SL = L} of . where L is a rational line. Proposition 5.2.6 Let .β : [0, 1] → T2 be a simple closed curve that is not null : [0, 1] → C of .β, then T is primitive. (0) = β (1) for a lift .β homotopic. If .T β u be the universal lift of .β given by (5.1). It follows from ProposiProof Let .β u (R) lies between parallel rational lines .L1 and .L2 such that tion 5.2.3 that .J = β  divides .C into two open .T Li = Li for .i = 1, 2. Proposition 5.2.5 implies that .J connected sets U and V such that .L1 ⊂ U and .L2 ⊂ V and .T U = U and .T V = V . Suppose T is not primitive, so .T = S j for some .j = ±1 and .S ∈ . We can assume that .j > 1. If .S J ∩ J = φ, then .S = T k by Proposition 5.2.2. It follows that .T = S j = T kj and .ι = T kj −1 , which is not possible because kj cannot equal 1 with .j > 1. So .S J ∩ J = φ. Clearly, .T = S j implies that .T 0 = S j 0 = j S0 or .(1/j )T 0 = S0. Since S is in ., it follows that .S0 = (1/j )T 0 is in .Ga for .Li . Thus .SLi = Li by part (b) of Proposition 5.1.3, and we can assume that both .J and .S J lie between .L1 and .L2 . Proposition 5.2.5 now implies that .SU ∩ U = φ and .SV ∩ V = φ. Since .C = S J∪SU ∪SV and .Jis connected, either .J ⊂ SU or .J ⊂ SV because   = φ. Suppose that .J ⊂ SU . It follows that .U ⊂ SU and that .U = SU . .S J ∩ J Therefore, T is primitive because U  SU  · · ·  S j −1 U  S j U = T U = U

.

is impossible. A similar argument works when .J ⊂ SV .



Proposition 5.2.7 If T is a primitive element of ., then there exists a primitive element S of . such that for every R in . there exist unique integers .α and .β satisfying .R = S β T α and . = [S] ⊕ [T ]. Proof Recall that T is primitive if and only if .T 0 = m + in with m and n relatively prime. From elementary number theory, we know that m and n are relatively prime if and only if there exist integers j and k such that .j m + kn = 1. Note that .−k and j are relatively prime because .j m + (−k)(−n) = 1. Suppose .Rz = z + (p + iq). For every element .p + iq of .Z[i], there exist unique integers .α and .β such that .p + iq = α(m + in) + β(−k + ij ) by Cramer’s rule because

 m −k .Det = 1. n j Consequently, .R = T α S β = S β T α .



5.2 Control Curves

127

Corollary 5.2.8 Let .β : [0, 1] → T2 be a simple closed curve that is not null : [0, 1] → C of .β and .J is the image of (0) = β (1) for a lift .β homotopic. If .T β u given by equation (5.1), then there exists .S ∈  such that every the universal lift .β   component of .π −1 β([0, 1]) has the form .S k J for a unique .k ∈ Z. Exercise 5.2.9 Prove Corollary 5.2.8. Proposition 5.2.10 Let x be a periodic point with minimal positive period .τ of a flow .(T2 , R) with lifted flow .(C, R). If the orbit of x is not null-homotopic, then the orbit of . x in .π −1 (x) is the image of the universal lift of the simple closed curve .β(t) = x(t/τ ) and .t →  x t is the type of a rational line. Exercise 5.2.11 Prove Proposition 5.2.10. Let x be a moving point of a flow .(X, R) on a compact connected surface. Given an open connected neighborhood U of x that is evenly covered by the universal  → X, there exists a local section .λ of length .2α at x such that covering map .π : X  of .π −1 (U ) contains a .λ[−α, α] ⊂ U by Corollary 4.1.9. Then each component .U , lift  .λ of .λ that is local section of length .2α of the lifted flow such that  .λ[−α, α] ⊂ U which proves to be a very useful property. Standing Assumption 2 A local section .λ of a flow on a compact connected surface and its flow box will always be contained in an open connected set that is evenly covered by the universal covering. Let .λ be a local section for a flow .(T2 , R), and let .xσ and .xτ with .σ < τ be distinct consecutive crossings of .λ. Equation (4.3) can now be written as J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ .

.

(5.5)

Note J is an embedded circle in .T2 . Although most of the time a parameterization of J will not be needed, it is useful to have an understanding of how J is to be parameterized when necessary. Standing Assumption 3 When an embedded circle J given by equation (5.5) is parameterized by a continuous function .β : [0, 1] → J , the function will always satisfy .β(0) = xσ = β(1), and as s increases .β(s) will follow .[xσ, xτ ]ϕ first and then move along .λ from .xτ to .xσ . Given an embedded circle J determined by distinct consecutive crossings of a local section as specified by equation (5.5), let .β : [0, 1] → J parameterize J according to Standing Assumption 3. If .β is not null-homotopic, it has a unique  starting at any . (1) = T  (0) in .π −1 (x) and ending at .β x σ for some lift .β xσ = β .T ∈  \ {ι}. The covering transformation T is primitive by Proposition 5.2.6. x σ , denoted By Standing Assumption 2, there exists a unique lift of .λ containing . by  .λ. Moreover,  .λ is a local section at . x σ , and .2α denotes the common length of .λ u can be obtained from equation (5.1), and  .λ. Then . x τ is in .T  λ. A universal lift .β u (R) is called a control curve. Using equation (5.2), and .J = β

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5 Flows on the Torus

Fig. 5.7 A schematic picture of a control curve .J

u (R) = J = β



.

  x σ,  x τ ]λ ∪ [ T m [ x τ, T  x σ ]T λ .

(5.6)

m∈Z

Because T is an automorphism of . ϕ, J =



.

m [T m x σ, T m x τ ] x τ, T m+1 x σ ]T m+1λ . ϕ ∪ [T 

(5.7)

m∈Z

Figure 5.7 provides a useful schematic picture of a control curve. Although the m x τ, T m+1 pieces .[T m x σ, T m x τ ] x σ ]T m+1λ are not usually line segments, ϕ and .[T  figures such as Figure 5.7 are sufficient for understanding the use of control curves in proofs. Note that .J divides .C into two components by Proposition 5.2.5. The next step is to adapt Proposition 4.3.1 to control curves. Set

m =  q t : 0 < t < α and  q ∈ (T m−1 x τ, T m x σ )T mλ P

(5.8)

m =  q t : −α < t < 0 and  q ∈ (T m−1 x τ, T m x σ )T mλ . N

(5.9)

.

and .

Proposition 5.2.12 If .J is a control curve for a flow on the torus and m is m and .N m given by equations (5.8) and (5.9) are in different an integer, then .P  components of .C \ J . m and .N m . Proof The proof of Proposition 4.3.1 works for any .P



Corollary 5.2.13 If .J is a control curve for a flow on the torus and U and V are m given by equation (5.8) is in U for all m or .P m is the components of .C \ J, then .P in V for all m. 0 ⊂ U , then .P m = T m P 0 ⊂ T m U = U . Proof If .P



0 will be called the positive side of the The component of .C \ J that contains .P   control curve .J and denoted by .JP . The other one will be called the negative side of the control curve and denoted by .JN . Proposition 5.2.14 If .J is a control curve for a flow .ϕ on the torus, then .JP is a positively invariant set of .C for the lifted flow . ϕ and .JN is a negatively invariant set of .C for the lifted flow . ϕ.

5.2 Control Curves

129

Proof Suppose . q is an element of .JP such that .O+ ( q ) ∩ (JN ∪ J) = φ. Set .s = inf{t > 0 :  q t ∈ JN ∪ J}. Clearly, . q s ∈ J because .JP and .JN are open sets. For some .n ∈ Z+ either . q s = T n x s for some .s such that .σ < s ≤ τ or . q s is in T n ( x τ, T  x σ ]T λ = (T n x τ, T n+1 x σ ]T n+1λ .

.

We can assume without loss of generality that .n = 0 by replacing . q with .T −n q. qs =  x s for some .s such that .σ < s ≤ τ or . q s is in So the two cases are . .( x τ, T  x σ ]T λ . Also note that the proof of Corollary 4.3.2 can be applied to show that .{ x (σ − t) : 0 < t ≤ α} ⊂ JN and .{ x (τ + t) : 0 < t ≤ α} ⊂ JP . Except for adding some tildes, we can borrow verbatim the last paragraph of the proof of Proposition 4.3.4 to complete the argument. In the first case, .O( q s) = O( x ) and . q (s − ε) =  x (s − ε) is in .J for small .ε, contradicting .s = inf{t > 0 :  q t ∈ JN ∪ J}. In the second case, . q (s − t) ⊂ JN for .0 < t < α, again contradicting the .s = inf{t > 0 :  q t ∈ JN ∪ J}. 

As before, J is a non-null-homotopic embedded circle determined by distinct consecutive crossings of a local section as specified by equation (5.5), .β : [0, 1] →  is the unique lift J parameterizes J according to Standing Assumption 3, and .β −1   x σ for some starting at some . x σ = β (0) in .π (x) and ending at .β (1) = T  primitive .T ∈ .  such that .T J = J, which is just a Let .J be the control curve constructed using .β universal lift of .β in this context. If Q is a covering transformation, then it follows from equations (5.3) and (5.4) that .QJ is a control curve. If .Q = T k for all .k ∈ Z, then Proposition 5.2.7 and Corollary 5.2.8 imply that .QJ = J. It follows from Proposition 5.2.2 that .QJ∩ J = φ. Now assume that S is given by Corollary 5.2.8. Then .S m J ∩ S n J = φ when  for some .Q ∈  equals .S k J for a .m = n, and every control curve of the form .QJ k  unique .k ∈ Z. Thus .S J provides an enumeration of the control curves. (0), T β (0)}. So By Proposition 5.2.3, .J is the type of the rational line .L = {β − +  .T L = L, and there exist lines .L and .L parallel to L such that .J lies between them, .L− ⊂ JN , and .L+ ⊂ JP . Clearly, .T L− = L− and .T L+ = L+ . Let .LS be the axis of S, that is, .LS is the line through the origin such that − < L ∩ L+ . Then S is an order preserving .SLS = LS . Order .LS such that .LS ∩ L S homeomorphism of .LS onto itself. Using infimum and supremum arguments, there exist unique points a and b in  ∩ LS such that one of the 2 open rays .Ra of .LS originating at a is contained .J in .JN and another open ray .Rb of .LS originating at b is contained in .JP . Then .Ra ∩ Rb = φ. Also .a ≤ b and .x < y when x is in .Ra and y is in .Rb . Note for future use that .L+ ∩ LS > b. By replacing S by .S −1 if necessary, it can be assumed that Sb is in .Rb . It follows by connectivity that .S J ⊂ JP because Sb is in .JP and .J ∩ S J = φ. For .k ≥ 1, the distance .|S k+1 b − S k b| is constant and .S k b increases to infinity in .LS . Likewise for .k ≤ −1, .S k a is an equidistant sequence decreasing to .−∞ in .Ra . In particular, k ⊂ J N for .k < 0, and .S k J ⊂ JP for .k > 0. .S J

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5 Flows on the Torus

Now .S JP must be the component of .C \ S J containing the ray .SRb = RSb . Since .J separates .S J and .S −1 J by the previous paragraph, .S J separates .J and 2 . Recall that the components of .C \ J  and .C \ S J are also the path components. .S J The point .S 2 b is in .S JP . Thus there is a path from every point in .S JP to .S 2 Rb that does not cross .S J and hence cannot cross .J. Therefore, every point in .S JP is contained in .JP , because .S 2 Rb is also in .JP . Two important set relations follow from .S JP ⊂ JP . They are: S k+1 JP ⊂ S k JP

(5.10)

.

holds for all .k ∈ Z. Similarly, for all .k ∈ Z, we have S k+1 JN ⊃ S k JN .

(5.11)

.

The closed set B consisting of all points between or on the lines .L+ and .L− will be called a control band, and the lines .L+ and .L− are called the positive and negative bounding lines of B, respectively. If T is an element of . such that .T L = L, then clearly .T B = B. Figure 5.8 shows not just one but two parallel control bands labeled accordingly for the next proposition. Proposition 5.2.15 If .J is a control curve for a flow on the torus and K is a compact subset of .C, then there exist control bands B and .B constructed from  with the following properties: .J (a) (b) (c) (d)

QB = B for some .Q ∈ . The region between B and .B , which we denote D, contains the compact set K. If z is in D, then .O+ (z) .{O− (z)} cannot cross B .{B }. If z is in D, then .O+ (z) .{O− (z)} can cross .B .{B} but cannot return.

.

Proof Let T be one of the two primitive elements of . such that .T J = J, and then let S be given by Corollary 5.2.8 such that equations (5.10) and (5.11) are valid, (0), T β (0)} such that is, Sb is in .Rb . Select lines .L+ and .L− parallel to .L = {β − +    that .J is between them, .L ⊂ JN and .L ⊂ JP from the infinitely many possible L+ B J

L−

D L+ x τ

B

x σ λ

Fig. 5.8 Parallel control bands B and .B

Tx σ Tλ

J

L−

5.3 Geometry of Recurrent Orbits

131

choices to form a control band .B0 . Then, clearly, .S k B0 is another control band for all k, but .B0 ∩ S k B0 need not be the empty set for small k. Let .ρ denote the usual Euclidean distance between .L+ and .L− . The choice of S guarantees that .S k L− for .k ≥ 0 eventually shifts from being on the same side of .L+ as .L− to being on the opposite side of .L+ and stays there. So there exists a smallest positive integer .κ such that .S κ L− and .L− are on opposite sides of .L+ . Clearly, .d(S κ L− , L− ) > ρ and then .d(S κ+m L− , S m L− ) > ρ. There exists a largest integer .μ in .Z such that .K ∩ S μ B0 = φ and .K ⊂ S μ JP . If n is a positive integer, then d(S μ L+ , S μ+nκ L− ) = d(L+ , S nκ L− ) > (n − 1)ρ.

.

Hence, there exists .ν ∈ Z+ such that K lies between .S μ L+ and .S μ+νκ L+ because K is bounded. Set .B = S μ B0 and .B = S μ+νκ B0 = S νκ B. Properties (a) and (b) follow from the construction, and properties (c) and (d) follow from Propositions 5.2.1 and 5.2.14. 

Corollary 5.2.16 Let .J be a control curve for a flow .(T2 , R) on the torus, and let . y be a point in .π −1 (y) for some .y ∈ T2 . If .O+ ( y ) crosses infinitely many distinct copies .QJ of .J with .Q ∈ , then .

lim | y t| = ∞.

t→∞

Proof Let r be a positive real number such that .| y | < r. It suffices to show that there exists .τ > 0 such that .| y t| > r when .t > τ . Apply the proposition with the same notation to .K = {z ∈ C : |z| ≤ r} to obtain 2 control bands such that K is D, the region between them. Letting S be given by Corollary 5.2.8, it is readily seen that the number of distinct copies of .J that intersect the region between .L− and .L + is finite. Then parts (c) and (d) of the proposition that imply that .O+ ( y ) must cross .B and cannot return to the region D between B and .B . 

5.3 Geometry of Recurrent Orbits In broad terms, the goal of this section is to show that the lift of a non-periodic recurrent orbit on the torus is the type of an irrational line. It is almost an immediate consequence of this result that a positively recurrent point on the Klein bottle is periodic. Along the way, we prove that a flow on the torus has at most one nonperiodic recurrent orbit closure. The first result already shows that the lift of a nonperiodic positively recurrent orbit must have a very different geometrical behavior than the lift of a periodic orbit.

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5 Flows on the Torus

Theorem 5.3.1 If x is a positively {negatively} recurrent non-periodic point of a flow .(T2 , R) on the torus and .(C, R) is the lifted flow of .(T2 , R), then .O+ ( x) − x )} does not lie between two parallel rational lines in .C for all . .{O ( x in .π −1 (x). Proof Suppose x is a positively recurrent and not periodic point and . x is a point in .π −1 (x) such that .O+ ( x ) lies between two parallel rational lines, .L and .L . There exists a primitive T in . such that .T L = L for every line L parallel to .L (Proposition 5.1.3, part √ (c)). Let d be the distance between .L and .L . Set .|T z−z| = |T 0| = c and .e = c2 + d 2 . Given .ζ between .L and .L , there exists a line L parallel to .L containing .ζ . Then L is between .L and .L , and there exists m such that .|T m ζ −  x | < e. Since the action of . on .C is proper and free, it follows from Proposition 1.1.20 that there exists a finite set of lines .L1 , . . . , Lq such that .ζ between .L and .L and .π(ζ ) = x implies that .ζ ∈ Li for some i. Because x is positively recurrent, there exists at least one i with the following property: given .ε > 0, there exists .ζ ∈ Li and .τ > 0 such that .| x τ − ζ | < ε and .π(ζ ) = x. We divide this situation into two possibilities and show that they are both impossible. The first case is eliminated using the bitransformation group structure .(, C, R) on .C and the second by a control curve argument. First assume . x ∈ L1 and .L2 has the required property. Let S be a covering transformation such that .SL1 = L2 . Obviously, .S m L1 is not between .L and .L for large m. To show that this case is impossible, it suffices to show by induction that for any positive integer m and .ε > 0 there exists .ζ ∈ S m L1 and .τ > 0 such that .| x τ − ζ | < ε and .π(ζ ) = x. Obviously, it is true for .m = 1. Suppose it holds for m, and let .ε > 0. Since it holds for .m = 1, there exists .ζ ∈ SL1 = L2 and .τ > 0 such that .| x τ − ζ | < ε and .π(ζ ) = x. We can assume that .S x = ζ . By continuity of the flow, there exists .δ > 0 such that .|zτ − S x| = x | < δ. Because every element of the abelian group . is |zτ − ζ | < ε when .|z −  both an isometry and an automorphism of the lifted flow, .|zτ − SR x | < ε when .|z − R x | < δ for all .R ∈ . By the induction hypothesis, there exists .ζ ∈ S m L1 and .τ > 0 such that .| x τ − ζ | < δ and .π(ζ ) = x. There exists .n ∈ Z such that .T n S m x = ζ . Setting n m .τ = τ + τ , it follows with .R = T S that | x τ − T n S m+1 x | = |( x τ )τ − ST n S m x | = |( x τ )τ − Sζ | < ε

.

because .| x τ − T n S m x | = | x τ − ζ | < δ. Obviously, .x = π(T n S m+1 x ) and the induction argument is complete, ruling out the first case. If the first assumption is not true, then, continuing to assume that . x ∈ L1 , there exists .r > 0 such that .| x τ − ζ | < r and .π(ζ ) = x implies .ζ ∈ L1 . Let U be an , the component of evenly covered open connected neighborhood of x such that .U −1 (U ) containing .  ⊂ {z ∈ C : |z −  .π x , satisfies . U x | < r}. Let .λ be a local section . It follows that of length .2α in U at x, and let  .λ be the lift of .λ in .U

5.3 Geometry of Recurrent Orbits

133

 ⊂ {z ∈ C : |z − R R λ ⊂ R λ[−α, α] ⊂ R U x | < r}

.

(5.12)

for all .R ∈ . x is not periodic and thus not positively recurrent or Since x is not periodic, . negatively recurrent by Theorem 4.3.16. Thus the consecutive crossings of  .λ by + x ) cannot get arbitrarily close to . .O ( x . By replacing .λ with a small enough subarc of .λ containing x, but not as an endpoint, we can assume that O+ ( x) ∩  λ = { x }.

.

The point x is positively recurrent and not periodic. Thus .O+ (x) must cross .λ arbitrarily close to x. So there exists .τ > 0 such that x and .xτ are distinct consecutive crossings of .λ. Then .J = [x, xτ ]ϕ ∪ [x, xτ ]λ is an embedded circle. Let .β be a simple closed curve that parameterizes J as specified by Standing  such that .β (0) =  x would Assumption 3. If .β is null-homotopic, then the lift .β (1) =  x , implying that . satisfy .β x τ was in  .λ, contrary to the construction of  .λ so that .O+ ( x) ∩  λ = { x }. Therefore, .β is not null-homotopic. (0) = β (1) (Proposition 5.2.6). There exists a primitive .S ∈  such that .S x = Sβ (1), it follows from (5.12) that .| Letting .ζ = β x τ −ζ | < r because both . x τ and .ζ are in .S λ. Therefore, .S x is in .L1 by the choice of r. Since . x is also in .L1 , it follows from Proposition 5.1.3 that .SL1 = L1 and that S generates the maximal cyclic subgroup ±1 from the beginning of the proof, .{R ∈  : RL1 = L1 } of .. (Of course, .S = T but that has no relevance here.) Moreover, .J, the component of .π −1 (J ) containing . x , is a control curve the type of .L1 by Proposition 5.2.3. x t ∈ JP for all .t > τ , but J was constructed Now Proposition 5.2.14 implies that . so that there exists .σ > τ such that .xσ ∈ (x, xτ )λ by Corollary 4.2.9. It follows that there exists R in . such that . x σ ∈ R λ and hence .| x σ − R x | < r by (5.12). x is in .L1 as is . x . The construction of .λ guarantees that . x σ is not Consequently, .R in  .λ and .R = ι. So .L1 = { x , R x } and .RL1 = L1 (Proposition 5.1.3). Therefore, k .R = S for some .k = 0, and it follows from equation (5.7)  x σ ∈ (S k−1 x τ, S k x )S k λ ⊂ J,

.

contradicting the positive invariance of .JP .



Corollary 5.3.2 If the flow .(T2 , R) has a periodic orbit that is not null-homotopic, then every positively .{negatively.} recurrent point is periodic. Exercise 5.3.3 Prove Corollary 5.3.2. The next theorem is the first example of the geometry of lifted orbits being used to prove dynamical results. Theorem 5.3.4 If x and y are positively or negatively recurrent but not periodic points of a flow .(T2 , R) on the torus .{(X, R) on a compact connected orientable − − bordered surface of genus 1.}, then .O(x) = O(y) .

134

5 Flows on the Torus

Proof By Theorem 2.3.2, it is sufficient to establish the result for flows on the torus. The result is trivial when x is in .O(y). So we can assume that .O(x) ∩ O(y) = φ. − Because the hypotheses on y and x are identical, it suffices to show that .O(x) ⊂ − − O(y) . (Note that .O− (x) and .O(x) are different things. The first is the negative − semi-orbit of x, and the second is the closure of the orbit of x.) Since .O(y) is a − closed invariant set, the proof is further reduced to showing that x is in .O(y) or equivalently, given an open neighborhood U of x, there exists .t ∈ R such that yt is in U . Without loss of generality, we can assume that U is an evenly covered open connected neighborhood of x. Let .λ be a local cross section at x of length .2α such that .λ[−α, α] ⊂ U as specified by Standing Assumption 2. Suppose x is positively recurrent. Then .{t ≥ 0 : xt ∈ λ} is a sequence .τk starting with .τ0 = 0 and increasing to infinity such that .xτk and .xτk+1 are consecutive crossings of .λ for every k and x −  be a component of .π −1 (U ), and let is in .{xτk : k > 0} (Proposition 4.1.1). Let .U   . x and .λ be the lifts of x and .λ contained in .U , respectively. λ for some .Tk ∈ . If .Tk = ι for all k, then . In the lifted flow, . x τk ∈ Tk x ∈ { x τk : − k > 0} and . x is positively recurrent, implying that . x is periodic (Theorem 4.3.16). This is impossible because x is not periodic. Set .κ = max{k : Tj = ι for 1 ≤ j ≤ k}. Then . x τκ is in  .λ and .Tκ+1 = ι. Replace x with .xτκ , set .τ = τκ+1 − τκ , set .T = Tκ+1 = ι, and use equation (5.5) with .σ = 0 to define an embedded circle J in .T2 . Because .T = ι, it follows that parameterizing J in the standard way is not null-homotopic and J =



.

m [T m x , T m x τ ] x τ, T m+1 x ]T m+1λ ϕ ∪ [T 

(5.13)

m∈Z

is a control curve that is the type of the line .L = { x, T  x }. There exist lines .L+ −  and .L parallel to L such that .J and L are in the open region between .L+ and .L− . Thus the closed set B consisting of all points between or on the lines .L+ and .L− is a control band. By Proposition 5.2.15, there exists a parallel control band .B = RB for some .R ∈  such that .d(B, B ) > 2. There exists .ζ ∈ D, the region between B and .B , such that .{z : |z−ζ | < 1} ⊂ D. So there exists . y , a lift of y, in D. By Theorem 5.3.1 and Propositions 5.2.15 and 5.2.1, .O+ ( y ) {.O− ( y )} crosses .B {B} and intersects   .R J {.J } when y is positively recurrent {negatively recurrent}. When y is positively recurrent, there exists .σ > 0 such that . y σ ∈ R J. This is only possible if  y σ ∈ R[T m x τ, T m+1 x ]T m+1λ = [RT m x τ, RT m+1 x ]RT m+1λ .

.

Applying .π yields .yσ ∈ [xτ, x]λ ⊂ U . Similarly, when y is negatively recurrent, there exists .σ < 0 such that .yσ is in U . Thus given an open neighborhood U of x, there exists .σ ∈ R such that .yσ is in U when y is either positively or negatively recurrent.

5.3 Geometry of Recurrent Orbits

135



Of course, the same overall proof works when x is negatively recurrent.

flow .(T2 , R)

Corollary 5.3.5 If x is an almost periodic but not periodic point of a on the torus .{(X, R) on a compact connected orientable bordered surface of genus 1.}, then every positively or negatively recurrent point of the flow that is not periodic is almost periodic. −

−

Proof For y positively or negatively recurrent but not periodic, .O(x) = O(y) by the theorem. Then apply Theorem 1.1.11. 

The proof of Theorem 5.3.4 can be modified to prove the following: Theorem 5.3.6 Let x be a positively .{negatively.} recurrent and not periodic point x ∈ π −1 (x), then of a flow .(T2 , R) and let .(C, R) be the lifted flow of .(T2 , R). If . .| x t| → ∞ as .t → ∞ .{t → −∞}. Proof It suffices to show that .| x t| → ∞ as .t → ∞ for one . x ∈ π −1 (x) because .| x t| = |T  x t − T 0| ≤ |T  x t| + |T 0|. This reduces the proof to considering an . x ∈ π −1 (x) such that .| x | < 1, and showing that given .r > 0, there exists .τ > 0 such that .| x t| > r when .t ≥ τ . As in the proof of Theorem 5.3.4, construct a control band B and apply Proposition 5.2.15 to obtain a parallel control band .B = RB for some .R ∈  such that .d(B, B ) > 2(r + 2) so there exists .ζ ∈ D, the region between B and .B , such that .{z : |z − ζ | < r + 2} ⊂ D. Then there exists .η ∈ Z[i] ∩ {z : |z − ζ | < 1} so that .{z : |z − η| < r + 1} ⊂ D. Define .S ∈  by setting .Sz = z + η. Hence, .|S x − η| = | x | < 1 and .{z : |z − S x | < r} ⊂ D. By Theorem 5.3.1 and Proposition 5.2.15, .S x t crosses .B and never returns to D. Thus there exists .τ > 0 such that when .t > τ it follows that .| xt −  x | = |S x t − S x | > r. 

To establish a uniform criterion on finite intervals for a curve to be the type of a line, we use Lemma 4 from [56]. The next proposition can be viewed as a criterion for a curve to be the “type of a ray.” It will then be used to establish a criterion for a curve to be the type of a line. Proposition 5.3.7 Let .f : [0, ∞) → C be an injective curve such that .|f (t)| → ∞ as .t → ∞. If there exists a constant .D > 0 such that for any closed interval .[α, β] with .0 ≤ α < β   d f (t), {f (α), f (β)} < D

.

for all .t ∈ [α, β], then there exists a line L containing .f (0) such that .d(f (t), L) ≤ D for all .t ≥ 0. Proof Let .Ln = {f (0), f (n)}. Since a nonzero complex number can always be written in the form .reiθ with .r > 0 and .θ ∈ R, a line in .C can always be parameterized in the form .g(s) = seiθ + b. In particular, .Ln can be parameterized in the form .fn (s) = seiθn + f (0). There exists a subsequence .θnk such that .eiθnk converges to some .eiθ because .S1 is compact. Let L be the line .g(s) = seiθ + f (0).

136

5 Flows on the Torus

Fix .τ > 0. Let .L⊥ be the line through .f (τ ) perpendicular to L, let .ζk = L⊥ ∩Lnk , and let .ζ = L⊥ ∩ L. So .d(f (τ ), L) = |f (τ ) − ζ | and .ζk converges to .ζ . Clearly, i(θ−θnk ) .e converges to 1, and hence, .cos(θ − θnk ) converges to 1. Using similar triangles, it is an exercise to show that for large k d(f (τ ), ζk ) cos(θ − θnk ) = d(f (τ ), Lnk ) ≤ D.

.

Letting k goes to infinity shows that .d(f (τ ), L) = |f (τ ) − ζ | ≤ D.



Exercise 5.3.8 Verify the claim in the proof of Proposition 5.3.7 that for large k, d(f (τ ), ζk ) cos(θ − θnk ) = d(f (τ ), Lnk ) ≤ D.

.

Proposition 5.3.9 Let .f : R → C be an injective curve such that .|f (t)| → ∞ as |t| → ∞. If there exists a constant .D > 0 such that for any closed interval .[α, β] in .R with .α < β

.

  d f (t), {f (α), f (β)} < D

.

for all .t ∈ [α, β], then f is the type of a line. Proof By Proposition 5.3.7, there exists a line L containing .f (0) such that d(f (t), L) ≤ D for all .t ≥ 0. If n is a negative integer, then the proposition applies to .g(s) = f (s +n) for .s ≥ 0, and there exists a line .L containing .g(0) = f (n) such that .d(g(s), L ) ≤ D for all .s ≥ 0 or equivalently .d(f (t), L ) ≤ D for all .t ≥ n. Because .|f (t)| → ∞ as .|t| → ∞, the line .L is parallel to L and .d(L , L) ≤ D. It follows that .d(f (t), L) ≤ 2D for all .t ≥ n. Since n was an arbitrary negative integer, .d(f (t), L) ≤ 2D for all .t ∈ R. So f satisfies the first two conditions in the definition of the type of a line on p. 5.2. It remains to prove that for every .z ∈ L there exists .t ∈ R such that .|f (t) − z| < 2D. Clearly,

.

.

lim

t→−∞

f (t) f (t) = ± lim . t→∞ |f (t)| |f (t)|

Suppose equality holds. Let .L⊥ be the line through .f (0) perpendicular to L. Obviously, .L⊥ divides the set .{z ∈ C : d(z, L) ≤ 2D}, which is the region between 2 parallel lines, into two pieces. Then there exists .C > 0 such that for all t such that for .|t| > C both .f (t) and .f (−t) are in the same region. Because .|f (t)| → ∞ as .|t| → ∞, there exists .C > C such that for .|t| > C both .d(f (t), f (0)) > 3D and .d(f (−t), f (0)) > 3D. Then there exist .σ > C and .τ < −C such that .{f (σ ), f (τ )} is perpendicular to L, which leads to the contradiction that .d({f (σ ), f (τ )}, f (0)) ≥ 3D. Therefore, .

lim

t→−∞

f (t) f (t) = − lim , t→∞ |f (t)| |f (t)|

5.3 Geometry of Recurrent Orbits

137

and for any z lying on L, the connected set .f (R) intersects the line perpendicular to L at z, proving that there exists t such that .|f (t) − z| < 2D. 

Theorem 5.3.10 If x is a non-periodic recurrent point of a flow .(T2 , R), then the orbit of . x ∈ π −1 (x) in the lifted flow .(C, R) is the type of an irrational line. Proof Recall that a lift of a non-periodic recurrent orbit in .C cannot lie between two rational lines (Theorem 5.3.1) and can only be the type of an irrational line. It follows from Theorem 5.3.6 that .| x t| → ∞ as .|t| → ∞, so Proposition 5.3.9 is applicable. The proof proceeds by assuming that there does not exist a constant .D > 0 such that for any closed interval .[α, β] in .R with .α < β   d  x t, { x α,  x β} < D

.

for all .t ∈ [α, β] and deriving a contradiction. It follows from this assumption that there exist a sequence .ζi in .O( x ) and a sequence of real numbers .ti such that .

    sup d ζi t, {ζi , ζi ti } : 0 ≤ t ≤ ti ≥ i.

Moreover, these sequences can be chosen so that (ζi , ζi ti ) ϕ ∩ {ζi , ζi ti } = φ

.

and Ci = (ζi , ζi ti ) ϕ ∪ s{ζi , ζi ti }

.

is a simple closed curve in .C. A further refinement of  this construction  is necessary. Fix .i > 2, and let .si be the first time in .[0, ti ] that .d ζi t, {ζi , ζi ti } achieves its supremum. Let .L be the line parallel to .L = {ζi , ζi ti } on the same side of L as .ζi si such that .d(L , L) = 2. Let .L1 be the line perpendicular to L at .ζi , and let H be the half plane determined by .L1 and containing .ζi ti . Set E = {t : t ∈ [0, si ], d(ζi t, L) ≤ 2, and ζi t ∈ H }.

.

There exists .σi ∈ E such that .d(ζi σi , L1 ) = sup{d(ζi t, L1 ) : t ∈ E}. Hence, d(ζi si , ζi σi ) ≥ d(ζi si , L ) ≥ i − 2.

.

Let .L2 be the line passing through .ζi σi parallel to .L1 , and let .L3 be the line parallel to .L1 such that .d(L3 , L1 ) = d(ζi σi , L1 ) + 2 = d(L2 , L1 ) + 2 (see Figure 5.9). The intersection of the closed regions between lines L and .L and between lines 2 .L2 and .L3 is a 2 by 2 square .. There exists .ζ ∈ C such that .[0, 1] + ζ ⊂  because every 2 by 2 square contains an inscribed circle of radius 1. Because .π

138 Fig. 5.9 An example of a simpled closed curve .Ci but not a complicated one

5 Flows on the Torus

L1

ζi si

L2

L3

L ζi σi

ζi

ζi τi ζi ti

L

maps .[0, 1]2 + ζ onto .T2 , it follows from Lemma 5.1.5 that . cannot be in the interior of .Ci . Therefore, there exists .τi in .(si , ti ] such that .ζi τi ∈ . So there are three additional sequences, .si , .σi , and .τi with the following properties: (a) .0 ≤ σi < si < τ√ i ≤ ti . (b) .d(ζi σi , ζi τi ) ≤ 8. (c) .d(ζi si , ζi σi ) ≥ i − 2. The remainder of the proof is devoted to using control bands to showing that these inequalities cannot hold for all i. Let .λ be a local cross section at x. Because x is recurrent, the set .{t ≥ 0 : xt ∈ λ} is an infinite discrete sequence of positive real numbers going to infinity. Let  .λ be the lift of .λ containing . x . Since .| x t| → ∞ as .t → ∞, there exists a largest positive s such that . xs ∈  λ. By replacing x and . x with xs and . x s, respectively, we can assume without loss of generality that . xt ∈ /  λ for all .t > 0. Let .η1 = inf{t > 0 : xt ∈ λ}, which is positive because the return times are discrete. Then 2 .J1 = [x, xη1 ]ϕ ∪ [xη1 , x]λ is a simple closed curve in .T , and .J1 is not null  homotopic because . x η cannot be in .λ. So . x η1 is in .S λ for some primitive S in 1 of .J1 with . . \ {ι}. Therefore, the universal lift .J x ∈ J1 is a control curve, and − + there exists a control band .B1 with bounding lines .L+ 1 and .L1 . (The lines .L1 and − .L have no connection with .L1 .) Note that .SB1 = B1 . 1 The next step is to construct a second control curve with primitive .T ∈  such that .T = S ±1 . Using Proposition 5.2.15, there exists another control band  such that .d(B1 , B ) > 8. The .B parallel to .B1 and containing a control curve .J 1 1 1 + bounding lines of .B1 will be denoted by .K1 and .K1− . For .ε, a small positive real number, . x (η1 + ε) is on the positive side of .J1 and between the rational lines .L− 1 + and .K1 . So .O+ ( x η1 ) must cross .B1 and can never return to .B1 . Therefore, there λ. exists a maximal .η2 ≥ η1 and .k ∈ Z such that . x η2 ∈ S k Since .xη2 is in .λ, there exists .η3 > η2 such that .xη2 and .xη3 are consecutive crossings of .λ and .J2 = [xη2 , xη3 ]ϕ ∪[xη3 , xη2 ]λ is a simple closed curve. From the λ x η3 ∈ T S k choice of .η2 , it follows that there exists T in . \ {S j : j ∈ Z} such that .

5.3 Geometry of Recurrent Orbits

139

K1+ K1−

P



P

B1

B +

+

L

K

2

2

−

L

2

2 −

K

L− 1

2

2

B

B1

L+ 1

Fig. 5.10 Parallelograms P and .P

and .J2 is not null-homotopic. Therefore, the universal lift .J2 of .J2 with . x η2 ∈ J2 is − a control curve, and there exists a control band .B2 with bounding lines .L+ 2 and .L2 . Moreover, T is primitive because .J2 is a simple closed curve (Proposition 5.2.6), and .B2 is not parallel to .B1 because the maximal cyclic subgroups of . generated by S and T are not equal (Proposition 5.1.3). By Proposition 5.2.15, there exists another control band .B2 parallel to .B2 with bounding lines denoted by .K2+ and .K2− such that the parallelogram P formed by − + − the parallel lines .L+ 1 and .K1 and the parallel lines .L2 , and .K2 contains a disk of radius 4 centered a point .w ∈ P . (See Figure 5.10.) If z is in P , then .O+ (z) can leave the larger parallelogram .P determined by .L− 1, + − + .K , .L , and .K only by crossing .B or .B , and if it does so, it cannot return to P . 1 2 2 1 2 Let r be sufficiently large so that .P is contained in the disk of radius r centered at w. Choose i such that .i > r + 3. Given .i > r + 3, there exists .R ∈  such that .|Rζi σi − w| < 1. On the one hand, .Rζi τi is in P because |w − Rζi τi | ≤ |w − Rζi σi | + |Rζi σi − Rζi τi |

.

< 1 + |ζi σi − ζi τi | √ ≤ 1 + 8 < 4. On the other hand, .Rζi si is not in .P because

140

5 Flows on the Torus

|Rζi si − w| + |w − Rζi σi | ≥ |Rζi si − Rζi σi |

.

= |ζi si − ζi σi | ≥ i−2>r +1 implies that .|Rζi si − w| > r. Since .σi < si < τi , this is impossible because O+ (Rζi σi ) cannot leave .P and then return to P . 

.

Corollary 5.3.11 If the flow .(T2 , R) has a non-periodic positively or negatively recurrent point, then every periodic orbit of the flow is null-homotopic. Exercise 5.3.12 Use Theorem 1.2.8 to prove Corollary 5.3.11. Corollary 5.3.13 If the flow .(T2 , R) has a non-periodic positively or negatively recurrent point, then the lift of every positively {negatively} recurrent orbit lies between 2 parallel irrational lines. Proof By Theorem 1.2.8 again, the flow .(T2 , R) has a recurrent point x and the lifts of .O(x) are the type of an irrational line by the theorem. Thus .O( x ) lies between 2 parallel irrational lines L and .L . The lines L and .L form an irrational band B that other orbits cannot cross (Proposition 5.1.1). Clearly, there exists .T ∈  such that the region between B and T B contains .[0, 1]2 + ζ for some .ζ ∈ C. Therefore, a positively recurrent point .y ∈ T2 has a lift . y lying between B and T B that trap + y ) between 2 parallel irrational lines. .O ( 

Theorem 5.3.14 Given a flow .(T2 , R) on the torus with lifted flow .(C, R), let x be a point in .T2 such that .ω(x) .{α(x)} contains a moving point y. If . x is a lift of x such that .| x t| → ∞ as .t → ∞ .{t → −∞}, then .O+ ( x ) .{O− ( x )} lies between two parallel lines. Proof Suppose .O+ ( x ) does not lie between two parallel lines. By Proposition 5.3.7, there does not exist a  constant .D > 0 such that for any closed interval .[α, β] with .α ≥ 0 we have .d  x t, {f (α), f (β)} < D for all .t ∈ [α, β]. It follows from this assumption that there exist a sequence .ζi in .O+ ( x ) and a sequence of real numbers .ti such that .

    sup d ζi t, {ζi , ζi ti } : 0 ≤ t ≤ ti ≥ i.

Now the sequences .si , .σi , and .τi can be constructed as in Theorem 5.3.10. Let .y ∈ ω(x). If y is in .O(x), then x is positively recurrent and Corollary 5.3.13 applies. Otherwise, let .λ be a local section at y and proceed as in the proof of Theorem 5.3.10. Because Theorem 5.3.1 does not apply, 2 contradictions are now used to complete the proof. It is possible that .O+ ( x ) remains between the rational + + x ) does not lie between and .K , contradicting the assumption that .O ( lines .L− 1 1 two parallel lines. In the remaining case, the same contradiction used to complete the proof of Theorem 5.3.10 is obtained. 

5.3 Geometry of Recurrent Orbits

141

In this context, one also has the following unused result: If .O+ ( x ) is the type of a rational ray for a flow .(T2 , R) and .ω(x) is locally connected, then .ω(x) contains an invariant simple closed curve that is not null-homotopic [56]. Theorem 5.3.15 Let .(T2 , R) be a flow and let x be a point in .T2 such that .

lim | x t| = ∞

t→∞

for . x ∈ π −1 (x). If there exists y in .ω(x) and a local section .λ at y such that + x ) is rational. .ω(x) ∩ λ = {y}, then the limit of the semi-orbit .O ( Proof If x is periodic, the conclusion is obvious, and we can assume that x is not periodic. Since Theorem 5.3.14 applies, we can assume that .O+ ( x ) lies between a pair of parallel irrational lines and obtain a contradiction. Let u and v be the endpoints of .λ. Then .O+ (x) intersects .(u, y)λ or .(y, v)λ infinitely many times. Assume it is .(u, y)λ . Then there exists a sequence .τn of positive real numbers increasing to infinity such that .O+ (x) ∩ (u, y)λ = {xτn : n ≥ 1}. It follows from the hypothesis that .xτn converges to y. Let . y be in .π −1 (y), let  .λ be the lift of .λ containing . y with endpoints . u and  .v , x be the lift of x such that . x τ1 ∈  λ. Because .| x t| → ∞ as .t → ∞, there and let . exists .τk such that . x τk is in .( u,  y )λ and . x τn is not in .( u,  y )λ for .n > k. Thus .xτk and .xτk+1 are consecutive crossings of .[u, y]λ , and .J = [xτk , xτk+1 ]ϕ ∪ [xτk+1 , xτk ]λ is a simple closed curve that is not null-homotopic. As usual, this produces a control curve J =



.

m [T m x τk , T m x τk+1 ] x τk+1 , T m+1 x τk ]T m+1λ . ϕ ∪ [T 

m∈Z

Apply Proposition 5.2.15 with .r = 1 and with .R ∈  satisfying .B = RB. Now consider the sequence of disjoint bands .R j B, .j ≥ 1. Observe that .O+ ( x τk+1 ) is x) contained in .JP ∪ J and thus cannot cross .L− . Since we are assuming that .O+ ( lies between a pair of parallel irrational lines, .O+ ( x τk+1 ) cannot remain between − and .R j L+ for any .j ≥ 1. Therefore, .O + ( .L x τk+1 ) must cross .R j B and intersect j  for all .j ≥ 1 by Proposition 5.2.1. .R J If . x t is in .R k [T m x τk , T m x τk+1 ] ϕ for .t > τk+1 , then .xt = xs for s satisfying .τk ≤ s ≤ τk+1 , making x periodic that is no longer under consideration. It follows that there exists a sequence .σj of positive real numbers increasing to infinity such that . x σj is in .R j (T m x τk+1 , T m+1 x τk )T m+1λ for some .m ∈ Z. Now the sequence of points .xσj lies in .(xτk , xτk+1 )λ and must be distinct because x is not periodic. It follows that .ω(x) ∩ [xτk , xτk+1 ]λ = φ contradicting the hypothesis that .ω(x) ∩ λ = {y}. 

142

5 Flows on the Torus

Theorem 5.3.15 completes the first step of a general theorem. In Section 8.3, we will prove that it and its converse hold for all compact connected orientable surfaces of genus at least 1. Recall that the Klein bottle, .P2 #P2 , has a 2-sheeted covering .π2 : T2 → P2 #P2 (p. 2.3). Using this double covering, we will show that the Klein bottles along with 2 2 .S and .P are the only compact connected surfaces whose recurrent points must be periodic; the others all have flows with nontrivial minimal sets. Theorem 5.3.16 If x is a positively or negatively recurrent point of a flow (P2 #P2 , R) .{(X, R) on a compact connected nonorientable bordered surface of genus 2.}, then x is periodic.

.

Proof By Theorem 2.3.2, it suffices to prove the theorem for .P2 #P2 . Let .π2 : T2 → P2 #P2 be the double covering of .P2 #P2 by .T2 . Then .π2 ◦ π : C → P2 #P2 is the universal covering of .P2 #P2 . Let y be a positively recurrent point of a flow − 2 2 .(P #P , R) on the Klein bottle. There exists a recurrent point x such that .O(x) = − O(y) by Theorem 1.2.8. So it suffices to prove that x is periodic. Suppose x is not periodic. Let .(T2 , R) and .(C, R) be the lifted flows of −1 2 2 .(P #P , R). Let .π 2 (x) = {x1 , x2 }. Then .x1 and .x2 are non-periodic recurrent x ∈ C of .x1 . Then .O( x ) is the type points of .(T2 , R) by Theorem 2.2.6. Fix a lift . of a specific irrational line L. Since .Sz = z + 1/2 is a covering transformation of the universal covering .π2 ◦ π and . x is also a lift of x, the orbit .O(S x ) is the type of the line SL. Obviously, L and SL intersect. Consequently, .O( x ) and .O(S x ) must be distinct orbits that intersect, a contradiction. 

Corollary 5.3.17 (Kneser) If a flow on the Klein bottle has no fixed points, then it has periodic orbit. Proof A flow .(P2 #P2 , R) contains a minimal set M (Proposition 1.1.7) and M is the orbit closure of an almost periodic point x (Theorem 1.1.11). Then x cannot be a fixed point by hypothesis and cannot be an almost periodic point that is not periodic by the theorem. Thus .O(x) is a true periodic point. 

Theorem 1.1.13 can now be refined and reorganized for surfaces. Theorem 5.3.18 Let X be a compact connected surface. If M is a minimal subset of a flow on X, then one of the following holds: (a) .M = X = T2 . (b) M is a nontrivial minimal set. (c) M is a periodic orbit (including a fixed point). Proof Taking advantage of Theorem 1.1.13, suppose that .M = X. Then there are no fixed points in X, and .χ (X) = 0 by Theorem 1.2.11. The only 2 compact connected surfaces of Euler characteristic 0 are the torus and Klein bottle. Now Theorem 5.3.16 implies that .M = T2 .

5.3 Geometry of Recurrent Orbits

143

Nontrivial minimal sets are nowhere dense but not periodic by definition. They will be the new part (b). Since the only finite orbits of a flow are fixed points, which are considered to be periodic orbits of period 0, it is natural to let part (c) be the periodic orbits. Because all 3 of these cases can occur, the list cannot be made shorter, and by Theorem 1.1.13, there are no other possibilities. 

Chapter 6

Hyperbolic Geometry

Euclidean geometry is not suitable for studying the lifts of orbits on compact connected surfaces of genus greater than 1, but hyperbolic geometry works well. This chapter provides an introduction to hyperbolic geometry focused on using it in the study of flows on compact connected surfaces of genus greater than 1. Non-Euclidean geometries were first studied independently by Bolyai and Lobachevsky. They replaced the parallel postulate with the postulate that through a point p not on a line L there are at least two lines that do not intersect L. To prove that such a geometry was as mathematically consistent as Euclidean geometry, Beltrami used Euclidean geometry to construct several models of Bolyai– Lobachevskian geometry. Known as hyperbolic geometries, they have the characteristic property that the sum of the angles of a triangle is less than .π or 180.◦ . The model we will use is usually referred to as Poincaré disk model. It uses the open unit disk .B2 as the hyperbolic plane. The Poincaré disk model has the advantage that the measure of non-Euclidean angles equals the measure of Euclidean angles and is sometimes called the conformal disk model. The nonEuclidean circles in the Poincaré disk model are also Euclidean circles, but the two centers need not coincide because Euclidean and hyperbolic distances are very different. Section 6.1 will provide an introduction to the basic facts from hyperbolic geometry that will be needed. In particular, the hyperbolic distance function and the rigid motions are important for the rest of the chapter. The rigid motions consist of the Möbius transformations or linear fractional transformations that preserve the open unit disk and their complex conjugates. Obviously, the unit circle is also invariant. The second section examines these rigid motions and their dynamical properties. To use hyperbolic geometry in the study of flows on surfaces, subgroups of hyperbolic rigid motions acting properly and freely on .B2 will be needed. This leads to a discussion of subgroups of hyperbolic rigid motions in Section 6.3. The goal is to gain an understanding of the dynamical behavior of these subgroups on the closed unit disk and especially on .S1 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_6

145

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6 Hyperbolic Geometry

6.1 Poincaré Disk Model The points in the Poincaré disk model are just the points in B2 = {(x, y) ∈ R2 : x 2 + y 2 < 1} = {z ∈ C : |z| < 1}.

.

In this section, we will develop the basic structure of the Poincaré disk model of hyperbolic geometry. (Although Poincaré made extensive use of this model, it appears to have been known earlier by Beltrami [19] and is sometimes referred to as the Beltrami–Klein model.) There is a natural interplay between these two geometries that will be used to our advantage. In the Poincaré disk model, a hyperbolic line or simply an h-line is the intersection of a Euclidean circle perpendicular to .S1 with .B2 or the intersection of a Euclidean line through the origin with .B2 . Notice that each h-line naturally has two points at infinity, namely, the points where its defining circle or line intersects 1 2 2 .S . (Viewing the sphere .S as the one-point compactification of .R in the usual way, 2 the intersection of a Euclidean line through the origin with .B is also the intersection of a circle perpendicular to .S1 with .B2 .) Proposition 6.1.1 A circle in .R2 intersects the unit circle perpendicularly if and only if it can be written in the form x 2 + y 2 + ax + by + 1 = 0

(6.1)

.

with .a 2 + b2 − 4 > 0. Proof Every circle in the plane can be written in the form x 2 + y 2 + ax + by + c = 0

(6.2)

.

with .a 2 + b2 − 4c > 0. So it suffices to show that a general circle intersects .S1 orthogonally if and only if .c = 1. The rotation .T z = eiθ z is also a linear transformation of .R2 onto itself. Specifically, if .T (x + iy) = u + iv, then 

cos θ sin θ .(u, v) = (x, y) − sin θ cos θ

 .

The matrix  A=

.

cos θ sin θ − sin θ cos θ



is orthogonal (.A−1 = At ) and preserves the inner product, that is, .(a, b) · (x, y) = (a, b)A · (x, y)A (see [59], Proposition 4.2.1, for example). Equation (6.2) can be

6.1 Poincaré Disk Model

147

written as .(x, y) · (x, y) + (a, b) · (x, y) + c = 0. Letting .(a  , b ) = (a, b)A, it follows that (u, v) · (u, v) + (a  , b ) · (u, v) + c =

.

(x, y)A · (x, y)A + (a, b)A · (x, y)A + c = (x, y) · (x, y) + (a, b) · (x, y) + c = 0 and .(a  )2 + (b )2 − 4c = (a, b)A · (a, b)A − 4c = (a, b) · (a, b) − 4c > 0, but c does not change. Therefore, by rotating the given circle, we can assume that its center is on the positive x-axis, that is, .a < 0 and .b = 0, without changing c. Moreover, the original circle clearly intersects .S1 perpendicularly if and only if its replacement intersects 1 2 2 .S perpendicularly. This reduces the proof to showing that .x + y + ax + c = 0 1 intersects .S perpendicularly if and only if .c = 1. Now the center is .(−a/2, 0), and the intersections with .S1 are ⎛

⎞  2  −c − 1 −c − 1 ⎠, .⎝ ,± 1 − a a and they intersect perpendicularly if and only if their radial vectors are perpendicular. So to complete the proof, we need to solve for c in the equation ⎛ .



⎝ −c − 1 + a , ± 1 − a 2



−c − 1 a

2

⎞ ⎛



⎠ · ⎝ −c − 1 , ± 1 − a



−c − 1 a

2

⎞ ⎠ = 0.

This simplifies to  .

−c − 1 a

2

 +

   −c − 1 2 −c − 1 a

+1− = 0, a 2 a

which is just .−c − 1 + 2 = 0 or .c = 1.

 

The following properties of the equation .x 2 + y 2 + ax + 1 = 0 are helpful: (a) If .a = a  , .a 2 − 4 > 0, and .(a  )2 − 4 > 0, then the circles .x 2 + y 2 + ax + 1 = 0 and .x 2 + y 2 + a  x + 1 = 0 do not intersect. (b) If .a 2 − 4 > 0, then the circle .x 2 + y 2 + ax + 1 = 0 does not contain any points on the y-axis. (c) If .a 2 − 4 > 0, then the circle .x 2 + y 2 + ax + 1 = 0 intersects the x-axis perpendicularly. (d) If .(u, v) ∈ R2 satisfies .u2 > 0, then there exists a unique a such that .a 2 − 4 > 0 and .(u, v) lies on the circle .x 2 + y 2 + ax + 1 = 0.

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6 Hyperbolic Geometry

Proposition 6.1.2 If (x,y) and .(u, v) are linearly independent vectors in .D2 , then there exists a unique circle passing through .(x, y) and .(u, v) that is perpendicular to .S1 . Proof It suffices by Proposition 6.1.1 to show that the system of equations x 2 + y 2 + ax + by + 1 = 0

.

u2 + v 2 + au + bv + 1 = 0 has a unique solution for a and b. Rewriting the system as ax + by = −x 2 − y 2 − 1

.

au + bv = −u2 − v 2 − 1, it is apparent by Cramer’s rule that there is a unique solution for a and b if and only if   xy .Det = 0, uv which is equivalent to the rows of the matrix being linearly independent.

 

Corollary 6.1.3 The Poincaré disk model has the following properties: (a) Two distinct points in .B2 uniquely determine an h-line. (b) A point in .B2 and a point on .S1 uniquely determine an h-line. (c) Two distinct points in .S1 uniquely determine an h-line. Proof Two distinct points .(x, y) and .(u, v) in .D2 are linearly dependent if and only if 0 lies on the line .L = {(x, y), (u, v)}, making .L ∩ B2 an h-line.   Proposition 6.1.4 Given a point in .B2 and a Euclidean line L passing through it, then there exists a unique h-line tangent to L at the given point. Proof Let .(r, s) be the given point. Then there exists .(u, v) = 0 such that .t → (r, s) + t (u, v) parameterizes L. If .(r, s) and .(u, v) are linearly dependent, then 2 .L ∩ B is the unique solution. Suppose .(r, s) and .(u, v) are linearly independent. Then .(u, v) is tangent to a circle of the form .x 2 + y 2 + ax + by + 1 = 0 at .(r, s) if and only if .(2r + a, 2s + b) · (u, v) = 0. (The gradient of .x 2 + y 2 + ax + by + 1 = 0 at (r,s) is .(2r + a, 2s + b).) Thus we have a system of 2 linear equations for the unknowns a and b, namely, ar + bs = −r 2 − s 2 − 1

.

au + bv = −2ru − 2bv.

6.1 Poincaré Disk Model

149

Fig. 6.1 Two h-lines through that do not intersect L

.ζ

ζ

p

L

q

As above, the linear independence implies the existence of a unique solution for a and b and hence a unique h-line tangent to L at .(r, s).   Using the points at infinity of an h-line L and a point .ζ not on L, it follows from part (b) of Corollary 6.1.3 that there are at least two h-lines passing through .ζ that do not intersect L (see Figure 6.1). Thus the Poincaré disk model does not satisfy the parallel postulate. The hyperbolic angle made by two h-lines intersecting at a point .ζ in .B2 is, by definition, the Euclidean angle made by the Euclidean tangent lines to the circles defining the h-lines at their point of intersection. In this sense, hyperbolic and Euclidean angles are synonymous. Consequently, conformal functions should be a natural source for hyperbolic rigid motions mapping .B2 onto itself. Recall that a Möbius or linear fractional transformation is a complex function of the form Tz =

.

az + b cz + d

(6.3)

such that .ad −bc = 0. The condition .ad −bc = 0 guarantees that the numerator and denominator do not vanish simultaneously and that the function is not a constant. It is easy to see that T is a homeomorphism of .C onto .C when .c = 0. A more careful analysis of the equation .T z = w when .c = 0 shows that T is a homeomorphism of .C \ {−d/c} onto .C \ {a/c} and its inverse is also a Möbius transformation. The Möbius transformation T naturally extends to the one-point compactification 2 .S of .C by setting .T ∞ = ∞ when .c = 0, and .T ∞ = a/c when .c = 0. And setting .T (−d/c) = ∞, when .c = 0. This extension is readily seen to be a homeomorphism of .S2 onto itself. Möbius transformations map circles {lines} in .C to circles or lines in .C. When .c = 0, circles map to circles and lines to lines. When .c = 0, circles map to circles unless they pass through .−d/c in which case they map to lines. When .c = 0, lines map to circles through .a/c, unless they also pass through .−d/c.

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6 Hyperbolic Geometry

When viewed as a homeomorphism of .S2 onto itself, a Möbius transformation maps circles to circles because the stereographic projection of a circle through the north pole, .(0, 0, 1), of .S2 is a line in .C and vice versa. Given two triples .(z1 , z2 , z3 ) and .(w1 , w2 , w3 ) of distinct points in .S2 , there exists a unique Möbius transformation T such that .T zi = wi . In particular, a Möbius transformation that fixes three distinct points is the identity map .ι. A Möbius transformation T is differentiable with respect to z on .C when .c = 0 and on .C \ {−d/c} when .c = 0. Moreover, .T  z = 0 also because .ad − bc = 0. Hence, T is conformal on .C when .c = 0 and on .C \ {−d/c} when .c = 0. The extension of a Möbius transformation to .S2 is also conformal on the sphere, but this is not needed here because we are only interested in the Möbius transformations that map .B2 onto itself. For an introduction to Möbius transformations, see [1] or almost any introduction to complex analysis. Another way of thinking about Möbius transformations is starting with A in .GL(2, C), the topological group of .2 × 2 invertible complex matrices. There exists a homeomorphism .TA of .S2 onto itself defined by equation (6.3). An easy calculation shows that .A → TA is an algebraic homomorphism (.TAB = TA ◦ TB ) of .GL(2, C) into the group of homeomorphisms of .S2 onto itself. It follows that the Möbius transformations are naturally a group of homeomorphisms of .S2 and will be viewed that way. As a result, it is often convenient to do matrix calculations instead functional ones. Letting .I denote the .2 × 2 identity matrix in .GL(2, C), the kernel of .A → TA is the uncountable group .{cI : c ∈ C\{0}}. There are, however, exactly two matrices in each coset of the kernel with determinant 1. So the kernel of the restriction of .A → TA to the special linear group .SL(2, C), matrices in .GL(2, C) with determinant 1, is .K = {I, −I}. Consequently, the algebraic homomorphism .A → TA is a 2-to-1 function of .SL(2, C) onto the group of Möbius transformations. In particular, for A and B in .SL(2, C), we have .TA = TB if and only if .A = ±B. So requiring that .ad − bc = 1 for a Möbius transformation given by equation (6.3) is a convenient normalizing condition that will be used. (For an introduction to matrix groups, see [59], Chapter 4.) Standing Assumption 4 When .T = TA is a Möbius transformation given by equation (6.3), .Det [A] = ad − bc = 1. Proposition 6.1.5 A Möbius transformation T maps the upper half plane .U2 onto itself if and only if there exists A in .SL(2, R), the topological group of .2×2 invertible real matrices of determinant 1, such that .T = TA . Proof Suppose T is a Möbius transformation mapping the upper half plane, denoted by .U2 , onto itself. Then .T −1 also maps .U2 onto itself. It follows that T and .T −1 map 2 2 .R ∪ {∞}, which is the boundary of .U in .S , onto itself. There exists .A ∈ SL(2, C) −1 such that .T = TA and .T = TA−1 . There are two cases to consider: either .c = 0 or .c = 0.

6.1 Poincaré Disk Model

151

Suppose .c = 0. Then .T (0) = b/d, .T (∞) = a/c, and .T −1 (∞) = −d/c are all real as is .(b/d)(−d/c) = −b/c. Thus Tz =

.

az/c + b/c z + d/c

has real coefficients and real determinant .(ac − bc)/c2 = 1/c2 . If it can be shown that if .1/c2 is positive, then a, b, c, and d are all real. Because T i is in .U2 , the imaginary part of T i is positive. To calculate it, simplify T i as follows:    −i + d/c ai/c + b/c .T i = i + d/c −i + d/c =

(a/c + bd/c2 ) + i(ad/c2 − b/c) . |i + d/c|2

Therefore, ad/c2 − b/c (ad − bc)/c2 .0 < = = |i + d/c|2 |i + d/c|2



1 c2



1 |i + d/c|2



and .1/c2 is positive. When .c = 0, a similar but easier argument works using .T (0) = b/d, .T (1) = (a + b)/d, .T −1 (0) = −b/a, and .ad = 1 to show that Tz =

.

(a/d)1/2 z + (b/d)(a/d)−1/2 (a/d)−1/2

has only real coefficients, which completes the first half of the proof. When .{a, b, c, d} ⊂ R and .ad − bc = 1, it is obvious that .R ∪ {∞} maps onto itself. Either T maps the upper half plane onto itself or T maps the upper half plane onto the lower half. An easy calculation shows that the imaginary part of T i is positive to complete the proof.   The Möbius transformations mapping .B2 onto itself, denoted by .G, clearly form a subgroup of the group of all Möbius transformations. Perhaps, the simplest examples of elements of .G are the rotations T z = eiθ z =

.

eiθ/2 z + 0 0z + e−iθ/2

with .θ ∈ R. Note that .ad − bc = 1 in the fractional form.

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6 Hyperbolic Geometry

Proposition 6.1.6 The subset of .SL(2, C) defined by    = A ∈ SL(2, C) : A = a c G ca

.

(6.4)

is a closed connected subgroup of .SL(2, C) homeomorphic to .C × S1 .  is a closed subgroup of .SL(2, C) Proof A straightforward argument shows that .G with  .

ac ca

−1

 =

a −c −c a

 .

(6.5)

 → C × S1 by Define .h : G  h

.

ac ca

 = (c, a/|a|).

Letting .ρ = (1 + cc)1/2 , h−1 (c, ζ ) =

.



ρζ c c ρζ

 .

Clearly, h and .h−1 are continuous and .C × S1 is connected.

 

If  A=

.

ac ca

 (6.6)

 then .aa = 1 + cc implying that .|a| ≥ 1 and .|a| > |c|. If .c = 0, then is in .G, | − a/c| = | − a|/|c| = |a|/|c| > 1,

.

and .TA−1 ∞ ∈ / D2 . If .c = 0, then a is in .S1 and .TA is a rotation about the origin. Proposition 6.1.7 A Möbius transformation S maps .B2 onto itself if and only if  such that .S = SA , that is, S can be written in the form there exists A in .G Sz =

.

with .aa − cc = 1. Proof The Möbius transformations

az + c cz + a

6.1 Poincaré Disk Model

153

iz + i −iz/2 − 1/2 and −1 z = −z + 1 −iz/2 + 1/2

z =

.

(6.7)

map .B2 onto .U2 and vice versa. Then  C=

.

i i −1 1

 and C

−1

 =

−i/2 −1/2 −i/2 1/2

 (6.8)

are matrices determining . and .−1 . Proposition 6.1.5 implies that S is in .G if and only if there exists B in .SL(2, R)  if such that .S = −1 ◦ TB ◦ . Therefore, it suffices to show that .C −1 BC is in .G and only if B is in .SL(2, R). Let   αβ .B = γ δ be an arbitrary element of .SL(2, R). Then C −1 BC =



.

 =  =  =

−i/2 −1/2 −i/2 1/2 −i/2 −1/2 −i/2 1/2





αβ γ δ



i i −1 1



iα − β iα + β iγ − δ iγ + δ



(α + iβ − iγ + δ)/2 (α − iβ − iγ − δ)/2 (α + iβ + iγ − δ)/2 (α − iβ + iγ + δ)/2 ac ca





with .a = (α + δ)/2 + i(β − γ )/2 and .c = (α − δ)/2 + i(β + γ )/2. Moreover, given a and c such that .aa − cc = 1, it is a straightforward exercise to solve for entries of   B in .SL(2, R). Corollary 6.1.8 Given T in .G, the following are equivalent: (a) T is a rotation about 0. (b) .T 0 = 0.   ac  such that .T = TA and .c = 0. (c) There exists .A = in .G ca Exercise 6.1.9 Prove Corollary 6.1.8.

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6 Hyperbolic Geometry

Corollary 6.1.10 Let A be a matrix satisfying equation (6.6) and .aa − cc = 0. The Möbius transformation .T = TA is in .G if and only if .aa − cc > 0. Exercise 6.1.11 Prove Corollary 6.1.10.  × D2 → D2 defined by .(A, z) → TA z is a Corollary 6.1.12 The function .G  on .D2 . continuous (left) group action of .G Proof Clearly, .(A, z) → TA z is a group action. If .(An , zn ) is a sequence converging  × D2 , then .an zn + cn and .cn zn + a n converge to .az + c and .cz + a, to .(A, z) in .G respectively. Since .| − a/c| > 1 when .c = 0 and .|a| = 1 when .c = 0, it follows that 2 .cz + a = 0 for z in .D and .TAn zn converges to .TA z.    and .G is algebraically Corollary 6.1.13 The matrix .K is a normal subgroup of .G  isomorphic to the quotient group .G/K.  and the .K coset of A in .G  is .{A, −A}. Proof Clearly, .K is a normal subgroup of .G Since .TA = TB if and only if .A = ±B, the kernel of .A → TA is .K.    Let .π be the natural projection Obviously, .K is also a closed subgroup of .G.    of .G onto .G/K. With the quotient topology on .G/K, it is a topological group with a metric topology and .π is a continuous homomorphism ([59], Theorem 1.4.12). Corollary 6.1.13 now implies that .G with the quotient topology from .A → TA is  and .A → TA is a a topological group isomorphic to the topological group .G/K surjective homomorphism of topological groups.  Proposition 6.1.14 If .G is given the quotient topology from .A → TA mapping .G onto .G, then the following hold: (a) .A → TA is a normal 2-sheeted covering. (b) .G is a 3-dimensional manifold. (c) The function .G × D2 → D2 defined by .(T , z) → T z is a continuous (left) group action of .G on .D2 . Proof Left multiplication by elements of .K defines a continuous (left) group  that is easily seen to be proper and free. Then part (a) follows from action on .G Propositions 2.2.1 and 2.2.7.  being a connected 3-manifold. Part (b) is now an immediate consequence of .G  × D2 → D2 defined Proposition 1.1.25 applies to the continuous group action .G by .(A, z) → TA z (Corollary 6.1.12) because  : (A, z) → z for all z ∈ D2 } K = {A ∈ G

.

to obtain part (c).

 

 the Möbius transformation .T = TA  with .c = 0 in .G, Since .|−a/c| > 1 for .A ∈ G is conformal on the disk .{z ∈ C : |z| < |−c/a|} that is included in .D2 and .B2 . When .c = 0, then T is a rotation about 0 and conformal on .C (Corollary 6.1.8). It follows that the restriction of .T ∈ G to .B2 , which for convenience will also be denoted by

6.1 Poincaré Disk Model

155

T , maps h-lines to h-lines, and preserves the h-angle between pairs of intersecting h-lines. What is missing is a hyperbolic distance function that is .G-invariant. The cross ratio of four complex numbers .z1 , z2 , z3 , z4 , which is defined by .

(z3 − z1 )(z4 − z2 ) , (z3 − z2 )(z4 − z1 )

is preserved by Möbius transformations (see [1], Chapter 3, for example). Consider ζ1 and .ζ2 in .B2 , and let .∞1 and .∞2 be the points in .S1 at the end of the h-line determined by .ζ1 and .ζ2 . We will assume that .ζ2 lies between .ζ1 and .∞1 on the h-line joining .ζ1 and .ζ2 or equivalently that .ζ1 lies between .ζ2 and .∞2 on the h-line joining .ζ1 and .ζ2 . A Möbius transformation in .G will also preserve this property. Roughly following [44] (p. 80), set

.

ξ(ζ1 , ζ2 ) =

.

(ζ1 − ∞1 )(ζ2 − ∞2 ) . (ζ1 − ∞2 )(ζ2 − ∞1 )

(6.9)

Note that .ξ(ζ, ζ ) = 1 for any choice of .∞1 and .∞2 in .S1 . It follows from the definition of .ξ that .ξ(T ζ1 , T ζ2 ) = ξ(ζ1 , ζ2 ) for all .T ∈ G and all .ζ1 , ζ2 ∈ B2 . Proposition 6.1.15 If .ζ1 and .ζ2 are distinct points in .B2 , then there exists .T ∈ G such that .T ζ1 = 0 and .T ζ2 = r, a positive real number. Proof Set .a = 1 and .c = −ζ1 . Then Sz =

.

z − ζ1 −ζ 1 z + 1

(6.10)

is a Möbius transformation such that .Sζ1 = 0. Since .aa − cc = 1 − |ζ1 |2 > 0, Corollary 6.1.10 implies that S is in .G. Writing .Sζ2 = reiθ where .0 < r < 1 and −iθ z, it is obvious that .T = R ◦ S is the required .0 ≤ θ < 2π and letting .Rz = e   Möbius transformation. Corollary 6.1.16 If .ζ1 and .ζ2 are distinct points in .B2 , then .ξ(ζ1 , ζ2 ) > 1. Proof It suffices to calculate .ξ(0, r) for .0 < r < 1 using (6.9). Clearly, .∞1 = 1, ∞2 = −1, and

.

ξ(0, r) =

.

1+r (0 − 1)(r − −1) = . (0 − −1)(r − 1) 1−r

Hence, .ξ(0, r) > 1 because .0 < r < 1.

(6.11)  

Corollary 6.1.17 The intersection of 2 distinct h-lines is either empty or a single point and is never a tangency. Proof By the proposition, it can be assumed that one of the h-lines is the open interval .(−1, 1) of the real axis. If the other line is also a Euclidean line through the

156

6 Hyperbolic Geometry

origin intersected with .B2 , the intersection of the 2 lines is the origin. If the other line has the form .x 2 + y 2 + ax + by + 1 = 0, then the solutions of the quadratic equation .x 2 + ax + 1 = 0 are .

−a ±

√ a2 − 4 . 2

There are 2 solutions if and only if .a 2 > 4. When .a < 0, one of the 2 solutions is obviously greater than 1. Since the product of the roots is 1, the other root must lie between 0 and 1. A similar argument works for .a > 0. Two h-lines cannot be tangent at an intersection by Proposition 6.1.4.   Corollary 6.1.18 If .ζ1 and .ζ2 are points in .B2 , then there exists .T ∈ G such that 2 2 .T ζ1 = ζ2 and .B is an orbit of the continuous action of .G on .D . Exercise 6.1.19 Prove Corollary 6.1.18. Now set   dh (ζ1 , ζ2 ) = ln ξ(ζ1 , ζ2 ) .

.

(6.12)

It follows from Corollary 6.1.16 that .dh (ζ1 , ζ2 ) ≥ 0 with equality if and only if ζ1 = ζ2 . For all .T ∈ G and all .ζ1 , ζ2 ∈ B2 , we have

.

dh (T ζ1 , T ζ2 ) = dh (ζ1 , ζ2 ),

.

(6.13)

making .dh a promising candidate for a hyperbolic metric on .B2 . Proposition 6.1.20 If r is a real number such that .0 < r < 1, then there exists S in G such that .S0 = r and .Sr = 0.

.

Proof Let Sz =

.

iz − ir . irz − i

Then S is in .G because .aa − cc = 1 − r 2 > 0. Clearly, .S0 = r and .Sr = 0.

(6.14)  

Corollary 6.1.21 If .ζ1 and .ζ2 are in .B2 , then .dh (ζ1 , ζ2 ) = dh (ζ2 , ζ1 ). Proof It follows from Proposition 6.1.15 and equation (6.13) that .dh (ζ1 , ζ2 ) = dh (0, r) and .dh (ζ2 , ζ1 ) = dh (r, 0) for some r such that .0 < r < 1. Thus it suffices to prove the corollary for .ζ1 = 0 and .ζ2 = r > 0. Now the proposition implies that .dh (0, r) = dh (S0, Sr) = dh (r, 0).   Corollary 6.1.22 If .ζ1 and .ζ2 are distinct points in .B2 , then there exists .R ∈ G such that .Rζ1 = ζ2 and .Rζ2 = ζ1 . Moreover, R maps the h-line determined by .ζ1 and .ζ2 to itself.

6.1 Poincaré Disk Model

157

Proof Let S and T be the Möbius transformations given by Propositions 6.1.20 and 6.1.15, respectively. Then set .R = T −1 ◦ S ◦ T .   Proposition 6.1.23 If .ζ1 , .ζ2 , and .ζ3 are distinct points on an h-line with .ζ2 lying between .ζ1 and .ζ3 , then dh (ζ1 , ζ3 ) = dh (ζ1 , ζ2 ) + dh (ζ2 , ζ3 ).

.

Proof Let T be an element of .G such that .T ζ1 = 0 and .T ζ3 = r > 0 (Proposition 6.1.15). It follows that .T ζ2 = r  such that .0 < r  < r because T maps h-lines onto h-lines homeomorphically. Using the defining equation (6.12), 

1+r .dh (0, r) = ln 1−r





1 + r and dh (0, r ) = ln 1 − r 

 (6.15)

follow from equation (6.11), and dh (r  , r) = ln

.



(r  − 1)(r − −1) (r  − −1)(r − 1)



 = ln

(1 − r  )(1 + r) (1 + r  )(1 − r)



follows from equation (6.9). Consequently, the simple logarithmic calculation (Exercise 6.1.24) completes the proof.   Exercise 6.1.24 Verify that .dh (0, r  ) + dh (r  , r) = dh (0, r) in the proof of Proposition 6.1.23. Since .dh (0, Rz) = dh (0, z) for every rotation .Rz = eiθ z, every Euclidean circle of radius r less than 1 centered at 0 is a hyperbolic circle of radius  .

ln

1+r 1−r



= 2 tanh−1 (r).

(6.16)

This observation provides a formula for .dh that is independent of .∞1 and .∞2 . Proposition 6.1.25 For all w and z in .B2 ,  dh (w, z) = ln

.

|1 − wz| + |w − z| |1 − wz| − |w − z|



   w−z   = 2 tanh−1  1 − zw 

(6.17)

and .dh (w, z) = dh (w, z). Proof Because Möbius transformations will be used in the proof, the formula will be derived using .ζ1 and .ζ2 instead of z and w. Letting S be the Möbius transformation given by equation (6.10) such that .Sζ1 = 0, the point .Sζ2 lies on the Euclidean circle of radius      ζ2 − ζ1   ζ2 − ζ1      .|Sζ2 | =  −ζ ζ + 1  =  1 − ζ ζ  < 1. 1 2 1 2

158

6 Hyperbolic Geometry

Therefore, 

  1 + (ζ2 − ζ1 )(1 − ζ 1 ζ2 )−1    , .dh (ζ1 , ζ2 ) = ln 1 − (ζ2 − ζ1 )(1 − ζ 1 ζ2 )−1  which can be written as  dh (ζ1 , ζ2 ) = ln

.

|1 − ζ 1 ζ2 | + |ζ2 − ζ1 |



|1 − ζ 1 ζ2 | − |ζ2 − ζ1 |

or as dh (ζ1 , ζ2 ) = 2 tanh

.

−1

  ζ2 − ζ1  1 − ζ ζ

1 2

   . 

Now replace .ζ1 and .ζ2 with z and w, respectively, and use .dh (w, z) = dh (z, w). The second equation follows from the properties of the function .z → z.   Proposition 6.1.26 Alternatively,  dh (z, w) = cosh

.

−1

 2|z − w|2   . 1+  1 − |z|2 1 − |w|2

(6.18)

Proof Let .dh (s, w) be the function defined by the right side of equation (6.18). The standard logarithmic formula for .cosh−1 is .

   cosh−1 x = ln x + x 2 − 1 .

Given .r ∈ R such that .0 < r < 1, ⎛

⎞  2   2 2r ⎜ ⎟   + 1+   − 1⎠ .dh (r, 0) = ln ⎝1 +  1 − r2 1 − r2 2r 2

 = ln

1+r 1−r



= dh (r, 0). Therefore, by Proposition 6.1.15, it suffices to show that .

|T z − T w|2 |z − w|2   =   1 − |T z|2 1 − |T w|2 1 − |z|2 1 − |w|2

for all .T ∈ G.

(6.19)

6.1 Poincaré Disk Model

159

Let Tz =

.

az + c cz + a

be an element of .G such that .aa − cc = 1. The calculations in Exercise 6.1.27 complete the proof.

 

Exercise 6.1.27 Complete the proof of Proposition 6.1.25 by using the formula for T to substitute into the left-hand side of equation (6.19). Theorem 6.1.28 The function .dh (w, z) defines a metric on .B2 equivalent to the usual metric .d(w, z) = |w − z| on .B2 . Proof All the properties of a metric have already been established except the triangle inequality. By Proposition 6.1.15, it suffices to prove that .dh (w, r) ≤ dh (w, 0) + dh (0, r) for .w ∈ B2 . When .r = 0 or .w = 0, the result is trivial. The proof of the triangle inequality uses Proposition 6.1.26 and follows Problem 12–8 on p. 289 in [42]. Using equation (6.18), it follows that  .

sinh dh (z, 0) =



=  = =

cosh2 dh (z, 0) − 1 2  2|z|2 1+ −1 1 − |z|2 4|z|2 (1 − |z|2 )2

2|z| . 1 − |z|2

Because .cosh(x + y) = cosh x cosh y + sinh x sinh y, it suffices to show that .

cosh dh (z, w) ≤ cosh dh (z, 0) cosh dh (0, w) + sinh dh (z, 0) sinh dh (0, w),

which is the same as the following: 

 2|z − w|2   . 1+  1 − |z|2 1 − |w|2       2|w|2 2|z| 2|w| 2|z|2 1 + + . ≤ 1+ 1 − |z|2 1 − |w|2 1 − |z|2 1 − |w|2 Using reversible steps, we will reduce this inequality to one that is obviously true. First multiplying by .(1 − |z|2 )(1 − |w|2 ) yields

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6 Hyperbolic Geometry

.

(1 − |z|2 )(1 − |w|2 ) + 2|z − w|2 ≤ (1 − |z|2 + 2|z|2 )(1 − |w|2 + 2|w|2 ) + 4|z| |w|.

Now simplifying and expanding produce the clearly valid inequality (|z − w|)2 ≤ (|z| + |w|)2 .

.

Since   1+r .{z : |z| < r} = z : dh (z, 0) < ln 1−r for .0 < r < 1, a subset of .B2 is a neighborhood of 0 in the usual metric topology for .B2 if and only if it is a neighborhood of 0 in the .dh metric topology on .B2 . Surjective isometries of a metric space are always homeomorphisms. Thus every element of .G is a homeomorphism of .B2 onto itself for the .dh metric topology on 2 2 .B as well as for the usual metric topology on .B . Using the Möbius transformation S defined by equation (6.10), it is readily shown that the two topologies must be equal.   The metric .dh on .B2 is called the hyperbolic metric. The interplay between the Euclidean and hyperbolic metric is often useful. Proposition 6.1.29 Let .ζn be a sequence in .B2 such that .limn→∞ |ζn | = 1. If .ρ > 0, then

sup{z − w : dh (z, ζn ) ≤ ρ and dh (w, ζn ) ≤ ρ} = 0, . lim n→∞

and .

lim d(S1 , {z : dh (z, ζn ) ≤ ρ}) = 0.

n→∞

Proof Given .0 < r < 1, the h-radius of .{z : |z| = r} is .ln[(1 + r)/(1 − r)], and clearly, there exists .1 > r  > r such that .

ln

1 + r 1+r + ρ. > ln  1−r 1−r

Then .|ζn | > r  and .{z : dh (z, ζn ) ≤ ρ} ⊂ {z : r < |z| < 1} for large n because when 2 h-circles intersect, the distance between their centers must be less than the sum of their radii. Since the Euclidean diameter of a circle lying in the annular region .{z : r < |z| < 1} is less than .1 − r, it follows that

6.2 Properties of Rigid Motions .

161

sup{|z − w| : dh (z, ζn ) ≤ ρ and dh (z, ζn ) ≤ ρ} < 1 − r

for large n. To complete the proof of the first limit, just let r go to 1. Similarly, the second limit follows from .d(S1 , {z : dh (z, ζn ) ≤ ρ}) ≤ 1 − r. Corollary 6.1.30 Let .ζn be a sequence .

in .B2

converging to .w ∈

S1 .

 

If .ρ > 0, then

lim d(w, {z : dh (z, ζn ) ≤ ρ}) = 0.

n→∞

For a more extensive introduction to hyperbolic geometry from a differentiable geometric point of view, see Chapter 1 of [38].

6.2 Properties of Rigid Motions We now know that all Möbius transformations in .G are rigid motions of the Poincaré disk model, that is, they are conformal isometries of .B2 with the hyperbolic metric 2 2 .dh defined by equation (6.17). They also map .D , a natural compactification of .B , onto itself. Understanding the dynamics of subgroups of .G on .S1 , the boundary of .D2 , will be a critical ingredient in using them to study flows on surfaces. The first step is to use the fixed points and their locations to classify Möbius transformations in .G and describe the dynamics of individual rigid motions when iterated. Assuming .T = ι, the fixed point equation .T z = z for a Möbius transformation T is a quadratic equation that leads to a natural decomposition of Möbius transformations based on its discriminant into four types: hyperbolic, elliptic, parabolic, and loxodromic. We will limit our attention to .T ∈ G for simplicity. Assuming .T = TA  the equation .T z = z quickly simplifies to with .A ∈ G, 0 = cz2 + (a − a)z − c,

.

(6.20)

and the quadratic formula provides the solution(s)

.

 (a − a)2 + 4cc = 2c  a − a ± (a + a)2 − 4 2c

a−a±

(6.21)

 is defined by .Tr [A] = a + a, which is because .aa − cc = 1. The trace of .A ∈ G twice the real part of a. It follows from (6.21) that the discriminant of the quadratic  If .A = ±B, then equation (6.20) is .Tr [A]2 − 4, which is real for all .A ∈ G. .Tr [A] = ±Tr [B] and .|Tr [A]| = |Tr [B]|. Thus .|Tr [A]| is an invariant of .TA when  because .TA = TB if and only if .A = ±B. A is in .G,

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6 Hyperbolic Geometry

 Given .T ∈ G, the trace of T is defined by .Tr [T ] = |Tr [A]| when .A ∈ G and .T = TA . A similar analysis works for all Möbius transformations. When the discriminant is not real, the transformation is said to be loxodromic. There are no loxodromic transformations in .G. A Möbius transformation T in .G is said to be hyperbolic, parabolic, or elliptic depending on whether .Tr [T ] > 2, .Tr [T ] = 2, or .0 ≤ Tr [T ] < 2. It follows from equation (6.5) that .Tr [T −1 ] = Tr [T ] for .T ∈ G. So T and .T −1 are always of the same type. The Möbius  transformation S defined by equation (6.10) is hyperbolic because .Tr [S] = 2/ 1 − |ζ1 |2 > 2. Similarly, S defined by equation (6.14) is elliptic √ √ because .Tr [S] = i/ 1 − r 2 − i/ 1 − r 2 = 0. Notice that in both cases we were careful to calculate the trace of a matrix with determinant 1. The simplest general parabolic Möbius transformation is a translation .Tβ z = z + β for .β ∈ R because the only fixed point is infinity, and it maps .U2 onto itself. It is not in .G, but we can use it to construct a parabolic element of .G. Using matrix multiplication, Sβ = −1 T z =

.

(1 + iβ/2)z − iβ/2 ∈G iβz/2 + 1 − iβ/2

(6.22)

and .Tr [Sβ ] = 2 for all .β ∈ R. (This notation is slightly ambiguous, but capital Latin subscripts as in .TA will always be matrices and lower case Greek subscripts as in .Sβ will always be real parameters.) The only fixed point of .Sβ is 1. The circles in 2 1 .D passing through 1, which are necessarily tangent to .S at 1, are mapped by . to the horizontal lines of .C above the real axis in .C. Hence, they are invariant circles of .Sβ , called horocycles. Proposition 6.2.1 If S and T are in .G, then the following hold: (a) T is hyperbolic if and only if .ST S −1 is hyperbolic. (b) T is parabolic if and only if .ST S −1 is parabolic. (c) T is elliptic if and only if .ST S −1 is elliptic.  such that .S = TA and .T = TB . Then .ST S −1 = Proof There exist A and B in .G  It follows that TABA−1 and .ABA−1 is in .G. Tr [ST S −1 ] = |Tr [ABA−1 ]| = |Tr [B]| = Tr [T ]

.

to complete the proof. Proposition 6.2.1 holds for the full group of Möbius transformations. Proposition 6.2.2 If T is an element of .G, then the following hold: (a) T is hyperbolic if and only if T has two fixed points, and they lie in .S1 . (b) T is parabolic if and only if T has one fixed point, and it lies in .S1 . (c) T is elliptic if and only if T has one fixed point in .B2 .

 

6.2 Properties of Rigid Motions

163

Proof Because the three fixed point conditions are mutually exclusive, it suffices to prove the “only if” part of each statement. We can assume that .T = TA for some  and use the fixed point formula from (6.21). .A ∈ G  If T is hyperbolic, then . (a + a)2 − 4 is real and .a − a is a multiple of i. Hence, the square of the absolute value of the numerator of equation (6.21) is .





± (a + a)2 − 4 + (a − a) ± (a + a)2 − 4 − (a − a) = (a + a)2 − 4 − (a − a)2 = 4(aa − 1).

Therefore, the square of the absolute value of either fixed point of a hyperbolic Möbius transformation equals .

4(aa − 1) 4(aa − 1) = = 1, 4cc 4(aa − 1)

and both fixed points lie in .S1 . If T is parabolic, then its only fixed point is .(a − a)/2c, and the square of its absolute value is .

4aa − (a + a)2 (a − a)(a − a) = 4cc 4aa − 4 4aa − 4 = 4aa − 4 = 1.

Suppose T is elliptic. If .c = 0, then T is a rotation about 0 with fixed points 0 and .∞, meeting the required conditions. We can now assume that .c = 0. Hence, .aa > 1 and .|a + a| < 2. It will be convenient to let .a = u + iv with u and v real so that .a + a = 2u  and√.a − a = 2vi. Then the numerator of (6.21) is a multiple of i equaling . 2v ± 4 − 4u2 i, and the square of the absolute value of the fixed points is √ √  2 2v ± 4 − 4u2 4v 2 ± 4v 4 − 4u2 + 4 − 4u2 = . 4cc 4(u2 + v 2 − 1) √ v 2 ± 2v 1 − u2 + 1 − u2 . = u2 + v 2 − 1 Replacing A by .−A if needed, we can assume that .v ≥ 0 and that choosing the minus sign yields the absolute value squared of the fixed point closest to 0. Suppose

164

6 Hyperbolic Geometry

√ v 2 − 2v 1 − u2 + 1 − u2 ≥1 . u2 + v 2 − 1 or equivalently  v 2 − 2v 1 − u2 + 1 − u2 ≥ u2 + v 2 − 1

.

because the denominator is positive. The latter inequality simplifies to  2(1 − u2 ) ≥ 2v 1 − u2

.

√ or more simply . 1 − u2 ≥ v, which implies that .1 ≥ u2 + v 2 = aa > 1. This contradiction proves that T has a fixed point .ζ in .B2 . By Corollary 6.1.18, there exists .S ∈ G such that .Sζ = 0. Hence, .ST S −1 is an element of .G such that .ST S −1 0 = 0, a rotation about the origin by Corollary 6.1.8.   The content of the first sentence of the previous paragraph stands alone as the proof of the following useful corollary: Corollary 6.2.3 If T is an elliptic Möbius transformation in .G, then there exists S ∈ G such that .ST S −1 is a rotation about 0.

.

The hyperbolic Möbius transformations in .G will be particularly important, and we need to understand the geometry of their invariant sets and their limiting behavior. The two fixed points of a hyperbolic Möbius transformation T in .G lie in .S1 and determine an h-line L called the axis of T . Because T is conformal and .T S1 = S1 , it follows that .T L = L. We begin by studying the hyperbolic transformations in .G with fixed points 1 and .−1. Their axis is the open interval .(−1, 1), which will be denoted by .LR . The function .B → C −1 BC used in the proof of Proposition 6.1.7 is obviously  proving that an automorphism of .SL(2, C) onto itself that maps .SL(2, R) onto .G, they are isomorphic topological groups. In particular, if H is a closed subgroup of −1 H C is a closed subgroup of .G  isomorphic to .R. .SL(2, R) isomorphic to .R, then .C This observation will be used to prove the first proposition about hyperbolic Möbius transformations.  defined by : R → G Proposition 6.2.4 The function .F (ρ) = .F



cosh(ρ/2) sinh(ρ/2) sinh(ρ/2) cosh(ρ/2)



 Furthermore, the function is an isomorphism of .R onto a closed subgroup of .G. .ρ → TF (ρ) = Tρ is an isomorphism of .R onto a closed subgroup of .G. Proof Using C and .C −1 from the proof of Proposition 6.1.7,

6.2 Properties of Rigid Motions

(ρ)C −1 = .C F



 =  =

165

i i −1 1



cosh(ρ/2) sinh(ρ/2) sinh(ρ/2) cosh(ρ/2)



−i/2 −1/2 −i/2 1/2



cosh(ρ/2) + sinh(ρ/2) 0 0 cosh(ρ/2) − sinh(ρ/2) eρ/2 0 0 e−ρ/2



 ,

or C −1

.



eρ/2 0 0 e−ρ/2



(ρ). C=F

Since  ψ(ρ) =

.

eρ/2 0 0 e−ρ/2



(ρ) = is clearly an isomorphism of .R onto a closed subgroup of .SL(2, R) and .F −1   C ψ(ρ)C, the function .F is an isomorphism of .R onto a closed subgroup of .G. For the second part, note that finite sheeted coverings are closed functions and (ρ) = −F (ρ  ) has no solution. .F   Corollary 6.2.5 The function defined by .(ρ, w) → Tρ w is a flow on .D2 . Proof Proposition 6.1.14 implies that .(ρ, w) → TF(ρ) w = Tρ w is a flow on .D2 (σ + τ ) = F (σ )F (τ ), which implies that .TF(σ +τ ) = TF(σ ) TF(τ ) . because .F   The Möbius transformation Tρ z = TF(ρ) z =

.

cosh(ρ/2)z + sinh(ρ/2) sinh(ρ/2)z + cosh(ρ/2)

(6.23)

is clearly a hyperbolic transformation such that .Tρ 1 = 1, .Tρ (−1) = −1, and Tρ 0 = tanh(ρ/2). So the axis of .Tρ is .LR when .ρ = 0, and .dh (0, Tρ 0) = |ρ| by equation (6.17). Since the range of .tanh is the open interval .(−1, 1), the function .ρ → Tρ 0 = tanh(ρ/2) parameterizes .LR like an axis by using plus and minus the hyperbolic distance from 0. It follows that .LR is the orbit of 0 for the .Tρ -flow. Using Proposition 6.1.23 with 0 as one of the 3 points, it is an exercise (Exercise 6.2.6) to show that when .σ < τ , then .

dh (Tτ 0, Tσ 0) = τ − σ.

.

(6.24)

Exercise 6.2.6 Verify that .dh (Tτ 0, Tσ 0) = τ −σ. in the proof of Proposition 6.1.14.

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6 Hyperbolic Geometry

Proposition 6.2.7 If .H is the subgroup of all .T ∈ G such that .T 1 = 1 and T (−1) = −1, then

.

 H = {Tρ : ρ ∈ R} = {TA : A ∈ SL(2, R) ∩ G}

.

is a closed subgroup of .G and .ρ → Tρ is an isomorphism of .R onto .H. Proof If T is in .H, then it is hyperbolic or equals .ι by Proposition 6.2.2. If .T ∈ H is hyperbolic, then its axis is .LR and .T = Tρ for some .ρ = 0. Thus .H ⊂ {Tρ : ρ ∈ R}.  It follows from the definition of .Tρ that .{Tρ : ρ ∈ R} ⊂ {TA : A ∈ SL(2, R)∩ G}.  ⊂ H. So it remains to show that .{TA : A ∈ SL(2, R) ∩ G}  then .T R = R because A is in .SL(2, R) and If T is in .{TA : A ∈ SL(2, R) ∩ G}, 1 1  It follows that .T [−1, 1] = [−1, 1]. So .T {−1, 1} = .T S = S because A is in .G. {−1, 1}, and it suffices to eliminate the possibility that .T (−1) = 1. Because A is real, .T (−1) = 1 implies that .−a + c = −c + a and .a = c, a contradiction.   Proposition 6.2.8 Every hyperbolic Möbius transformation S in .G is conjugate in G to a unique .Tρ given by equation (6.23) with .ρ > 0.

.

Proof Let L be the axis of S, and let .ζ be an element of L. Then .Sζ = ζ because the only fixed points of S are in .S1 . Let .ρ = dh (ζ, Sζ ). By Proposition 6.1.15, there exists .R ∈ G such that .Rζ = 0 and .RSζ = r, a positive real number. Clearly, −1 is a hyperbolic element of .G with .RL = LR and .dh (0, r) = ρ. Observe that .RSR −1 fixed points .−1 and 1 such that .RSR 0 = r. Since .Tρ has the same properties, −1 = T or .S = R −1 T R. Obviously, .ρ is unique. .RSR   ρ ρ Essentially, the same argument just used to prove Proposition 6.2.8 can also be used to prove the following: Proposition 6.2.9 If .ζ1 and .ζ2 are distinct points in .B2 , then there exists a unique hyperbolic Möbius transformation .T ∈ G such that .T ζ1 = ζ2 and the axis of T is the h-line determined by .ζ1 and .ζ2 . Exercise 6.2.10 Prove Proposition 6.2.9. Proposition 6.2.11 If S is a hyperbolic Möbius transformation in .G with axis L, then .dh (z, Sz) = dh (w, Sw) > 0 for all z and w in L. Proof By Proposition 6.2.8, there exists .R ∈ G such that .S = R −1 Tρ R for a unique positive .ρ > 0. Since R is an isometry for the hyperbolic metric, it suffices to prove the proposition for .Tρ with .ρ > 0. From the construction of .Tρ , we know that .dh (0, Tρ 0) = ρ, and given .w ∈ LR , there exists a unique .δ ∈ R such that .w = Tδ 0. Then .Tρ w = Tρ Tδ 0 = Tρ+δ 0 (Corollary 6.2.5). Then apply equation (6.24) with .σ = δ and .τ = δ + ρ to show   that .dh (w, Tρ w) = δ + ρ − δ = ρ for all .w ∈ LR . Note that there can be at most one h-line perpendicular to another h-line at a specified point because a point and a direction uniquely determine an h-line

6.2 Properties of Rigid Motions

167

(Proposition 6.1.4). It will be convenient to let .LI = {iy ∈ C : −1 < y < 1}, which is clearly the unique h-line perpendicular to .LR at 0. Now the family of circles passing through both .−1 and 1 in .C enters the picture. So their centers are on the y-axis. Analytically, they are the circles of the form 2 2 .x + y + by − 1 = 0, which is not hard to demonstrate. Proposition 6.2.12 Given an h-line L with its points at infinity denoted by a and b, the following hold: (a) The line L divides .B2 into two open .h−convex sets. (b) If .ζ lies on L, then there exists an h-line .L⊥ perpendicular to L at .ζ . (c) If .ζ is in .B2 \ L, then there exists an h-line .L⊥ passing through .ζ and perpendicular to L. (d) If .ζ lies on .L⊥ perpendicular to L and .ζ  = L⊥ ∩ L, then dh (ζ, L) ≡ inf{dh (ζ, z) : z ∈ L} = dh (ζ, ζ  ).

.

(6.25)

(e) If C is a circle passing through a and b and .C = S1 , then .dh (z, L) = dh (w, L) for all z and w in .E = C ∩ B2 . Proof By Proposition 6.1.15, there exists .T ∈ G such that .T L = LR . So it suffices to prove the results for the h-line .LR because transformations in .G are conformal isometries. This will be used to prove all 5 parts of the proposition. Part (a): It is clear from Euclidean geometry that .LR divides .B2 into two open connected sets. Let .ζ1 and .ζ2 be 2 distinct points in .U2 ∩ B2 , that is, in the upper component of .B2 \ LR . The h-line segment joining .ζ1 and .ζ2 can intersect .LR at most once and cannot do so tangentially (Corollary 6.1.17). If it does intersect .LR , then .ζ1 and .ζ2 are in different components of .B2 \ LR , a contradiction. Thus .U2 ∩ B2 is h-convex. Part (b): Given .0 < |r| < 1, there exists .ρ = 0 such that .tanh(ρ/2) = r. Hence, the h-line .Tρ LI is perpendicular to .LR at r. Part (c): Given .ζ is in .B2 \ LR , let C be the unique circle determined by .−1, 1, and .ζ and set .E = C ∩ B2 , which is an open arc of C. There exists .y ∈ R such that .0 < |y| < 1 and .{iy} = E ∩ LI . Then there exists a unique Möbius transformation T such that .T iy = ζ , .T 1 = 1, and .T (−1) = −1. Then .T = Tρ for some .ρ = 0 by Proposition 6.2.7 because T is in .H. Clearly, .L⊥ = Tρ LI . Part (d): First consider iy on .LI with .y = 0, and show that .dh (iy, 0) ≤ dh (iy, r) for all .r ∈ LR . A slightly tedious calculation using equation (6.17) (natural logarithm version) shows that assuming .dh (iy, 0) > dh (iy, r) for some .r ∈ LR leads to a contradiction. Therefore, .dh (iy, LR ) ≡ inf{dh (iy, z) : z ∈ LR } = dh (iy, 0), and the usual hyperbolic metric space distance from iy to .LR is the hyperbolic length of the h-line segment perpendicular to .LR and passing through iy. Given .ζ ∈ B2 such that .ζ is not in .LR , there exists an h-line .L⊥ passing through .ζ and intersecting .LR perpendicularly by part (c). Moreover, the proof of (c) shows that there exists .ρ ∈ R and .0 < |y| < 1 such that .Tρ LI = L⊥ , .Tρ 0 = L ∩ L⊥ , and

168

6 Hyperbolic Geometry

Tρ iy = ζ . The following calculation now completes part (d):

.

dh (ζ, ζ  ) = dh (Tρ iy, Tρ 0)

.

= dh (iy, 0) = inf{dh (iy, z) : z ∈ L} = inf{dh (Tρ iy, Tρ z) : z ∈ L} = inf{dh (ζ, w) : w ∈ L}. Part (e): Set .E = C ∩ B2 , so .E ∩ LI = iy for some y such that .0 < |y| < 1. It suffices to show that .dh (ζ, LR ) = dh (iy, LR ) for all .ζ in E. The proof of part (c) showed that there exists .Tρ such that .Tρ iy = ζ and that .Tρ LI is perpendicular to  .LR . Setting .ζ = Tρ LI ∩ LR , part (d) implies that dh (ζ, L) = dh (ζ, ζ  ) = dh (Tρ iy, Tρ 0) = dh (iy, 0) = dh (iy, L)

.

to complete the proof.

 

Let L be an h-line with points at infinity denoted by a and b. Given any circle C passing through a and b, the set .E = C ∩ B2 is naturally called an equidistant curve of L because .dh (z, L) is constant for all .z ∈ E. Given .r > 0, there exist 2 equidistant curves .E1 and .E2 on opposite sides of L such that .dh (z, L) = r for all .z ∈ Ej , .j = 1, 2. It follows from the proof of part (c) that the equidistant curves are also the .Tρ -orbits of the points in .B2 \ L. Suppose f and g are homeomorphisms of topological space X onto itself. By definition, the cascades .(X, f ) and .(X, g) are isomorphic if and only if there exists another homeomorphism h of X onto itself such that .f ◦ h = h ◦ g or equivalently −1 . (This is just a special case of the general definition of an isomorphic .f = h◦g ◦h continuous group action.) Consider T , an elliptic Möbius transformation in .G. By Corollary 6.2.3, there exists a rotation R about 0 and S in .G such that .R = ST S −1 and .T = S −1 RS. Then S is an isomorphism of .(D2 , T ) onto .(D2 , R) where .Rz = e2π ia and .S −1 is its inverse. From Section 1.1, we know that .(D2 , R) is either a periodic transformation group or the circles centered at 0 are all minimal sets. The h-circles centered at 0 are the same as the Euclidean circles of radius less than 1 centered at 0. Consequently, every h-circle centered at the fixed point of T equals .S −1 C for some h-circle C centered at 0 and is T -invariant because .T S −1 C = S −1 RSS −1 C = S −1 RC = S −1 C. Since S is an isomorphism of .(D2 , T ) onto 2 2 .(D , R), it follows that either .(D , T ) is a periodic transformation group or the hcircles centered at the fixed point of T in .D2 are all minimal sets. Thus we have a complete description of the dynamical behavior on .D2 of every elliptic Möbius transformation in .G, and the following theorem: Theorem 6.2.13 Given an elliptic Möbius transformation T in .G with a fixed point a in .B2 , the cascade .(D2 , T ) has the following properties:

6.2 Properties of Rigid Motions

169

(a) Every h-circle centered at the fixed point a is T -invariant as is .S1 . (b) Either .(D2 , T ) is a periodic transformation group or the h-circles centered at a are all minimal sets as is .S1 . The next step is to obtain similar theorem for a hyperbolic T in .G. Theorem 6.2.14 Given a hyperbolic Möbius transformation T in .G with axis L and fixed points a and b in .S1 , the cascade .(D2 , T ) has the following properties: (a) The equidistant curves, L, and the 2 arcs of .S1 \ {a, b} are invariant sets. (b) When z is not a fixed point, the biinfinite sequence .T n z with .n ∈ Z is a monotone sequence contained in an invariant set. (c) Either .limn→∞ T n z = a or .limn→−∞ T n z = b for all z in .D2 \ {a, b} or n n 2 .limn→∞ T z = b and .limn→−∞ T z = a for all z in .D \ {a, b}. Proof By Proposition 6.2.8, there exists .Q ∈ G such that .T = QTρ Q−1 for some 2 2 .σ > 0. So the cascade .(D , T ) is isomorphic to .(D , Tσ ) for some .σ > 0, and it will suffice to prove the theorem for .Tσ . Since .Tσ is part of the flow .(ρ, z) → Tρ z on .D2 whose orbits are precisely the fixed points .{1, −1}, and the sets listed in part (a), the first part is obvious and the others follow because .(ρ, z) → Tρ z is a homeomorphism of .R onto the orbit of z when z is not a fixed point.   Proposition 6.2.15 Every parabolic Möbius transformation S in .G is conjugate to a unique .Sβ given by equation (6.22) with .β = 0. Proof By replacing S with .RSR −1 for a suitable rotation R about 0, it can be  where assumed that 1 is the fixed point of S. Of course, .S = TA for some A in .G, A has the usual form (equation (6.6)). Because S is parabolic, .a + a = ±2, and the real part of a is .±1. Replacing A by .−A, if necessary, we can assume that the real part of a is 1 and .a = 1 + iβ/2 for some nonzero real number .β. It follows from equation (6.21) that the only fixed point of S is .(a − a)/2c.   Solving .(a − a)/2c = 1 for c shows that .c = iβ/2, the imaginary part of a. Beginning with a subgroup of .SL(2, R) and following the discussion of the hyperbolic transformations in .G, the function .β → Sβ is also an isomorphism of 2 1 .R into .G and .(β, w) → Sβ w is a flow on .D . Other than the fixed point and .S \ {1}, the orbits of this flow are of the form .C \ {1}, where C is a circle inside .D2 and tangent to .S1 at 1. Furthermore, .α(w) = {1} = ω(w) for all .w ∈ D2 . Exercise 6.2.16 State and prove an analogous result to Theorem 6.2.14 for parabolic elements of .G. Given .T ∈ G, it is now clear that if .T = ι and .T n = ι for some .n ≥ 2, then T is an elliptic transformation. Since the 3 types of Möbius transformations along with .ι decompose .G into 4 disjoint sets, one naturally observes the following fact: Proposition 6.2.17 Given .T ∈ G and .|n| ≥ 2, T is hyperbolic {parabolic} {elliptic} if and only if .T n is hyperbolic {parabolic} {elliptic or .ι }.

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6 Hyperbolic Geometry

Proposition 6.2.18 Let T be a hyperbolic transformation in .G. A Möbius transformation S in .G commutes with T if and only if it is hyperbolic with the same fixed points as T . Proof Let R be an element of .G. Since S and T commute if and only if .RSR −1 and .RT R −1 commute, we can assume without loss of generality that .T = Tρ (Proposition 6.2.8) and hence .T = TB , where  B=

.

αγ γ α

 ∈ SL(2, R)

(Proposition 6.2.7). Then .|α| > 1 because .Tr [T ] > 2. It follows that .γ = 0 because α 2 − γ 2 = 1.  given by (6.6) such that .S = TA . If S and T As usual, there exists .A ∈ G commute, then .TAB = TBA and either .AB = BA or .AB = −BA. The latter implies that .ABA−1 = −B and .Tr [B] = −Tr [B], which is impossible because .Tr [B] = 0. Thus A and B commute, and .

 .

ac ca



αγ γ α



 =

αγ γ α



ac ca



or  .

aα + cγ aγ + cα cα + aγ cγ + aα



 =

aα + cγ cα + aγ aγ + cα cγ + aα

 .

It follows that .aα +cγ = aα +cγ and .aγ +cα = cα +aγ . Hence, .c = c and .a = a.  and .S = SA = Tσ is hyperbolic (Proposition 6.2.7). Therefore, A is in .SL(2, R) ∩ G   Thus the fixed points of both T and S are 1 and .−1. Exercise 6.2.19 Prove the converse of Proposition 6.2.18. For more results of this type, see [44], Chapter II, Section 9F. Proposition 6.1.25 shows that .z → z is an isometry of .dh . One can show that .z → z preserves angles but reverses the direction of rotation from one side to the other. Thus it is a rigid motion not in .G, and we need to construct a larger group of rigid motions of the Poincaré disk model that contains .z → z. → G  defined by The function . : G  (A) = 

.

ac ca



 =

ac ca

 =A

 onto itself such that .2 = ι so that .{ι, } is is a continuous automorphism of .G    isomorphic to .Z2 . Let .G = G × Z2 . Then .(A, μ) ∗ (B, ν) = (A μ(B), μ + ν)  , which is called the semi-direct product and written defines a group structure on .G    .G = G  Z2 . The identity is .(I, 0) and

6.2 Properties of Rigid Motions

171

 (A, μ)−1 =

(A−1 , 0)

.

(A

−1

, 1)

when μ = 0 when μ = 1

.

 = G  × Z2 , it With the discrete topology on .Z2 and the product topology on .G   is a topological group because .G  and .Z2 are topological is readily shown that .G  , however, is not connected, groups and . is continuous. The topological group .G  that is  is an index 2 closed connected subgroup of .G but .{(A, 0) : A ∈ G}   So .G  has 2 components. We will routinely identify .(A, 0) and isomorphic to .G. A. (See pp. 245–247 in [59] for a parallel Euclidean result.) Now set  TA z when μ = 0 .T(A,μ) z = , (6.26) TA z when μ = 1  to .HB2 , the and verify that .(A, μ) → T(A,μ) is an algebraic homomorphism of .G 2 group of homeomorphisms of .B onto itself. The kernel of .(A, μ) → T(A,μ) is K  = {(I, 0), (−I, 0)},

.

and    and μ ∈ Z2 G  = T(A,μ) : A ∈ G

.

is a subgroup of .HB2 . Note that .G ⊂ G  , and .G  includes .T(I,1) z = z as desired. In fact, .G  is the smallest subgroup of .HB2 containing .G and .z → z. Using T(A,μ) T(B,ν) = T(A,μ)∗(B,ν)

.

is helpful in calculating compositions of functions in .G  . The ideas used to prove Corollaries 6.1.12 and 6.1.13 and Proposition 6.1.14 can also be used to prove the next two propositions. Proposition 6.2.20 If .G  is given, the quotient topology from .π(A, μ) = T(A,μ)  onto .G  , then the following hold: mapping .G (a) .π is a two-to-one function. (b) Every point of .G  is contained in an open connected subset that is evenly covered by .π . (c) .G  is a 3-dimensional manifold.  /K  . (d) .G  is a topological group isomorphic to .G  (e) .G is an index 2 subgroup of .G . (f) .G is the component of the identity of .G  . Proposition 6.2.21 Both .(A, μ, z) → T(A,μ) z and .(T(A,μ) , z) → T(A,μ) z define  and .G  on .D2 . continuous actions of .G

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6 Hyperbolic Geometry

Exercise 6.2.22 Prove Propositions 6.2.20 and 6.2.21. Proposition 6.2.23 If S and T are in .G  \ G and .Sz = T z for 3 points in .C, then .S = T . Proof The hypothesis implies that .ST −1 has 3 fixed points. Since .ST −1 is in .G, the classical 3 point theorem for Möbius transformations implies that .ST −1 = ι and .S = T .   Although not needed here, .G  is the group of all isometries of the hyperbolic metric .dh on .B2 . The function .z → z restricted to .B2 is the reflection of the Poincaré disk model of hyperbolic geometry in the h-line .LR just as .(x, y) → (x, −y) is the reflection of the Euclidean plane in the x-axis. The restriction of .z → z to .B2 can also be used to construct the reflection in any h-line of .B2 . Given an h-line L, let .ζ1 and .ζ2 be 2 distinct points of L. By Proposition 6.1.15, there exist .T ∈ G such that .T ζ1 = 0 and .T ζ2 = r, a positive real number. Therefore, T L must be the unique  such that h-line determined by 0 and r, that is, .LR . Letting A be an element of .G .T = TA , the reflection in L is −1 T(A,0) ◦ T(I,1) ◦ T(A,0) = T(A−1 A,1) .

.

 (Proposition 6.2.4).  is an isomorphism of .R to .G Recall that .Tρ = TF(ρ) , where .F The model paddle motion will be .Tρ z with .ρ = 0. It reflects the point z in the axis .LR and then pushes it toward .±1 along an equidistant curve depending on whether .ρ is positive or negative. The fixed points of .Tρ z are 1 and .−1, and .Tρ z has an axis in the sense that it maps .LR onto itself. Observe that Tρ z = TF(ρ) z = T(F(ρ),0) T(I,1) z = T(F(ρ),1) z.

.

Thus .T(F(ρ),1) is the formal form of the model paddle motion and T(2F(ρ),1) = T(F(ρ),1)∗(F(ρ),1) = T(F(2ρ),0) = T2ρ

.

(ρ) is a real matrix. It follows by induction that because .F  T(nF(ρ),1) =

.

T(F(nρ),0)

when n is even

T(F(nρ),1)

when n is odd

.

 A paddle motion is a conjugate of .T(F(ρ),1) in .G  for some .ρ = 0. Given A in .G and .ρ = 0, −1 T(A,0) T(F(ρ),1) T(A,0) = T(A−1 ,0) T(F(ρ)A,1) = T(A−1 F(ρ)A,1)

.

6.3 Groups of Rigid Motions

173

−1 is a typical paddle motion with axis .T(A,0) LR . Letting .ρ = 0 is a typical reflection in an h-line. With care, it is also readily shown that −1 T(A,1) T(F(ρ),1) T(A,1) = T(A−1 F(ρ)A,1) .

.

A paddle motion S has the following properties: (a) It is in .G  \ G. (b) It has 2 fixed points, and they are in .S1 . (c) The unique h-line L determined by the fixed points is an invariant set of S called the axis of S. (d) S interchanges the 2 components of .B2 \ L. (e) .S 2 is a hyperbolic transformation in .G with the same axis as S. (f) For every hyperbolic transformation R in .G, there exists a paddle motion S such that .S 2 = R. Actually, every transformation T in .G  \ G is either a reflection in an h-line or a paddle motion. Using the notation .a = α + iβ and .c = γ + iδ, the fixed point equation .az + c = z(cz + a) simplifies to real and imaginary equations: γ = γ (x 2 + y 2 )

.

0 = δ(x 2 + y 2 ) − 2βx + 2αy + δ. Then T is a paddle motion or a reflection in an h-line depending on whether .γ = 0 or .γ = 0. The second equation specifies the invariant h-line of T .

6.3 Groups of Rigid Motions In this section, we will be primarily concerned with the dynamics of subgroups of the full group of rigid motions. Letting . be a subgroup of .G  , consider the (left) transformation group .(D2 , ). Clearly, .B2 and .S1 are automatically invariant sets of 2 1 .(D , ), and .S is the more interesting of the two. −

Proposition 6.3.1 If . is a subgroup of .G  , then .(z)− ∩ S1 = (w) ∩ S1 for all z and w in .B2 . −

Proof Let .Tn be a sequence in . such that .zn = Tn z converges to .ζ ∈ (z) ∩ S1 , and let C be an h-circle of h-radius r centered at z such that .dh (z, w) < r. Clearly, .Tn C is an h-circle of h-radius r centered at .zn , and .Tn w is inside .Tn C. Now apply Corollary 6.1.30.   −

Given a subgroup . of .G  , the limit set of . is defined by .() = (z) ∩ S1 for any z in .B2 . Note that .() is a closed .-invariant subset of .S1 . If .z ∈ S1 is a fixed point of a parabolic, hyperbolic or paddle motion in ., z is in .().

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6 Hyperbolic Geometry

When T is an element of .G  , the cyclic subgroup generated by T is the subgroup of .G  defined by .[T ] = {T n : n ∈ Z}. It follows from the discussion in Section 6.2 that the cardinality of .([T ]) is 2 when T is a hyperbolic transformation or a paddle motion, 1 when T is parabolic, and 0 when T is elliptic or a reflection in an h-line. Proposition 6.3.2 Let E be a finite subset of .S1 , let . be a subgroup of .G  , and let ζ be an element of .(). If U is an open interval of .S1 centered at .ζ , then there exists .T ∈  such that .T E ∩ (S1 \ U ) contains at most one point of T E.

.

Proof The result is trivial when E contains only one point. Assuming E contains at least two points, let .β be the least angle formed at the origin by distinct pairs of h-rays with points at infinity in E. Let .θ (z) be the angle subtended by h-rays starting at .z ∈ B2 with the endpoints of U at infinity. Observe that .limz→ζ θ (z) = 2π as shown in Figure 6.2. Because .ζ is in .(), there exists a sequence .Tn in . such that .Tn 0 converges to .ζ by Proposition 6.3.1. Thus there exists .Tn ∈  such that .2π > θ (Tn 0) > 2π − β, so −1 preserves angles, .T −1 (S1 \ U ) subtends an angle .2π − θ (Tn 0) = δ < β. Since .Tn n of size .δ < β at 0 and contains at most one point of E. Therefore, .Tn E ∩ (S1 \ U ) contains at most one point of .Tn E.   Corollary 6.3.3 Let . be a subgroup of .G  . If .ζ ∈ () is not a fixed point of ., − then .(ζ ) = (). Proof There is at least one other point .ζ  in .ζ . Let .E = {ζ, ζ  } and apply the proposition to any open interval centered at a point in .(). It follows that .ζ is   dense in .() because either .T ζ or .T ζ  is in .() ∩ U for some .T ∈ .

Fig. 6.2 .Tn−1 (S1 \ U ) subtends an angle of size .δ < β at 0 and contains at most one point of E

T −n (S1 /U ) δ

0

δ

U

ζ

T n0

6.3 Groups of Rigid Motions

175

Proposition 6.3.4 Let . be a subgroup of .G  . If .() is finite, then the cardinality of .() is 0, 1, or 2. Proof It was pointed out earlier in this section that the cyclic groups .[T ] have limit sets of cardinality 0, 1, or 2. Suppose .() is a finite set containing more than 2 elements. Let .ζ be in .(), and let U be an open interval of .S1 centered at .ζ such that .U ∩ () = {ζ }. By Proposition 6.3.2, there exists .T ∈  such that all but at most one point of .T () is in U . In particular, U contains more than one point of .(), a contradiction.   Theorem 6.3.5 Let . be a subgroup of .G  . If .() is infinite and contains no fixed points of ., then .() is a minimal subset of .(D2 , ) and is either .S1 or a Cantor subset in .S1 . Proof By Corollary 6.3.3, .() is a minimal set. And it is Cantor set or all of .S1 by Theorem 1.1.13.   A subgroup . of .G  is said to be mobile provided that . has no fixed points in that is, there does not exist .w ∈ S1 such that .Pw = . Since the subgroups of .G  that are needed to study flows on compact surfaces are all mobile, the assumption that . is mobile will be used to present a streamlined version of the discussion of the dynamics of the . action on .S1 in [31] beginning with 13.14.

1 .S ,

Corollary 6.3.6 Let . be a mobile subgroup of .G  such that .() is infinite. If . contains at least one hyperbolic Möbius transformation, then the set of endpoints of the axes of hyperbolic elements of . is dense in .(), and there are infinitely many distinct axes of hyperbolic elements of .. Proof Let T be a hyperbolic element of ., and let L be its axis with endpoints a and b. Obviously, a and b are in .(). For every S in ., the Möbius transformation −1 is a hyperbolic element of . with axis SL. Thus the orbit .a is a subset .ST S of .() consisting entirely of endpoints of the axes of hyperbolic elements of .. It is dense in .() by the theorem and hence infinite. It follows that there must be infinitely many axes.   Even if . is not mobile and there is a fixed point of . in .S1 , it is still true that () is either .S1 or a Cantor subset of .S1 , when .() is infinite ([31], Theorem 13.15). Furthermore, if .() is infinite, it contains at most one fixed point of . ([31], Theorem 13.18). These results, however, will not be needed.

.

Proposition 6.3.7 If . is a mobile subgroup of .G  containing at least one hyperbolic Möbius transformation and with .() infinite, then . contains two hyperbolic elements whose axes have no common endpoints. Proof Let L be an axis of .T ∈  with endpoints a and b. Suppose all the axes of the hyperbolic transformations in . have a as an endpoint. Because . is mobile, there exists .S ∈  such that .Sa = a. Then Sa is also the endpoint of infinitely many distinct axes of . (Corollary 6.3.6), and only one of them can have a as an endpoint.

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6 Hyperbolic Geometry

Fig. 6.3 The axes L and have no common endpoints

.R1 L2

b L1

R1 L2

R1 a

L2 a

c

Therefore, there must exist an axis .L1 of .R1 ∈  such that a is not an endpoint of L1 , and likewise an axis .L2 of .R2 ∈  such that b is not an endpoint of .L2 . If the theorem is false, then, as shown in Figure 6.3, .L1 must have b as one of its endpoints, .L2 must have a as one of its endpoints, and .L1 and .L2 must have a common endpoint c. Then .R1 L2 , the axis of .R1 R2 R1−1 , has endpoints c and .R1 a, which are unequal to a and b. This contradicts the assumption that there are no two hyperbolic elements whose axes have no common endpoints.  

.

Recall that a continuous function .f : [0, 1] → [0, 1] has a fixed point. Obviously, the fixed point is in the open interval .(0, 1) when .f ([0, 1]) ⊂ (0, 1). Since by definition an arc is homeomorphic to .[0, 1], an arc has the same fixed point property as .[0, 1]. Theorem 6.3.8 Let . be a mobile subgroup of .G  such that . contains a hyperbolic Möbius transformation and .() is infinite. If .U1 and .U2 are open intervals of .S1 such that .Ui ∩ () = φ for .i = 1, 2, then there exists a hyperbolic transformation in . with one endpoint of its axis in .U1 and the other in .U2 . −

−

Proof Without loss of generality, it can be assumed that .U1 ∩ U2 = φ. By Proposition 6.3.7, there exist hyperbolic transformations .T1 and .T2 in . with axes .L1 and .L2 , respectively, such that all four of their endpoints are distinct. Since .SL1 is the axis of .ST1 S −1 , it can be assumed that one endpoint of .L1 is in .U1 by the proof of Corollary 6.3.6. Then Proposition 6.2.14 implies that there exists .n ∈ Z such that both the endpoints .T1n L2 = Lˆ 2 are in .U1 and .Lˆ 2 is the axis of .R2 = T1n T2 T1−n . The above construction can be carried out with 1 and 2 interchanged, to show that there exist hyperbolic transformations .R1 and .R2 in . such that both endpoints of their axes are in .U2 and .U1 , respectively. (See Figure 6.4.) Let .Ci = S1 \ Ui , which is a closed interval of .S1 and hence an arc. Note that − − − − .U 1 ⊂ C2 and .U2 ⊂ C1 because .U1 ∩ U2 = φ. There exists a positive integer m

6.3 Groups of Rigid Motions

177

Fig. 6.4 The endpoints of the axes of .R1 and .R2 are in .U2 and .U1

Axis of R1

U2

Axis of R2

U1 such that R1±m C2 ⊂ U2 and R2±m C1 ⊂ U1 .

.

Set .S = R2m R1m . It follows that .SC2 ⊂ U1 and S has a fixed point in .U1 because −1 has a fixed point in .U . Since S has fixed points in the .U1 ⊂ C2 . Likewise, .S 2 disjoint intervals .U1 and .U2 of .S1 , it is hyperbolic (Proposition 6.2.2), and the endpoints of its axis are in .U1 and .U2 .   The hypothesis that . contains hyperbolic Möbius transformation is not really necessary because Theorem 13.20 in [31] implies that . contains hyperbolic elements if it is mobile. Since Theorem 6.3.8 will only be applied to groups consisting entirely of hyperbolic transformations and paddle motions except for .ι, there was no reason to include the more general result. Anytime .(X, G) is a transformation group, there is a natural transformation group .(X × X, G) defined by .g(x, y) = (gx, gy) or .(x, y)g = (xg, yg) for .g ∈ G depending on whether the continuous action of . on X is a left or right action. Theorem 6.3.9 If . is a mobile subgroup of .G  such that .() is infinite and . contains a hyperbolic Möbius transformation, then the transformation group .(() × (), ) has a residual set of transitive points. Proof Given open sets .U1 , .U2 , .V1 , and .V2 of .S1 having nonempty intersections with .(), it suffices to show that there exists .R ∈  such that   φ = R (U1 × U2 ) ∩ (() × ()) ∩ (V1 × V2 ) ∩ (() × ())

.

= [RU1 × RU2 ] ∩ [() × ()] ∩ [V1 × V2 ] = [RU1 ∩ () ∩ V1 ] × [RU2 ∩ () ∩ V2 ]

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6 Hyperbolic Geometry

by Proposition 1.1.3, which is equivalent to showing that RU1 ∩ () ∩ V1 = φ and RU2 ∩ () ∩ V2 = φ.

.

Without loss of generality, it can be assumed that .U1 , .U2 , .V1 , and .V2 are open intervals of .S1 having nonempty intersections with .(). By Theorem 6.3.8, there exists a hyperbolic transformation S in . such that the fixed points of S are in .U1 and .V2 . Then there exists .m ∈ Z \ {0} such that .S m U2 ∩ () ∩ V2 = φ (Proposition 6.2.14). It follows that there exists an open interval .U3 of .S1 such that .U3 ⊂ S m U2 ∩ V2 and .U3 ∩ () = φ. Now there exists a hyperbolic transformation .T ∈  with fixed points in .U3 and .V1 . Hence, φ = T k U3 ∩ () ∩ U3 ⊂ T k S m U2 ∩ () ∩ V2

.

for all .k ∈ Z. Because T has a fixed point in .V1 , there exists .n ∈ Z such that T n S m U1 ∩ () ∩ V1 = φ. Thus .R = T n S m meets the required conditions.  

.

The next result is a generalization of Weil’s theorem (Theorem 5.1.4). It was obtained independently by Pupko [69] in Russia and the author [47] in the United States in the mid-1960s. Theorem 6.3.10 Let . be a mobile subgroup of .G  such that .() = S1 and . contains at least one hyperbolic transformation. Let .π : B2 → B2 /  be covering map. Let .γ : [0, ∞) → B2 be a continuous curve such that .π ◦ γ is a simple curve on .B2 / . If .

lim |γ (t)| = 1,

t→∞

then .

lim γ (t)

t→∞

exists and lies in .S1 . Proof For .t > 0, let .Rt be the h-ray originating at .γ (0) and passing through .γ (t), and let .ut be its point at infinity. Analogous to the .ω-limit set of an orbit, define a subset . of .S1 by .u ∈  if and only if there exists a sequence .ti of positive real numbers such that .limi→∞ ti = ∞ and .limi→∞ γ (ti ) = u. It suffices to show that . is a point, and this is done by contradiction. Like an .ω-limit set for a flow on a compact space, it is easily seen that . is a closed and connected subset of .S1 . So if . is not a point, it must be a closed interval or .S1 . If . is a closed interval with interior . o , then the curve .γ (t) must intersect every h-ray with its point at infinity in . o infinitely often. If . = S1 , then there exists a closed interval with the property that the curve .γ (t) must intersect every

6.3 Groups of Rigid Motions Fig. 6.5 This configuration is guaranteed when . o = φ

179

T γ(t2 ) γ(t2 ) b = T 2a Ta T γ(t1 )

a γ(t1 )

h-ray with its point at infinity in the interior of this interval infinitely often. For simplicity, denote this open interval by U in either case. By Theorem 6.3.8, there exists a hyperbolic transformation T in . whose fixed points are elements of U . Let L be the axis of T and let a be point of L and let 2 .b = T a. Then .dh (a, b) = 2dh (a, T a). Let V be the open h-line segment joining a and b. Then .L\V is the union of two h rays .A1 and .A2 with endpoints in U . Because .limt→∞ |γ (t)| = 1, the curve .γ (t) intersects .A1 and .A2 for arbitrarily large t, but not V . Therefore, there exist .t1 and .t2 such that .γ (t1 ) ∈ Ai for .i = 1, 2 and .γ (t) is not in L for t between .t1 and .t2 . For convenience, assume that .t1 < t2 . The choice of V guarantees that .T γ (t1 ) is not in .A2 and .T γ (t2 ) is not in .A1 . (See Figure 6.5.) Therefore, {γ (t) : t1 ≤ t ≤ t2 } ∩ T {γ (t) : t1 ≤ t ≤ t2 } = φ

.

and {γ (t) : t1 ≤ t ≤ t2 } = T {γ (t) : t1 ≤ t ≤ t2 }

.

contradicting the hypothesis that .π ◦ γ was a simple curve.

 

Chapter 7

Flows and Hyperbolic Geometry

It is now time to connect planar hyperbolic geometry with compact connected surfaces by restricting our attention to groups of rigid motions whose actions on 2 .B are free and proper. The subgroups of .G that act properly on .B2 can be characterized as classical Fuchsian groups or as the discrete subgroups of the topological group .G. The action of a discrete group of .G is free if and only if it does not contain any elliptic Möbius transformations. Consequently, discrete subgroups of .G that do not contain elliptic elements are the central topic of the chapter. The first section is used to develop the theory of Fuchsian covering groups starting from a general viewpoint and gradually specializing to the covering groups of compact surfaces. Fundamental regions and especially the Dirichlet regions are the tools used to gain an understanding of the Fuchsian covering groups of compact surfaces. Section 7.2 is devoted to the construction of specific covering groups of hyperbolic rigid motions for compact connected surfaces. The orientable case uses Möbius transformations, but the nonorientable case must include paddle motions that have the effect of embedding Möbius bands in the covered surface. The last section of the chapter (Section 7.3) adapts the theory of control curves and extends it to the more general weak control curves. Although many of the ideas carry over easily to the hyperbolic setting, there are differences that need to be examined.

7.1 Fuchsian Groups The action of .G itself on .B2 is dynamically not very interesting because it has only one orbit. (This follows from Proposition 6.2.9.) At the other end of the spectrum, discrete subgroups of .G have interesting orbit spaces and are the central subject of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_7

181

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7 Flows and Hyperbolic Geometry

this section. These groups are classically known as Fuchsian groups and are closely tied to the study of automorphic functions in complex variables (see [30, 38] or [44]). For the moment, let G be an arbitrary Hausdorff topological group. To determine whether or not a subgroup H of G is discrete, it suffices to check the identity e. Specifically, H is a discrete subgroup of G if and only if there exists an open subset U of G such that .U ∩H = {e}. Moreover, a discrete subgroup of a topological group is closed. (See Proposition 1.4.1 in [59].) So whenever H is a discrete subgroup of a metric group G and .hn is a sequence in H converging to .g ∈ G, then g is in H and there exists .N > 0 such that .hn = g for .n > N.  : TA ∈ }. Clearly, .  = {A ∈ G  is a subgroup Let . be a subgroup of .G and set .   of .G. Because the natural projection of .G onto .G is a 2-to-1 covering, it is easy to   is a discrete subgroup of .G. see that . is a discrete subgroup of .G if and only if .  So we can work with either the Möbius transformations in . or the matrices in . . Furthermore, if S is in .G, then the subgroup .SS −1 is a discrete subgroup of .G if and only if . is a discrete subgroup of .G because .T → ST S −1 is an automorphism of .G. For the rest of this section, the classical notion of a group of homeomorphisms being discontinuous at a point ([44], p. 86) will be useful, although the name is awkward. The subgroup . of .G is said to be discontinuous at .z ∈ C provided that there does not exist a sequence .Tn of distinct transformations in . and .w ∈ C such that .Tn w converges to z. Note that if z is in .B2 , then .Tn w could only converge to z if w is also in .B2 because .B2 is an open invariant set of .G. Theorem 7.1.1 Let . be a subgroup of .G. The following are equivalent: (a) . is a discrete subgroup of .G. (b) . is discontinuous at every z in .B2 . (c) The action of . on .B2 is proper. Proof Suppose . is a discrete subgroup of .G and . is not discontinuous at .ζ ∈ B2 . There exists a sequence .Tn of distinct transformations in . and .w ∈ B2 such that −1 is .Tn w converges to .ζ . There also exists .R ∈ G such that .Rw = 0. Then .RR −1 a discrete subgroup of .G such that .Sn 0 = RTn R 0 = RTn w converges to .Rζ . Writing .Sn z = (an z + cn )/(cn z + a n ) with .an a n − cn cn = 1, we have .Sn 0 = cn /a n converging to .Rζ . If .|cn | or .|an | has a bounded subsequence, then the same subsequence of .|an | or 2 2 .|cn | is also bounded because .|an | = |cn | +1. In this case, it can be assumed that .an and .cn converge to a and c, respectively. Setting .Sz = (az + c)/(cz + a), it follows that .Sn converges to S in .G because  .

lim

n→∞

an cn cn a n



 =

ac ca



7.1 Fuchsian Groups

183

 Consequently, S is in . and .Sn = S for .n > N, contradicting the assumption is in .G. that the transformations .Tn were distinct. Therefore, .|cn | → ∞ and .|an | → ∞ as .n → ∞. Since .Sn 0 = cn /a n converges to .Rζ , the sequence .cn cn /an a n converges to 2 .|Rζ | . Because .

cn cn an a n − 1 |an |2 − 1 = lim = lim = 1, n→∞ an a n n→∞ n→∞ an a n |an |2 lim

|Rζ | = 1 and .Rζ ∈ S1 . It follows that .ζ ∈ S1 , contrary to the assumption that .ζ was in .B2 . Therefore, . is discontinuous at every .ζ in .B2 to complete the proof that (a) implies (b). Let .ζ1 and .ζ2 be points in .B2 , but not necessarily distinct. To prove that (b) implies (c), it must be shown that there exist neighborhoods .Ui of .ζi , .i = 1, 2, such that the set .{T ∈  : T U1 ∩ U2 = φ} is finite. Let .Ui = {z ∈ B2 : dh (z, ζi ) < 1}. Because . is discontinuous at every z in .B2 , the set .F = {T ∈  : dh (T ζ1 , ζ2 ) ≤ 2} is finite. Suppose z is a point in .U1 such that T z is in .U2 . Then

.

dh (T ζ1 , ζ2 ) ≤ dh (T ζ1 , T z) + dh (T z, ζ2 ) = dh (ζ1 , z) + dh (T z, ζ2 ) < 2,

.

proving that T is in the finite set F and that (b) implies (c). Suppose . acts properly on .B2 . If . is not a discrete subgroup of .G, then there exists a sequence .Tn of distinct transformations in . such that .Tn converges to .ι in .G. Because the action of .G on .D2 is continuous, .Tn ζ converges to .ζ for every .ζ ∈ B2 . Since the action of . on .B2 is proper at a given .ζ , there exist open neighborhoods .U1 and .U2 of .ζ such that .F = {T ∈  : T U1 ∩ U2 = φ} is finite. Then .F = {T ∈  : T (U1 ∩ U2 ) ∩ (U1 ∩ U2 ) = φ} is also finite. Consequently, .Tn ζ is in .U1 ∩ U2 for large n, implying that .Tn is in .F for large n. Therefore, the .Tn are not all distinct, and (c) implies (a). 

A Fuchsian group is by definition a subgroup of .G that satisfies one, and hence all, of the conditions in Theorem 7.1.1 with (b) being the classical definition. A Fuchsian group is of type I or II depending on whether .() equals .S1 or is not equal to .S1 (and thus nowhere dense in .S1 ). The action of a Fuchsian group on .B2 is free if and only if there are no elliptic transformations in .. A number of other basic properties of Fuchsian groups are corollaries of Theorem 7.1.1. Corollary 7.1.2 If . is a Fuchsian group, then the orbit .z is a discrete subset of B2 for all .z ∈ B2 .

.

Proof Proposition 1.1.18 applies because the action of . on .B2 is proper by part (c) of the theorem. 

Corollary 7.1.3 If . is a Fuchsian group, then . is not discontinuous at every point of .() and .(z)− = z ∪ () for all .z ∈ B2 .

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7 Flows and Hyperbolic Geometry

Proof If w is in .(), then given .z ∈ B2 there exists a sequence .Tn in . such that 2 1 .Tn z converges to w. Since z is in .B and w is .S , there exists a subsequence .Tnk of .Tn consisting of distinct elements of ., proving that . is not discontinuous at w. Similarly, it follows from part (b) of the theorem that .(z)− \ z ⊂ S1 . Thus − = z ∪ () for all .z ∈ B2 by Proposition 6.3.1. .(z) 

Corollary 7.1.4 If . is a Fuchsian group, then there exists at most a countable set of points in .B2 such that .z = {T : T z = z} = {ι}. Proof It suffices to show that the set z such that .z = {ι} is finite when .|z| ≤ r < 1. If not, then for some .0 < r < 1 there exists a convergent sequence .zn of distinct points such that .|zn | < r and .zn = {ι}. So for each .zn there exists .Tn ∈  \ {ι} such that .Tn zn = zn . It follows from Proposition 6.2.2 that every .Tn is an elliptic Möbius transformation and .Tm = Tn when .m = n. Consequently, . does not act properly at .z = limn→∞ zn because for every open neighborhood U of z the set .{T ∈  : T U ∩ U = φ} is infinite. 

Corollary 7.1.5 If . is a Fuchsian group containing no elliptic transformations, then the natural projection .π : B2 → B2 /  is the universal covering of a surface and . is both the covering group and an isomorphic copy of .1 (B2 / ). Proof Proposition 2.2.1 applies because the action is proper by the theorem and 

free because there are no elliptic transformations in .. A Fuchsian group that does not contain any elliptic transformations will be called a Fuchsian covering group. So whenever . is a Fuchsian covering group, 2 → B2 /  is a universal covering of a surface by Corollary 7.1.5. Since .π : B −1 )Sz for .S ∈ G, the homeomorphism S of .B2 onto itself passes to a .S(z) = (SS homeomorphism of .B2 /  onto .B2 /(SS −1 ), and the surface depends only on the conjugacy class of . in .G. Fuchsian covering groups will become essential. The proof of the next result, which will be needed, uses the same idea used to prove a more general result in [44] on p. 94. Theorem 7.1.6 Let T be a hyperbolic transformation in ., a subgroup of .G. If some .S ∈  has exactly one fixed point in common with T , then . is not a Fuchsian group. Proof Replacing T with .T −1 if necessary, it can be assumed that the positive fixed point of T is the common fixed point of T and S. Applying Proposition 6.2.8, it can be assumed without loss of generality that .T = Tρ given by equation (6.23) with .ρ > 0 and 1 is the common fixed point of T and S. Then . ◦ Tρ ◦ −1 and . ◦ S ◦ −1 have exactly one common fixed point at .∞. (See equation (6.7) for the definition of . .) Furthermore, in the proof of Proposition 6.2.4, it is shown that . ◦ Tρ ◦ −1 z = eρ z. It follows that  A=

.

eρ/2 0 0 e−ρ/2



 and B =

α β 0 α −1



7.1 Fuchsian Groups

185

with .αβ = 0 are the matrices of . ◦ Tρ ◦ −1 and . ◦ S ◦ −1 , respectively. A calculation shows that   1 αβ(1 − enρ ) n −1 −n .BA B A = , 0 1 which is a sequence of distinct matrices in .SL(2, R) such that  lim

.

n→−∞

1 αβ(1 − enρ ) 0 1



 =

1 αβ 0 1

 .

Therefore, .  −1 is not a discrete subgroup of .SL(2, R), and consequently, . is 

not a discrete subgroup of .G. Before proceeding, we need a little more hyperbolic geometry. Given distinct points .ζ1 and .ζ2 in .B2 , there exists an h-line .L⊥ perpendicular to the h-line L determined by .ζ1 and .ζ2 at the point of L midway between them (Propositions 6.1.23 and 6.2.12) called the h-perpendicular bisector of .ζ1 and .ζ2 . Proposition 7.1.7 If .L⊥ is the h-perpendicular bisector of the distinct points .ζ1 and .ζ2 in .B2 , then L⊥ = {z ∈ B2 : dh (z, ζ1 ) = dh (z, ζ2 )}.

.

Proof It suffices to prove the result for real numbers r and .−r such that .−1 < −r < 0 < r < 1. In this case, .L⊥ = LI = {is : −1 < s < 1}. Since .z → −z, the reflection in .LI is an h-isometry, .L⊥ ⊂ {z ∈ B2 : dh (z, r) = dh (z, −r)}. For the converse, suppose .dh (w, −r) = dh (w, r) = ρ. Let .C1 and .C2 be hcircles with centers .−r and r and with h-radius .ρ. Clearly, .w ∈ C1 ∩ C2 . They are also symmetric with respect to the real axis because .z → z, the reflection in .LR , is an h-isometry. Thus .C1 ∩ C2 = {w, w}. If .w = w, then w is in .ŁR and must be 0 because .dh (w, −r) = dh (w, r) = ρ. If .w = w, then .C1 ∩ C2 = {w, w}. The h-isometry .z → −z interchanges the half circles of .C1 and .C2 lying above .LR and likewise for the half circles below .LR . Hence, .z → −z fixes w and .w. Therefore, w and .w are in .LI = L⊥ . 

Let G be a group of homeomorphisms of a metric space X. With the discrete topology, G is a topological group and the natural action of G on X is continuous, producing a transformation group .(X, G). In this context, suppose the action of G on X is proper. A subset F is a fundamental region for .(X, G) provided that: o − (a) .F = (F  ) . (b) .X = T ∈G T F . (c) .T F o ∩ F o = φ when T is not the identity of G.

The family of sets .{T F : T ∈ G} is called a tessellation of X. Clearly, for every T ∈ , the set T F is a fundamental region when F is a fundamental region.

.

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7 Flows and Hyperbolic Geometry

The unit square .[0, 1] × [0, 1] is a fundamental region for the left action of the Gaussian integers .Z[i] on .C by addition. For a more general example, recall the construction of the suspension flow .(S(X, h), R) from a cascade .(X, h) (p. 60). In this case, .G = Z, the set .F = X×[0, 1] is a fundamental region, and the tessellation of .X × R consists of all sets of the form .X × [n, n + 1] with .n ∈ Z. Let .ζ be a point in .B2 that is not a fixed point, and define the Dirichlet region of . centered at .ζ by Dζ () = {z ∈ B2 : dh (z, ζ ) ≤ dh (z, T ζ ) for all T ∈ }

.

(7.1)

or, equivalently, Dζ () = {z ∈ B2 : dh (z, ζ ) ≤ dh (T z, ζ ) for all T ∈ }

.

(7.2)

because .dh (z, T ζ ) = dh (T −1 z, ζ ) for all .T ∈ . Observe that SDζ () = DSζ (SS −1 ).

(7.3)

.

In particular, when S is in ., we have SDζ () = DSζ ().

(7.4)

.

Proposition 7.1.8 Let . be a Fuchsian group. If .Dζ () is a Dirichlet region centered at .ζ ∈ B2 , then .Dζ () is a closed h-convex set of .B2 containing an open neighborhood of .ζ . Proof By Corollary 7.1.2, the orbit .ζ is discrete, and there exists .ε > 0 such that dh (ζ, T ζ ) > ε for all .T ∈  \ {ι}. If .dh (z, ζ ) < ε/2, then

.

ε < dh (ζ, T ζ ) ≤ dh (ζ, z) + dh (z, T ζ )
0.

.

Because .S → T ST −1 is a bijective map of . \ {ι} to itself, it follows that η(T z) = inf{dh (T z, T ST −1 T z) : S ∈  \ {ι}} = inf{dh (z, Sz) : S ∈  \ {ι}} = η(z).

.

Thus .η is constant on the .-orbits of points in .B2 . It will be shown that .η is continuous on the open set .U = {w : dh (w, z) < r} such that .η(z) = dh (z, Sz) = r > 0. Since .dh (w, Sw) is continuous on .B2 and .U − is compact, .sup{dh (w, Sw) : w ∈ U − } = ρ is finite. Then .η(w) ≤ ρ on U . Set .V = {w : dh (w, z) < r + ρ} and .W = {w : dh (w, z) < 2r + ρ}. Let .G = {T ∈  \ {ι} : T U ∩ V = φ}, which includes S because Sz is in V . Then T w is in W when .T U ∩ V = φ by the triangle inequality because dh (z, T w) ≤ dh (z, w) + dh (w, Sw) + dh (Sw, T z) < r + ρ + r.

.

If G is infinite, there exists a sequence .Tj of distinct elements of . such that Tj U ∩ V = φ; .Tj z converges to .z ∈ W − ⊂ B2 ; and . is not discontinuous at .z , contradicting the assumption that . is a Fuchsian group. Therefore, G is finite, and .η(w) is the minimum of a finite number of continuous functions on U . It follows that .η is continuous on U and hence on .B2 . By hypothesis, there exists a compact fundamental region F . The function .η assumes a minimum .c > 0 on F , which is minimum for .B2 because .η is .-invariant. The proof will show that .η cannot have a positive minimum on .B2 . It suffices to show that .

.

lim dh (Sβ r, r) = 0

r→1−

for r such that .−1 < r < 1. This is further reduced to showing that

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7 Flows and Hyperbolic Geometry

.

lim

r→1−

Sβ r − r =0 1 − rSβ r

by equation (6.17). A careful calculation shows that .

Sβ r − r iβ(r − 1)/2 = 1 − rSβ r (r + 1) + iβ(r − 1)/2

and its limit as .r → ∞ is clearly 0 and completes the proof.



Exercise 7.1.20 In the proof of Theorem 7.1.19, verify that .

Sβ r − r iβ(r − 1)/2 = . 1 − rSβ r (r + 1) + iβ(r − 1)/2

Corollary 7.1.21 If . is a Fuchsian covering group with a compact fundamental region, then the Dirichlet region centered at .ζ for all .ζ in .B2 is an h-convex fundamental region for .. Proof Clearly, the theorem implies that .E = {ζ ∈ B2 : ζ = {ι}} = B2 , and then 

Theorem 7.1.12 completes the proof. Proposition 7.1.22 Let T be a hyperbolic Möbius transformation in a Fuchsian covering group . with axis .LT , and let K be a compact subset of .B2 . If .B2 /  is compact, then .{SLT : S ∈  and SLT ∩ K = φ} is a finite set of hyperbolic lines. Proof It can be assumed that .K = {z ∈ B2 : dh (z, 0) ≤ κ} for some .κ > 0, and Corollary 6.2.11 implies that .dh (T z, z) = c > 0 for all .z ∈ LT . If S is in ., then −1 and .SLT is the axis of .ST S dh (ST S −1 Sz, Sz) = dh (ST z, Sz) = dh (T z, z) = c.

.

Thus .dh (ST S −1 z, z) = c for all .z ∈ SLT and .S ∈ . Fix .ζ in .LT . Let I be the closed h-line segment from .ζ to .T ζ of .LT . It follows that LT =



.

T n I.

n∈Z

If the proposition is false, there exist a sequence .Sk in . such that .Sk LT = Sj LT when .k = j and a sequence .zk in .LT such that .Sk zk ∈ K. Then there also exist sequences .mk ∈ Z and .wk ∈ I such that .zk = T mk wk . So it can be assumed that .Sk zk and .wk converge to .z ∈ K and .w ∈ I , respectively. m k Set .Rk = Sk T . Then .Rk LT = Sk LT and .Rk = Rj when .k = j because .Sk LT = Sj LT when .k = j . If U and V are open neighborhoods of .z and w, respectively, then .{Q ∈  : U ∩ QV = φ} is infinite because .Rk wk is in .U ∩ Rk V for large k. So . does not act properly on .B2 , contradicting Theorem 7.1.1. 

7.1 Fuchsian Groups

193

Theorem 7.1.23 If . is a Fuchsian covering group for a compact connected orientable surface, then . is a mobile type I Fuchsian group. Moreover, .S1 is a minimal set of .(D2 , ). Proof It must be shown that . has no fixed points in .S1 (mobile) and that .() = S1 (type 1). Then .(D2 , ) will be minimal by Theorem 6.3.5. If .() is finite, then its cardinality is 0, 1, or 2 by Proposition 6.3.4. Since every element of . \ {ι} is hyperbolic by Theorem 7.1.19, 2 is the only possible finite cardinality for .() and then .D0 () would not be compact (see Figure 7.1). Therefore, . is infinite and either .S1 or a Cantor subset of .S1 . By Corollary 7.1.18, .D0 () is compact, and hence, it is contained in an h-ball of radius, say r. It follows that every h-disk of radius 2r contains .T D0 () for some 1 .T ∈ . Since there are h-balls of radius 2r close to every point in .S in the Euclidean − 1 1 metric, .(0) contains .S and .() = S (Proposition 6.3.1). If . has a fixed point in .S1 , then there exists a common fixed point .w ∈ S1 for all .T ∈ , which is impossible for a Fuchsian group by Theorem 7.1.6. 

Recall from Proposition 6.2.20 that .G is both the component of the identity of .G and a subgroup of index 2. In particular, .G is the disjoint union of .G and .RG when R is in .G \ G, and they are the components of .G . Also RS is in .G whenever R and S are both in .G \ G. Given a subgroup . of .G , the subgroup . ∩ G = c will be called the conformal subgroup of .. When . = c , the index of .c in . is 2 with the other coset being .Rc for any R in . \ c . Theorem 7.1.24 Let . be a subgroup of .G such that . = c . If .c is a Fuchsian group, then: (a) . is a discrete subgroup of .G . (b) . is discontinuous at every z in .B2 . (c) . acts properly on .B2 . Proof Because .c is a Fuchsian group, it is a discrete subgroup of .G, it is discontinuous at every point of .B2 , and it acts properly on .B2 by Theorem 7.1.1. These properties will be used to prove parts (a), (b), and (c), respectively. Because .c is discrete, there exists an open neighborhood U of .ι in .G such that .c ∩ U = {ι}. Since .G is an open subset of .G , it follows that U is an open subset of .G such that . ∩ U = {ι}, proving that . is a discrete subgroup of .G . Turning to part (b), suppose .Tn is a sequence of distinct transformations in . such that .Tn w converges to z for some w and z in .B2 . To prove that this is impossible, it suffices to consider the cases: every .Tn is in .c and no .Tn is in .c . The first case is impossible because .c is discontinuous at every z in .B2 . In the second case, every .Tn is in . \ c . Then .T1 Tn is a sequence of elements in .c such that .T1 Tn w converges to .T1 z, which is also impossible because .T1 z is in .B2 . Thus . is discontinuous at every z of .B2 . For part (c), let w and z be points in .B2 , and let S be a specific element of . \ c . Because .c acts properly on .B2 , there exist open neighborhoods U and V of w

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7 Flows and Hyperbolic Geometry

and z, respectively, such that the set .D = {T ∈ c : T U ∩ V = φ} is finite. Again, there exist open neighborhoods .U and .V of w and Sz, respectively, such that .E = {T ∈ c : T U ∩ V = φ} is finite. Suppose .T U ∩ S −1 V = φ and T is in . \ c . Then .ST U ∩ V = φ, and ST is in E because ST is in .c . Letting .F = {T ∈  \ c : T U ∩ S −1 V = φ}, it follows that .SF ⊂ E. Thus F is finite because .T → ST is injective. Therefore,  .

T ∈  : T (U ∩ U ) ∩ (V ∩ S −1 V ) = φ ⊂ D ∪ F

and is finite.



Corollary 7.1.25 Let . be a subgroup of .G . The following are equivalent: (a) . is a discrete subgroup of .G. (b) . is discontinuous at every z in .B2 . (c) The action of . on .B2 is proper. Proof When . = c , the result is Theorem 7.1.1. Then the following hold: when  is discrete, so is .c ; when . is discontinuous at every z in .B2 , so is .c ; when the action of . on .B2 is proper, so is the action .c . Now Theorems 7.1.1 and 7.1.24 imply that (a) and (b) are equivalent, and (b) and (c) are equivalent. 

.

Corollary 7.1.26 If . is a discrete subgroup of .G , then the orbit .z is a discrete subset of .B2 for all .z ∈ B2 . Proof Same proof as Corollary 7.1.2.



Thus a subgroup . of .G is a covering group of a surface if and only if .c is a Fuchsian group and . acts freely on .B2 . This is a larger class of groups than the Fuchsian covering groups. The covering groups contained in .G , of course, do not contain any elliptic transformations or reflections in h-lines. Proposition 7.1.27 Let . be a subgroup of .G acting freely and properly on .B2 . If L is the axis of a paddle motion P in ., then .{R ∈  : RL = L} is a maximal cyclic subgroup of ., and there exist exactly 2 primitive paddle motions T and .T −1 in . such that .[T ] = [T −1 ] = {R ∈  : RL = L}. Proof Since .P 2 is a hyperbolic Möbius transformation with L as its axis, .{R ∈ c : RL = L} is a maximal cyclic group of .c with generators T and .T −1 (Proposition 7.1.15). There exists S in .G such that .SL = LR and .ST S −1 = Tρ for a unique .ρ > 0 (Proposition 6.2.8). Then . = SS −1 is a discrete subgroup of 2 −1 is a paddle motion with axis .L , .G acting freely and properly on .B . Also .SP S R and .Tρ is a generator of the maximal cyclic group of .c with axis .LR . This reduces the proof to showing that .{R ∈  : RLR = LR } is a maximal cyclic subgroup of . . Since .Qz = Tρ/2 z is a paddle motion such that .QLR = LR and .Q2 = Tρ , it suffices to prove that every paddle motion R of . with axis .LR has the form .R = Q2n+1 with .n ∈ Z. Suppose the paddle motion .Rz = Tσ/2 z with axis .LR is in . . It can be assumed that .σ > 0 because R is in . if and only if .R −1 is in . . It will be shown that .σ/2 = (2n + 1)ρ/2 for some .n ∈ Z.

7.2 Constructing Compact Dirichlet Regions

195

If .σ/2 = nρ for .n ∈ Z \ {0}, then .Rz = Tρn z and .T −n Rz = z is in . , which is impossible because the elements of . have no fixed points in .B2 . Thus .σ/2 = nρ + δρ such that .n ∈ Z and .0 < δ < 1. If .δ = 1/2, then 2 2n 2 2n .R = Tρ T2δρ and either .0 < 2δ < 1 or .0 < 2δ − 1 < 1. Then either .R = Tρ T2δρ 2 2n+1 or .R = Tρ T(2δ−1)ρ implies that either .T2δρ or .T(2δ−1)ρ is in .c depending on whether .0 < 2δ < 1 or .0 < 2δ − 1 < 1. Both possibilities contradict that .Tρ is a generator of the maximal cyclic group of .c with axis .LR because .ρ = min{α > 0 : Tα ∈ c }. Therefore, .δ = 1/2 and Q along with .Q−1 are the generators of . . 

The proofs of the key theorems for Fuchsian groups can be reused for subgroups of .G such that .c is a Fuchsian group. First note that Proposition 7.1.7 is a basic fact of hyperbolic geometry and the definition of a Dirichlet region, equation (7.1), makes sense for any subgroup of .G . Since we are assuming . acts freely on .B2 and the orbits of . are discrete, the proofs of Propositions 7.1.8 and 7.1.9 can be used in this more general setting. With Corollary 7.1.25 replacing Theorem 7.1.1, the proof of Proposition 7.1.16 works when .c is a Fuchsian group and . acts freely on .B2 . Finally, a careful reread of Theorems 7.1.12 and 7.1.17 now shows that their proofs require only that the conformal subgroup of . is Fuchsian and the action of . on .B2 is free. Thus, we have the following result: Theorem 7.1.28 Let . be a subgroup of .G such that . = c . If . acts freely on 2 .B and .c is a Fuchsian group, then the following hold: (a) For all .ζ ∈ B2 , the Dirichlet region centered at .ζ is an h-convex fundamental region for .. (b) . has a compact Dirichlet region if and only if .B2 /  is compact.

7.2 Constructing Compact Dirichlet Regions Section 7.1 leaves one fundamental question unanswered. Given a compact connected surface, does there exist a discrete subgroup  of G such that B2 /  is homeomorphic to the given surface? This section is devoted to answering this question, starting with the orientable case. Theorem 7.2.1 If M is a compact connected orientable surface of genus n ≥ 2, then there exists a Fuchsian covering group  with the following properties: (a) M is homeomorphic to the orbit space B2 / . (b) B2 is a universal covering space for M with covering group . (c) The Dirichlet region D0 () is a regular hyperbolic polygon with 4n sides. A version of this result is discussed in [42]. On the one hand, the construction of the group  in [42] is more geometric and does not make use of the analytic tools at our disposal. On the other hand, repeating the substantive topological argument

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7 Flows and Hyperbolic Geometry

needed to complete the proof that is clearly presented in [42] makes little sense. Consequently, we will start with a more analytic construction of the edge pairing transformations that generate the group  and then refer the reader to Lee’s proof to complete the proof of the theorem. In doing so, we will be careful to make sure the notation coincides with Lee’s notation in [42]. This includes the use of M instead of X to denote the compact connected orientable surface of genus at least two in this section. To carry out this plan for a compact connected surface M, we need to have a hyperbolic polygon with the right number of sides for edge pairings that will produce M. These edge pairings need to be constructed using Möbius transformations in the orientable case and elements of G \G in the nonorientable case. For the group  generated by the edge pairings to have the property that B2 /  is homeomorphic to M, it must at least satisfy the algebraic condition in the group theoretic presentation of the fundamental group of M. These conditions can be found in Example 10.10 of [42] and are the central issue in the construction of the edge pairings. We begin with a discussion of regular hyperbolic polygons with an even number of edges starting with their construction. The equation x 2 + y 2 − ηx + 1 = 0 can be written as

.

x−

η 2 η 2 η2 − 4 . + y2 = −1= 2 2 4

It is the equation of acircle orthogonal to S1 with center at (η/2, 0) (Proposition 6.1.1) and radius η2 − 4/2 when η > 2. The intersection of this circle with B2 is an h-line and will be denoted by Lη . It is easy to see that Lη is perpendicular to LR and crosses LR at 

η−

.

 η2 − 4 ,0 . 2



The points at infinity for Lη are  .

     2 2 η2 − 4 4 ,± 1 − 2 = ,± . η η η η

Since the equation x 2 +y 2 −ηx +1 = 0 can also be written as r 2 −ηr cos θ +1 = 0, the angle the ray emanating from 0 and passing through  .

2 , η

  η2 − 4 η

makes with the x-axis satisfies 1 − η cos θ = 1 and θ = cos−1 (2/η).

7.2 Constructing Compact Dirichlet Regions

197

Let m be an integer greater than 1. Consider the 2m evenly spaced points on S1 given by eiπ k/m for k = 0, 1, . . . , 2m − 1. Let Hη be the closed h-half plane containing 0 determined by Lη , η > 2. Set Rz = eiπ/m z, which is a rotation of period 2m. Note that R m z = −z. In general, Lη ∩ RLη can be empty. If, however, π/2m < cos−1 (2/η), then Lη ∩ RLη is a point and Pη =

2m−1 

.

R k Hη

(7.5)

k=0

is a regular 2m-sided compact h-polygon centered at 0 such that RPη = Pη . The radian measure of its interior angles will be denoted by μ(η). Clearly, Eη = Lη ∩ {reiθ : |θ | ≤ π/2m} is the unique edge of Pη containing the point Łη ∩ LR . The other edges of Pη are R k Eη for k = 1, . . . , 2m − 1. Set u0 = Eη ∩ R −1 Eη and uk = R k u0 for k = 1, . . . , 2m − 1. (We are suppressing the dependency of the vertices on η for convenience of notation.) The following fact will be useful later in the construction of covering groups: Proposition 7.2.2 Given an integer m > 1, the function μ(η) is continuous on the infinite open interval {η : 2/ cos(π/2m) < η},

.

and the range of μ(η) is the open interval   π(m − 1) . . μ : 0 < μ < m Proof Observe that the condition π/2m < cos−1 (2/η) is equivalent to η > 2/ cos(π/2m). To ensure that Pη is a compact regular polygon, only η that satisfy this condition will be considered. The vertex u1 = Ru0 lies on the ray {reiπ/2m : r ≥ 0}, and there exists a positive function r(η) such that u1 (η) = r(η)eiπ/2m . Since u1 (η) lies on Lη , the function r(η) satisfies r(η)2 − ηr(η)cos(π/2m) + 1 = 0

.

and yields the following continuous dependence of r(η) on η: r(η) =

.

η cos(π/2m) −

 [η cos(π/2m)]2 − 4 . 2

By the symmetry of Pη , the ray {reiπ/2m : r ≥ 0} passes through u1 and bisects the interior angle of Pη . So the angle between the ray {reiπ/2m : r ≥ 0} and the Euclidean tangent line Lu1 to Lη at u1 is μ(η)/2 as shown in Figure 7.2.

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7 Flows and Hyperbolic Geometry

Tangent Line RLη κ

u1

η) μ( 2

e−iπ/2m η

η/2 1 Lη

2 Lη

S1 Fig. 7.2 The geometry used to calculate μ(η)

Rather than working with the Euclidean tangent line Lu1 to Lη , we will make use of the Euclidean line L η determined by the points u1 and (η/2, 0), the center of Lη that is perpendicular to Lu1 . Let κ denote the radian measure of the angle between L η and the ray {reiπ/2m : r ≥ 0} as indicated in Figure 7.2. Then μ(η)/2+κ = π/2, and we can use the dot product to determine cos(κ). Specifically, κ is the radian measure of the angle between the vector from u1 to (0, 0) and minus the vector from u1 to (η/2, 0), that is, the angle between the vectors .

  −r(η) cos(π/2m), sin(π/2m) and − η/2−r(η) cos(π/2m), −r(η) sin(π/2m) . The lengths of these vectors are r(η) and the radius of Lη , respectively. Thus  r(η) η cos(π/2m)/2 − r(η)[cos2 (π/2m) + sin2 (π/2m)]  . cos κ = r(η) η2 − 4/2 =

η cos(π/2m) − 2r(η)  η2 − 4

η cos(π/2m) − η cos(π/2m) +  = η2 − 4  =

η2 cos2 (π/2m) − 4  η2 − 4



[η cos(π/2m)]2 − 4

7.2 Constructing Compact Dirichlet Regions

199

by substituting the formula for r(η) from the second paragraph of the proof. Clearly, cos κ is a continuous function. It is now easy to see that .

lim cos κ = cos(π/2m) and

η→∞

lim

η→2/ cos(π/2m)

cos κ = 0,

and the range of κ is the open interval from π/2m to π/2. Consequently, the range of μ(η)/2 is the open interval from 0 to π/2 − π/2m = π(m − 1)/2m, and the range of μ(η) is 0 to π(m − 1)/m. Obviously, μ(η) is continuous. 

Corollary 7.2.3 For m > 2, there exists a unique η such that μ(η) = π/m. Proof An algebraic calculation shows that η → cos κ is an injective function, and cos−1 is injective on the range of η → cos κ. The rest is trivial. 

To construct a fundamental region for a compact orientable surfaces of genus at least 2, first let n denote the desired genus. Then m = 2n is the number of pairs of edges required, and 2m = 4n is the total number of edges required. Note that 2m is a sequence of integers divisible by 4 starting with 8. For future use, note that Corollary 7.2.3 applies to π/m because m > 2 in this context. Given n ≥ 2, let Pη be the h-polygon with 4n = 2m sides given by equation (7.5) for 2/ cos(π/2m) < η. Of course, Pη is a natural candidate for a Dirichlet region of a Fuchsian covering group. The first step is to construct an edge pairing using Möbius transformations in G and the construction of regular h-polygons. Let Rz = eiπ/2n z = eiπ/m z, the rotational symmetry of Pη , and let  ρ = 2 tanh

.

−1

η−

 η2 − 4 , 2



the distance from 0 to Lη . Define Sη ∈ G by Sη = Tρ ◦ R m ◦ Tρ−1 , where Tρ is defined by equation (6.23). So Sη is just an h-rotation through π radians about the point Lη ∩ LR such that Sη u0 = u1 . Note that Sη2 = ι and Sη−1 = Sη . We will use R and Sη to construct the edge pairings. Proposition 7.2.4 The Möbius transformation R −1 ◦ Sη is an h-rotation about u0 conjugate to z → eiμ(η) z. Proof Clearly, R −1 ◦ Sη is in G and R −1 ◦ Sη u0 = u0 . So R −1 ◦ Sη is an elliptic element of G (Proposition 6.2.2) and conjugate to an h-rotation about the origin by Corollary 6.2.3. It remains to determine the angle of rotation. The interior angle μ(η) of Pη at u0 has sides Eη and R −1 Eη . Let Q be the clockwise rotation about u0 through μ(η) radians. It suffices to show that R −1 Sη = Q. Clearly, Qu0 = R −1 Sη u0 , and Qu1 = u2m−1 = R −1 Sη u1 . Using Figure 7.3, one sees that Qu2m−1 = u1 = R −1 Sη u2m−1 . Therefore, R −1 Sη = Q since they agree at 3 points. 

Letting Eη be the zero edge and going around counterclockwise, R k Eη is the kth edge. The ends of the kth edge are the vertices uk and uk+1 . Next, for j = 1, . . . , n,

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7 Flows and Hyperbolic Geometry

u1 Sη (μ(η)) 2

μ(η)

u0

u2m−1

R−1 Sη (μ(η)) Fig. 7.3 R −1 Sη is a μ(η) counterclockwise rotation about u0

set αj = R 4j −4 ◦ Sη ◦ R −(4j −2)

.

and βj = R 4j −1 ◦ Sη ◦ R −(4j −3) .

.

Observe that the αj Möbius transformations first rotate the edges R 2 Eη , R 6 Eη , R 10 Eη etcetera to Eη . Then they all rotate Eη through π radians about the point Lη ∩ LR so that Sη R −(4j −2) Pη ∩ Pη = Eη . Finally, they rotate Eη to R 0 Eη , R 4 Eη , R 8 Eη , etcetera, to identify R −(4j −2) Eη with R 4j −4 Eη . It follows that Pη ∩ αj Pη = R 4j −4 Eη . A similar analysis applies to βj to identify (4j − 3)th and (4j − 1)th edges so that Pη ∩ βj Pη = R 4j −1 Eη . The h-polygon with these identifications is shown in Figure 7.4 for genus 2, that is, m = 4. Note that the transformations αj and βj reverse the order of the vertices as required in the standard edge pairing for a compact connected orientable surface. Using the functions αj and βj for j = 1, . . . , n to identify points in Pη produces a quotient map p : P → M and quotient space M, which is a compact connected orientable surface of genus n.

7.2 Constructing Compact Dirichlet Regions

201

Fig. 7.4 The hyperbolic edge pairing for an orientable compact connected surface of genus 2

a b

b

a

c

d

d c

Letting η be the subgroup of G generated by αj and βj for j = 1, . . . , n, the big question is whether or not B2 / η is homeomorphic to M. There are two major obstacles to the plausibility that B2 / η is homeomorphic to M. First, Pη should be a fundamental region for η . Second, the generators of η should satisfy the defining relationship of the fundamental group of M. We will show that the first requirement is certainly possible and there is a choice of η for which the second condition is satisfied. Proposition 7.2.5 If η is a Fuchsian covering group, then D0 (η ) ⊂ Pη . Proof From the definitions of αj and βj , it follows that the h-lines R k Lη , k = 0, . . . , 4n − 1 are the h-perpendicular bisectors of 0 and the 4n points α ±j 0 and β ±j 0 for j = 1, . . . , n. Recalling that Hη is the closed h-half plane containing 0 determined by Lη , it follows that D0 (η ) ⊂

4n−1 

.

R k Hη = Pη

k=0



to complete the proof. Proposition 7.2.6 For all real numbers η > 2 and integers n > 1, βn ◦ αn ◦ βn−1 ◦ αn−1 ◦ · · · ◦ β1 ◦ α1 ◦ β1−1 ◦ α1−1 = (R −1 Sη )4n .

.

Proof Using Sη−1 = Sη , βj ◦ αj ◦ βj−1 ◦ αj−1 =

.

R 4j −1 ◦ Sη ◦ R −(4j −3) ◦ R 4j −4 ◦ Sη ◦ R −(4j −2)

(7.6)

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7 Flows and Hyperbolic Geometry

◦R 4j −3 ◦ Sη ◦ R −(4j −1) ◦ R 4j −2 ◦ Sη ◦ R −(4j −4) = R 4j −1 ◦ Sη ◦ R −1 ◦ Sη ◦ R −1 ◦ Sη ◦ R −1 ◦ Sη ◦ R −(4j −4) = R 4j ◦ (R −1 Sη )4 ◦ R −4(j −1) . So the left side of equation (7.6) equals R 4n ◦ (R −1 Sη )4 ◦ R −4(n−1) ◦ · · · ◦ R 4 ◦ (R −1 Sη )4 ◦ R 0 ,

.

which simplifies to (R −1 Sη )4n .



Corollary 7.2.7 Given n ≥ 2, there exists η in the domain of μ(η)such that .

βn ◦ αn ◦ βn−1 ◦ αn−1 ◦ · · · ◦ β1 ◦ α1 ◦ β1−1 ◦ α1−1 = ι

(7.7)

α1 ◦ β1 ◦ α1−1 ◦ β1−1 ◦ · · · ◦ αn ◦ βn ◦ αn−1 ◦ βn−1 = ι.

(7.8)

and .

Proof Note that the second equation is the inverse of the first. By Proposition 7.2.4, it now suffices to prove that there exists η in the domain of the function μ(η) such that μ(η) = 2π/4n = π/2n. Since m = 2n, Corollary 7.2.3 implies there exists η such that μ(η) = π/2n. 

For the rest of the discussion of compact connected orientable surfaces, η will be given by Corollary 7.2.7 so that μ = μ(η) = π/2n = π/m so that equation (7.8) holds. And we will speak of P and  instead of Pη and η . Furthermore, Theorem 7.2.1 now follows from the next theorem, which is proved in [42] and its corollary. Theorem 7.2.8 The group  is discrete and acts freely and properly on B2 , the quotient B2 /  is homeomorphic to M, and the restriction of the quotient map B2 → B2 /  to P is the quotient map p : P → M. Proof Since equations (12.5) and (12.6) in [42] are the same as our equations (7.7) and (7.8), we can pick up the proof of Theorem 12.17 in [42] with the first full paragraph on p. 277. Although the notational match is good, a few cautions about Lee’s notation are in order. On p. 275, he rewrites the left-hand side of equation (12.6) as σ4n ◦ · · · ◦ σ2 ◦ σ1 . This is immediately followed by introducing a different vertex scheme than the one we have been using. His scheme, denoted by vj instead of uj , has the advantage that ηj maps vj −1 to vj and is clearly illustrated by Figure 12.8 on p. 275 of [42]. 

Corollary 7.2.9 The Dirichlet region D0 () equals P . Proof It follows from the theorem that D0 () is a fundamental region by Theorem 7.1.12. Moreover, D0 () is contained in P by Proposition 7.2.5. It follows

7.2 Constructing Compact Dirichlet Regions

203

from the last statement of the theorem that there are no identifications in the interior of P . Therefore, the interior of D0 () and the interior of P are equal. Obviously, their closures are D0 () and P . 

Corollary 7.2.10 The elements of \{ι} are all hyperbolic Möbius transformations. 

Proof Theorem 7.1.19 now applies to .

What is important about Theorem 7.2.1 is not the specific group , but that using bitransformation groups (, B2 , R), where  is a Fuchsian covering group with a compact Dirichlet region to study flows on the compact connected orientable surfaces of genus at least 2, is a comprehensive approach. Not surprisingly, the same is true for compact connected nonorientable surfaces. Theorem 7.2.11 If M is a compact connected nonorientable surface of genus m ≥ 3, then there exists a covering group  in G with the following properties: (a) M is homeomorphic to the orbit space B2 / . (b) B2 is a universal covering space for M with covering group . (c) The Dirichlet region D0 () is a regular hyperbolic polygon with 2m edges. Following the construction for the compact connected orientable case, let Pη be the regular 2m-sided h-polygon that is a natural candidate for a fundamental region of a nonorientable surface of genus m ≥ 3. Set Qη z = RSη z, a paddle motion in G (see p. 171). Note that Qη u0 = u1 , Qη u1 = u2 and Qη P ∩ P = REη . So δ0 = Qη can be used to identify the edges Eη and REη . Let δ1 = R 2 Qη R −2 , δ2 = R 4 Qη R −4 , and so forth until finally δm−1 = R 2m−2 Qη R −2m+2 = R −2 Qη R 2 because R 2m = ι. Thus the transformations δ0 , . . . , δm−1 can be used as edge pairings for P as illustrated in Figure 7.5 for m = 4. Using the δj edge pairings to identify points of P produces a quotient map p : P → M and quotient space M such that M is a compact connected nonorientable surface of genus m. For the subgroup  of G generated by δ0 , . . . , δm−1 to be a covering group of a nonorientable surface of genus m, it is necessary that 2 δm−1 . . . δ12 δ02 = ι.

.

(7.9)

A routine calculation shows that m

2 δm−1 . . . δ12 δ02 = R −2 Q2η .

.

(7.10)

The next proposition is then the first step in proving that there exists η such that equation (7.9) holds. Proposition 7.2.12 The equation R −2 Q2η = (R −1 Sη )2 is now valid. Proof For this proof, it is helpful to use functional notation for the complex conjugate by letting Cz = z, for example, Qη = RSη C. Obviously, R m z = −z

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7 Flows and Hyperbolic Geometry

Fig. 7.5 The hyperbolic edge pairing for a nonorientable compact connected surface of genus 4

b a

b

a

c

d

c d

commutes with C, that is, R m C = CR m . Since the matrix for Tρ is real, Tρ C = CTρ and Tρ R m Tρ−1 C = CTρ R m Tρ−1 . A simple calculation shows that CRC = R −1 . Putting the pieces together, we have R −2 Q2η = R −2 RSη CRSη C

.

= R −1 Sη CRTρ R m Tρ−1 C = R −1 Sη CRCTρ R m Tρ−1 = (R −1 Sη )(R −1 Sη ) = (R −1 Sη )2 to complete the proof.



Corollary 7.2.13 Given an integer m ≥ 3, there exists a real number η > 2 such that equation (7.9) holds. Proof Applying Proposition 7.2.4 to the right side of R −2 Q2η = (R −1 Sη )2 shows that R −2 Q2η is an h-rotation about u0 conjugate to z → ei2μ(η) z. It is apparent from equation (7.10) that equation (7.9) holds when μ(η) = π/m. Then Corollary 7.2.3 again completes the proof. 

For the rest of this section,  will denote the subgroup of G generated by δ0 , . . . , δm−1 such that equation (7.9) holds. The construction of  reduces the proof of Theorem 7.2.11 to proving the following analog of Theorem 7.2.8 by referring again to [42]: Theorem 7.2.14 The group  is discrete and acts freely and properly on B2 , the quotient B2 /  is homeomorphic to the compact connected nonorientable surface

7.3 Lifts of Closed Curves Again

205

M of genus m, and the restriction of the quotient map B2 → B2 /  to P is the quotient map p : P → M. Proof We again want to use the argument in [42] beginning with the first paragraph on p. 277. Since this is a slightly different context, the essential ingredients for this argument must be in place. First, we have an h-polygon P in B2 and edge pairings δ0 , . . . , δm−1 in G such that δj P ∩ P is an edge of P . Using these edge pairings to identify points of P produces a compact connected nonorientable surface of genus m, that is, there exists a quotient map p : P → M given by the edge pairings. 2 Second, we have shown that δ02 · · · δm−1 = ι and .

  2 δ0 , . . . , δm−1 | δ02 · · · δm−1 =1

is a presentation of the fundamental group of M [42, Example 10.10].



Corollary 7.2.15 The Dirichlet region D0 () equals P . Although it is possible to construct a Fuchsian covering group  such that B2 /  is homeomorphic to any specified compact connected orientable surface of genus at least 2, there is no Fuchsian covering group  such that B2 /  is homeomorphic to the torus (see [38] pp. 35 and 36). Consequently, there is also no discrete subgroup  of G such that B2 /  is homeomorphic to the Klein bottle.

7.3 Lifts of Closed Curves Again Let X be a compact connected orientable surface of genus at least two. From Section 7.2, we know that there exists a Fuchsian covering group . with a compact Dirichlet region such that .B2 /  is homeomorphic to X. Since . is a Fuchsian covering group, the natural projection .π : B2 → B2 /  is a universal covering space of .B2 /  and hence also of X. In particular, . is algebraically isomorphic to the fundamental group of X. Although . is not unique, it has the following several important properties that will be used: (a) Every element of . \ {ι} is a hyperbolic Möbius transformation and . is a type I Fuchsian group (Theorem 7.1.19 and Proposition 7.1.23). (b) If the axes of two elements of . \ {ι} are not equal, then all four endpoints of these two axes are distinct points of .S1 (Theorem 7.1.6). (c) Two elements of . \ {ι} commute if and only if they have the same axis (Proposition 6.2.18). (d) If L is an axis of a hyperbolic transformation in ., then .{S ∈  : SL = L} is a maximal cyclic subgroup of . isomorphic to .Z (Proposition 7.1.15), and its two generators are primitive elements of ..

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7 Flows and Hyperbolic Geometry

(e) If .U1 and .U2 are open intervals of .S1 , then there exists a hyperbolic transformation in . with one endpoint of its axis in .U1 and the other in .U2 (Theorem 6.3.8). Recalling that the covering group for the torus was the group of parabolic Möbius transformations .T z = z + m + in such that .m + in ∈ Z[i], a Euclidean line L in .C is rational if and only if .T L = L for some .T z = z + m + in such that .m + in is in .Z[i] \ {0}. (See part (e) of Proposition 5.1.3.) This suggests that given a Fuchsian covering group . with a compact Dirichlet region, a hyperbolic line should be called a rational h-line (with respect to .) provided that .T L = L for some .T ∈  \ {ι}. In other words, the rational h-lines are just the axes of the hyperbolic transformations in .. An endpoint of a rational h-line will be called a rational point of .S1 (with respect to .). As with the torus, universal lifts and control curves will be useful, but we must first adapt these ideas to surfaces of higher genus, starting with loops in X and their lifts. Consider a loop .β : [0, 1] → X that is not null-homotopic, and let  : [0, 1] → B2 be a lift of .β to the universal covering space. There exists a unique .β u is (0) = β (1). The universal lift .β .T ∈ , the covering group of X, such that .T β again defined by equation (5.1), which is restated below: u (s) = T [s] β (s − [s]). β

.

u (s) is given by equation (5.2), which is Recall that the image of .β u (R) = β



.

  [0, 1) . T nβ

n∈Z

u = S β u is just the universal lift for the lift .S β  of .β, so Given .S ∈ , the curve .S ◦ β  every lift of .β is part of a universal lift. The universal lifts .S βu are most useful when .β is a simple closed curve. As in Chapter 2, .γ (X) denotes the genus of a compact surface. Like the torus, we have the following test for simplicity: Proposition 7.3.1 When .β is a loop that is not null-homotopic on a compact connected orientable surface X with .γ (X) ≥ 2, the loop .β is a simple closed curve u has the following properties: if and only if a universal lift .β u is an injective curve. (a) .β u (R) or the empty set for all .S ∈ . u (R) ∩ β u (R) is either .β (b) .S β Exercise 7.3.2 Prove Proposition 7.3.1.

 When .β is a simple closed curve that is not null-homotopic, .J = β [0, 1] is an u (R) is a component of .π −1 (J ). Every component of embedded circle and .J = β −1 .π (J ) has the form .S J for some .S ∈ , and .π |J is a universal covering of J . There are also properties of universal lifts that are different when .γ (X) ≥ 2. Since hyperbolic Möbius transformations will play a central role in what follows, it will be convenient to denote the axis of T by .LT , which is the same notation we used when T was a covering transformation of the torus. The endpoints, .a + and .a − ,

7.3 Lifts of Closed Curves Again

207

of .LT are defined by .

u (t) = a + and lim β

t→∞

u (t) = a − , lim β

t→−∞

and .J ∪ {a + , a − } = J∞ is an arc in .D2 with endpoints .{a + , a − }. Then .J∞ with either of the arcs of .S1 determined by .a + and .a − is an embedded circle in .C. The interiors of these two embedded circles will be particularly useful. A simple curve .f : R → B2 is said to be the type of an h-line provided that there exists an h-line L in .B2 and a constant .D > 0 satisfying: (a) .dh (f (s), f (0)) → ∞ as .|s| → ∞. (b) .dh (f (s), L) < D for all .s ∈ R. (c) For every .z ∈ L, there exists .s ∈ R such that .dh (f (s), z) < D. Using equidistant curves (see p. 168), the statement and proof of Proposition 5.2.1 can be easily modified to establish the following hyperbolic version: Proposition 7.3.3 Let .f : R → B2 be a simple curve that is the type of the h-line L, and let .E1 and .E2 be equidistant curves to L such that L and .f (R) are contained in the region between .E1 and .E2 . If .γ : [0, 1] → B2 is a simple curve such that .γ (0) ∈ E1 and .γ (1) ∈ E2 , then .γ ([0, 1]) ∩ f (R) = φ. Exercise 7.3.4 Prove Proposition 7.3.3. Similarly, the basic properties of universal lifts of simple closed curves on the torus that are not null-homotopic (Propositions 5.2.3, 5.2.5, and 5.2.6) can also be recast and proved in a straightforward way for compact connected orientable surfaces with .γ (X) ≥ 2 by using hyperbolic geometry. The hyperbolic statements of these results follow with their proofs left to the reader. Proposition 7.3.5 Let .β : [0, 1] → X be a simple closed curve that is not null : [0, 1] → B2 be a lift of .β. If T is the element of . such homotopic, and let .β u of .β is the type of the axis of T . (0) = β (1), then the universal lift .β that .T β Furthermore, the points .

u (t) and lim β

t→∞

u (t) lim β

t→−∞

of .S1 are the points at infinity of the axis of T . Proposition 7.3.6 If .β : [0, 1] → X is a simple closed curve that is not null : [0, 1] → B2 is a lift of .β, and T is the covering transformation homotopic, .β u has the following (0) = β (1), then the image .J of the universal lift .β such that .T β properties: (a) .T J = J. (b) There exist open connected sets U and V of .B2 such that .B2 \ J = U ∪ V , − =U ∪J , and .V − = V ∪ J. .U

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(c) If .E1 and .E2 are equidistant curves to .LT , the axis of T , such that .LT and .J lie between them, then .E1 ⊂ U and .E2 ⊂ V or vice versa. (d) .T U = U and .T V = V . Proposition 7.3.7 Let .β : [0, 1] → X be a simple closed curve that is not null : [0, 1] → B2 of .β, then T is primitive. (0) = β (1) for a lift .β homotopic. If .T β Exercise 7.3.8 Use hyperbolic geometry to prove Propositions 7.3.5 and 7.3.6. In preparation for Proposition 7.3.10, we need the following proposition of hyperbolic geometry: Proposition 7.3.9 If L and .L are disjoint h-lines, then there exist .ζ ∈ L and .ζ ∈ L such that the h-line determined by .ζ and .ζ is perpendicular to both L and .L . Moreover, dh (ζ, ζ ) = inf{dh (z, z ) : z ∈ L and z ∈ L } ≡ dh (L, L ).

.

Proof If .L = LR and the Euclidean center of .L is on the y-axis, then obviously .ζ = 0 and .ζ = L ∩ LI . It is an exercise to verify the distance formula using Proposition 6.2.12. So it suffices to reduce the general case to this special case using hyperbolic rigid motions. There exists .T ∈ G mapping L to .LR . Then .L is in the upper or lower hyperbolic plane determined by .LR . Using .z → −z, if needed, we can assume that .L is in the upper half plane. There exists an equidistant curve E to .LR tangent at .L at some point .(a, b). Then there exists .Tρ such that .Tρ (a, b) lies on .LI and .Tρ L is tangent to E, forcing the Euclidean center of .Tρ L to lie on the positive y-axis. 

Proposition 7.3.10 Given T in . with axis .LT , let .BT be the family of closed curves u that is the β : [0, 1] → X that are not null-homotopic and have a universal lift .β type of .LT . There exists .ρ > 0 such that .β ∈ BT is not a simple closed curve when u (t), LT ) > ρ for all .t ∈ R. .dh (β .

(R) lies in one of the two half u (t), LT ) > 0 for all .t ∈ R, then .β Proof If .dh (β planes determined by .LT . Thus it suffices to show that the result holds for all .β ∈ BT that lie in one of these half planes, say H . Let I be the open interval of .S1 bounding H . By Theorem 6.3.8, there exists .S ∈  such that the endpoints of its axis are both in I . Note that S can be chosen so that there exists a positive integer m such that both endpoints of .S m LT are in I and so that the hyperbolic region between .LT and .S m LT is .H ∩ S m H because m is a hyperbolic transformation. By Proposition 7.3.9, there exists an h-line .L⊥ .S perpendicular to .LT at .ζ and perpendicular to .S m LT at .ζ such that dh (ζ, ζ ) = inf{dh (z, z ) : z ∈ LT and z ∈ S m LT }.

.

Set .ρ = 1 + dh (ζ, ζ )/2, and suppose .β ∈ BT is a simple closed curve such that (R) ⊂ H and .dh (β u (t), LT ) > ρ for all .t ∈ R. Let E be the equidistant curve .β

7.3 Lifts of Closed Curves Again

209

to .LT at distance .ρ and lying in H . There exists another equidistant curve .E at u (R) lies between E and .E . Let .μ be the midpoint of distance .ρ > ρ such that .β the h-line segment joining .ζ and .ζ . Let R and .R denote the rays of .L⊥ starting at m .μ and passing through .ζ and .ζ , respectively. Observe that .E ∩ R and .S E ∩ R are points such that dh (E ∩ R , ζ ) > dh (ζ, ζ )/2 and dh (S m E ∩ R, ζ ) > dh (ζ, ζ )/2

.

as illustrated in Figure 7.6. It follows that .E and .S m E have these properties: dh (E ∩ R , ζ ) > dh (ζ, ζ )/2 and dh (S m E ∩ R, ζ ) > dh (ζ, ζ )/2.

.

(R) and .S m β (R) intersect in both Now Proposition 7.3.3 can be used to prove that .β of the small regions bounded by 4 circular arcs shown in Figure 7.6. So .β is not a simple closed curve by Proposition 7.3.1. 

Proposition 7.3.11 If .(B2 , R) is the lift of a flow .(X, R) on a compact connected orientable surface with .γ (X) ≥ 2, then .(B2 , R) extends to a flow on .(D2 , R) by setting .wt = w for all .w ∈ S1 and .t ∈ R. Proof Given a sequence .wn in .B2 converging to .w ∈ S1 and a sequence .sn converging to s in .R, it suffices to prove that .wn sn converges to w. There exists .σ > 0 such that .−σ ≤ sn ≤ σ for all n. Since .D0 (), the Dirichlet region centered at 0, is a compact subset of .B2 by Theorem 7.1.17, the set .D0 ()[−σ, σ ] is also compact in .B2 . So there exists .ρ > 0 such that .dh (0, z) ≤ ρ for all z in .D0 ()[−σ, σ ]. Because .D0 () is a fundamental region, for each .wn , there exists .Tn ∈  such that .Tn wn is in .D0 (). Hence, .dh (wn sn , wn ) = dh (Tn wn sn , Tn wn ) ≤ 2ρ and .wn sn is in the hyperbolic ball .Bn = {z : dh (z, wn ) ≤ 2ρ} of radius .2ρ. With d denoting (R) must intersect Fig. 7.6 .β mβ (R)

.S

S m LT ζ E



E LT

SmE

μ ζ L

SmE

H

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7 Flows and Hyperbolic Geometry

the Euclidean metric, it follows from Corollary 6.1.30 that .

lim d(w, {z : dh (z, wn ) ≤ 2ρ}) = 0,

n→∞

and hence, .d(wn sn , w) converges to 0 to complete the proof.



2 .(, B , R)

Corollary 7.3.12 If is the bitransformation group associated with the flow .(X, R) on a compact connected orientable surface with .γ (X) ≥ 2, then 2 2 .(, B , R) extends to a bitransformation group .(, D , R) by setting .wt = w for 1 all .w ∈ S and .t ∈ R. Proposition 7.3.13 Let x be a periodic point with minimal period .τ of a flow on a compact connected orientable surface X with .γ (X) ≥ 2, and let .(B2 , R) be the lifted flow. If the orbit of x is not null-homotopic and . x is in .π −1 (x), then: (a) The orbit of . x is the image of a universal lift of the simple closed curve .β(t) = x(tτ ). (b) The function .t →  x t is the type of an axis L of a primitive T in .. (c) The limits .limt→∞  x t and .limt→−∞  x t exist and are the points at infinity of the axis L. Exercise 7.3.14 Prove Proposition 7.3.13. Proposition 7.3.15 Let .(B2 , R) be the lift of a flow .(X, R) on a compact connected orientable surface with .γ (X) ≥ 2. If .x ∈ X is a non-periodic point and lim | x t| = 1

.

t→∞

for . x ∈ π −1 (x), then there exists .a ∈ S1 such that .

lim  x t = a.

t→∞

Proof Since x is not periodic, .t → xt = π( x t) is a simple curve on X. By Proposition 7.1.23, . is a mobile type I Fuchsian group. All the elements of . except .ι are hyperbolic transformations by Theorem 7.1.19. The result now follows from Theorem 6.3.10. 

Corollary 7.3.16 Let .(D2 , R) be the extension of .(B2 , R), the lift of a flow .(X, R) on a compact connected orientable surface with .γ (X) ≥ 2. If .ω(z) ⊂ S1 , then .ω(z) = {a}. Recall from Standing Assumption 2 (page 127) that the flow box of a local section for a flow on compact connected surface will always be contained in an open connected set that is evenly covered by the universal covering. Frequently, multiple lifts of a local section will play a role in a proof, and it will be useful to understand how they can be arrayed in .B2 with respect to an axis of a Möbius transformation in ., a Fuchsian covering group with a compact Dirichlet

7.3 Lifts of Closed Curves Again

211

region. If T is a primitive element of . with axis .LT , then .[T ] = {T n : n ∈ Z} is a maximal cyclic subgroup of . and . is the disjoint union of the left cosets .[T ]S. Since . is countable and the index of the subgroup .[T ] of . is not finite, there exists a sequence of coset representatives .Qk , .k ∈ Z+ , that is, every .S ∈  is in .[T ]Qk for a unique .Qk . The following general result applies to lifts of local sections and their flow boxes. Proposition 7.3.17 Let . be a Fuchsian covering group for a compact connected orientable surface X, let T be a primitive transformation in . with axis .LT , and let .Qk , .k ∈ Z+ be a sequence of coset representatives for the maximal cyclic group  is a lift of a compact connected subset C of an evenly covered connected .[T ]. If .C open subset of X, then the following hold:  LT ) = dh (Qk C,  (a) Given k, .dh (T n Qk C,  LT )nfor all .n ∈ Z.  −1   (b) .π C = {S C : S ∈ } = k∈Z+ n∈Z T Qk C.  (c) .limk→∞ dh (Qk C, LT ) = ∞. Proof The first two parts are exercises. For the third, recall that .dh (z, T z) = c is a constant for .z ∈ LT . Let .L1 and .L2 be h-lines perpendicular to .LT at .w1 and .w2  Then for such that .dh (w1 , w2 ) > c + 2δ where .δ is the hyperbolic diameter of .C.  lies between .L1 and .L2 . every .Qk , there exists n such that .T n Qk C Given .D > 0, let .E1 and .E2 be equidistant curves to .LT on opposite sides of 2 .LT such that .dh (Ej , LT ) = D. Because . is discontinuous at every point of .B ,  can lie in the region bounded by .L1 , .L2 , .E1 , and only a finite number of copies of .C  LT ) > D when .E2 (see Figure 7.7). Thus there exists .K > 0 such that .dh (Qk C, n  LT ) = dh (Qk C,  LT ) for .n ∈ Z+ . .k > K, since .dh (T Qk C, 

Exercise 7.3.18 Prove parts (a) and (b) of Proposition 7.3.17. Let X be a compact connected orientable surface such that .γ (X) ≥ 2, and let (X, R) = (X, R, ϕ) be a flow where .ϕ : X × R → X defines the flow. Recall that

.

Fig. 7.7 The region bounded by .E1 , E2 , L1 and .L2

L2

L1 E1

LT

w1

w2

E2

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7 Flows and Hyperbolic Geometry

[xσ, xτ ]ϕ = {xt : σ ≤ t ≤ τ } and .(xσ, xτ )ϕ = {xt : σ < t < τ }. Naturally, . ϕ will denote the function that defines the lifted flow .(B2 , R). Let .λ be a local section of length .2α for the flow .(X, R), and let .xσ and .xτ with .σ < τ be consecutive crossings of .λ. As before,

.

J = [xσ, xτ ]ϕ ∪ [xτ, xσ ]λ

.

(7.11)

is an embedded circle in X. Following Standing Assumption 3 on p. 127, a parameterization .β of J follows .[xσ, xτ ]ϕ first and then moves along .λ from .xτ to  starting at .β (0) =  x σ and ending .xσ . When .β is not null-homotopic, it has a lift .β  x σ , where T is a primitive covering transformation ( Proposition 7.3.7). at .β (1) = T  x τ is in Because .λ is assumed to be in an evenly covered open connected set, . u can be obtained from equation (5.1), and .J = β u (R) is again .T  λ. A universal lift .β called a control curve. Using equation (5.2), as in Chapter 5, equation (5.6) provides a description of a control curve, that is, J =



.

 x σ,  x τ ] T m [ x τ, T  x σ ]T λ , ϕ ∪ [

m∈Z

where . x ∈ π −1 (x),  .λ is the lift of .λ containing . x , and T is the element of . such (1) = T  x σ . Because T is an automorphism of . that .β ϕ , equation (5.7) is valid in the current context, that is,

m = .J [T m x σ, T m x τ ] x τ, T m+1 x σ ]T m+1λ . ϕ ∪ [T  m∈Z

Proposition 7.3.19 Let .Jbe a control curve as defined by (5.6) for two consecutive crossings of the local cross section .λ for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 2. If .O+ (˜z) crosses .Tm J for a sequence of distinct covering transformations .Tm ∈  and an increasing sequence of crossing 1 times .tm , then  .zt → a ∈ S as .t → ∞. Proof Note that each .Tm J is also a control curve, and hence, .O+ (˜z) can cross .Tm J at most once. In particular, .Tm J = Tn J for .m = n. Consequently, .Tm J ∩ Tn J = φ when .m = n because .J = π(J) is a simple closed curve in X. It follows that + ⊂ Tm J+ when .m < n because .tm < tn when .m < n. .Tn J Let .αm and .βm denote the fixed points of .Tm . There are two closed intervals of 1 + . It follows .S with endpoints .αm and .βm ; let .Im denote the one that bounds .Tm J  is properly that .Im+1 ⊂ Im for all m. Therefore, because the action of . on .X discontinuous, ∞  .

m=0

Im = {a}.

7.3 Lifts of Closed Curves Again

213

From the construction of .J, it follows that there exists .c > 0 such that dh (w, LT ) ≤ c for all .w ∈ J, where .LT is the axis of T . Because .Tm is an isometry, we have .dh (Tm w, Tm LT ) ≤ c for all .w ∈ J and .m ≥ 0, and hence,

.

∞  .

(Tn J+ )− = {a}.

n=1

+ for .t > tm . Finally,  .zt → a as .t → ∞ because  .zt ∈ Tm J



Proposition 7.3.20 Let .λ be a local section for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, let .yτ1 and .yτ2 be distinct consecutive crossings of .λ by .O(y) with .τ1 < τ2 , and let J be given by .J = [yτ1 , yτ2 ]ϕ ∪ [yτ2 , yτ1 ]λ . If y is not periodic and is in .ω(x) for some x, then: (a) J is not a null-homotopic loop. (b) The semi-orbit .O+ ( x ) goes to infinity for . x ∈ π −1 (x). Proof Let . y be a lift of y. If J is null-homotopic, then . y τ1 and . y τ2 are crossings of a lift  .λ of .λ. Hence, there exist consecutive crossings of  .λ by the positive orbit of . y. So it suffices to assume that . y τ1 and . y τ2 are consecutive crossings of  .λ and derive a contradiction with a Poincaré–Bendixson style argument. It follows that .J = [ y τ2 ] y τ2 ,  y τ1 ]λ is a simple closed curve in .C or y τ1 ,  ϕ ∪ [ 2 2 .B that separates either .C or .B into positively invariant and negatively invariant components denoted by .JP and .JN , respectively. Moreover, . y (τ2 + α) is in .JP and . y (τ1 − α) is in .JN , where .2α is the length of the local section .λ (Corollary 4.3.2). Either .JP− = JP ∪ J or .JN− = JP ∪ J is a compact subset of either .C or .B2 . By the continuity of the flow, there exists a neighborhood U of . y (τ1 − α) contained in .JN such that .V = U (τ2 − τ1 + 2α) is a neighborhood of . y (τ2 + α) contained in .JP . Because .y ∈ ω(x), there exist lifts . xn of x and positive real numbers .σn such that . xn σn converges to . y (τ1 − α) and .σn → ∞. Without loss of generality, . xn σn is in U and .σn+1 − σn > τ2 − τ1 + 2α for all n. It follows that . xm σn ∈ JP and . xm =  xn when .m < n. Thus . xm =  xn when .m = n, and there exists a sequence .Sn of distinct elements of . such that . xn = Sn x1 . When .γ (X) ≥ 2, the Fuchsian group . is discontinuous at every point of .B2 by definition (p. 183). When .γ (X) = 1, it is easily seen that the covering group . is discontinuous at every point of .C. The proof will be completed by showing that assuming .JN− is compact leads to a contradiction of the discontinuity of . at every point of either .C or .B2 as does assuming .JP− is compact. If .JN− is compact, then . xn has a convergent subsequence . xnj because . xn = ( xn σn )(−σn ) is in .JN for all n. Thus .Snj  x1 converges to .ζ ∈ JN− , and . is not discontinuous at .ζ . (See Figure 7.8.) If .JP− is compact, then .ω( xn ) = ω( xn (σn + τ2 − τ1 + 2α)) is a nonempty subset of .JP− because .O+ (( xn (σn + τ2 − τ1 + 2α))) ⊂ JP . Consequently, Sn (ω( x1 )) = ω(Sn ( x1 )) = ω( xn ) ⊂ JP− ,

.

214

7 Flows and Hyperbolic Geometry

V yτ2 yτ1

U ζ

y(τ1 − α)

J Fig. 7.8 The case when .JN− is compact

and .ω( x1 ) = φ. Let z be a point in .ω( x1 ). Then a subsequence of .Sn z converges to a point .ζ in .JP− and . is not discontinuous at .ζ , which is in .JP− . For part (b), it can also be assumed that .yτ1 and .yτ2 are consecutive crossings of .λ. By Corollary 4.2.9, there exists an increasing sequence .ti of positive real numbers such that .xti is in .(yτ1 , yτ2 )λ for all i. For a lift . y of y, there exists a primitive .T ∈  and a lift  .λ such that

 = y τ1 ,  .J T n [ y τ2 ] y τ2 , T  y τ1 ]T λ ϕ ∪ [ n∈Z

is a control curve. Given a lift . x of x, the positive orbit of . x must cross .Si J at .ti for some .Si ∈ . Since .O+ ( x ti ) is in .(Si J)+ , it follows that .Si is a sequence of distinct elements of .. Depending on whether .γ (X) = 1 or .γ (X) ≥ 2, Corollary 5.2.16 or x ) goes to infinity. 

Proposition 7.3.19 implies that the positive orbit .O+ ( Theorem 7.3.21 Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 2: (a) If x is positively {negatively} recurrent and not periodic, then  .

lim  xt

t→∞

 lim  xt

t→−∞

exists and is in .S1 for all . x in .π −1 (x). (b) If x is recurrent and not periodic, then the limits .

lim  x t and

t→∞

lim  xt

t→−∞

exist and are distinct points of .S1 . (c) If x is positively {negatively} recurrent and not periodic and x is in .ω(y) {.α(y)}, then

7.3 Lifts of Closed Curves Again

215

 .

lim  yt

t→∞

 lim  yt

t→−∞

exists and is in .S1 for all . y in .π −1 (y). Exercise 7.3.22 Prove Theorem 7.3.21 using Propositions 7.3.20 and 7.3.19. In Chapter 8, a generalization of control curves will be needed. Since control curves are the universal lifts of simple closed curves on the surface that are not null-homotopic, they must satisfy both of the conditions in Proposition 7.3.1. Weak control curves (defined below) need only satisfy the condition that the universal lifts are injective. Some of the useful properties of control curves still hold for weak control curves. The discussion of weak control curves will apply to both the torus and compact connected orientable surfaces with genus greater than one. Given a flow .(X, R) on a compact connected orientable surface X, let .λ be a local section of the flow, and let x be a point on the surface X that is not periodic. Suppose .xσ and .xτ with .σ < τ are distinct crossings of .λ such that the loop J  is a lift of J starting at . given by equation (7.11) is not null-homotopic. If .β x σ , then (1) = S x σ for some .S ∈  \ {ι}. Although S need not be primitive, there exists a .β primitive T in . and .κ ∈ Z+ such that .S = T κ . In this context, u (R) = J = β



.

mκ [T mκ  x σ, T mκ  x τ ]  x τ, T (m+1)κ  x σ ]T (m+1)κλ ϕ ∪ [T

(7.12)

m∈Z

is called weak control curve provided that for n in .Z nκ ( x σ,  x τ ) x τ, T κ  x σ ]T κλ = φ. ϕ ∩ T [

.

(7.13)

Proposition 7.3.23 If .J is a weak control curve given by equation (7.12), then u such that .J = βu (R) is injective. T κ J = J and the universal lift .β

.

Proof The equation .T κ J = J is an immediate consequence of equation (7.12). For the second part, note that .O(T m x ) ∩ O(T n x ) = φ when .m = n because .x = π( x) is not periodic (Corollary 2.2.9). So nκ T mκ [ x σ,  x τ ] x σ,  x τ ] ϕ ∩ T [ ϕ =φ

.

λ ∩ T nκ λ = φ whenever whenever m and n are distinct integers. In addition, .T mκ m and n are distinct integers by Standing Assumption 2. Combined with equation u is injective because it consists of a biinfinite sequence (7.13), it is now clear that .β of arcs meeting only consecutively at endpoints. 

Corollary 7.3.24 If .J is a weak control curve given by equation (7.12), then the u such that .J = β u (R) is the type of a rational (Euclidean or universal lift .β hyperbolic) line. Exercise 7.3.25 Use Propositions 5.2.3 and 7.3.5 to prove Corollary 7.3.24.

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7 Flows and Hyperbolic Geometry

It follows from the Jordan curve theorem applied to the one-point compactification of .C or .B2 that a weak control curve divides these universal covering spaces into two open connected sets U and V . Using ideas from the proofs of Propositions 5.2.5 and 7.3.6, it follows that .T κ U = U and .T κ V = V where .κ is the degree of the weak control curve. A weak control curve need not, however, have the property that .R J ∩ J is either .J or the empty set for all .R ∈ . Obviously, every control curve is a weak control curve, and results about weak control curves also apply to control curves. Like control curves, weak control curves are really only useful when x is non-periodic. Proposition 4.3.1 adapts to weak control curves on compact connected surfaces of genus greater than one with the same proof used for Proposition 5.2.12 on the torus. Setting  m =  q t : 0 < t < α and  q ∈ (T (m−1)κ  x τ, T mκ  x σ )T mκλ P

.

and  m =  q t : −α < t < 0 and  q ∈ (T (m−1)κ  x τ, T mκ  x σ )T mκλ , N

.

we have: Proposition 7.3.26 If .J is a weak control curve for a flow .(X, R) on a compact m and .N m are in different connected orientable surface X with .γ (X) ≥ 1, then .P 2  components of .B \ J . Corollary 7.3.27 If .J is a weak control curve and U and V are the components of 2 , then .P m is in U for all m or .P m is in V for all m. .B \ J  \ J that contains .P 0 will be called the positive side of the The component of .X   control curve .J and denoted by .JP . The other one will be called the negative side of the control curve and denoted by .JN . Given Propositions 5.2.12 and 7.3.27, we can reuse the proof of Proposition 5.2.14 to prove the more general result: Proposition 7.3.28 If .J is a weak control curve for a flow on a compact connected orientable surface X with .γ (X) ≥ 1, then .JP is a positively invariant set of .B2 and N is a negatively invariant set of .B2 for the lifted flow. .J

Chapter 8

Lifts and Limits

Beginning in this chapter, we present a more unified treatment of flows on compact connected surfaces of genus at least one. Chapters 5 and 7 provide the similar geometric structures that lead to unified statements about both geometric properties of the lifted flow on the universal covering space and purely dynamical results about flows on compact connected surfaces. Since the underlying metric spaces of both the Euclidean and hyperbolic planes are not compact, a curve mapping the positive reals into either of these geometric spaces can be bounded, unbounded, or go to infinity as the parameter increases to infinity. In particular, this applies to the lifts of positive orbits. The focus of this chapter is the interplay between the dynamical behavior of a positive orbit and the geometric behavior of its lifts, especially when and why they go to infinity. This involves lifts of limits and limits of lifts. The first section is devoted to some of Anosov’s theorems about classes of flows whose lifted positive orbits are either bounded or go to infinity. In this situation, Poincaré–Bendixson theory can be applied to the dynamical behavior of the bounded lifts of positive orbits and geometric methods can be used when the lift of a positive orbit goes to infinity. Homotopy theory provides a way of working with winding orbits. Every control curve represents an orbit winding once around the surface in a specific way that is not null-homotopic. We will be especially interested in disjoint windings in Section 8.2. This section ends with the result that the lift of a positively recurrent nonperiodic orbit converges to a point in .S1 that is not an endpoint of an axis of .. The third section of this chapter focuses on the .ω-limit sets of positive orbits of a flow on a compact connected orientable surface that have lifts converging to a point in .S1 . A point of .S1 is rational if it is an endpoint of an axis of . and irrational if it is not an endpoint of an axis of .. In the case of a rational point, there are two special types of .ω-limit points called remote and isolated .ω-limit points. The study of these two concepts is intertwined. The remote limit points are shown to be fixed points. This result is then used to prove a characterization of isolated .ω-limit points. This © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_8

217

218

8 Lifts and Limits

characterization is useful because it links the existence of isolated .ω-limit points of a positive orbit going to infinity to its lifts converging to rational points of .S1 as time increases. In the case of an irrational limit point, the moving points in the .ω-limit set are one of two types called Cantor and open recurrent .ω-limit sets at infinity. The fourth section of this chapter focuses on lifts of almost periodic points that are not periodic on a surface of genus greater than one. The main result is that the lift of such an orbit is the type of an h-line with irrational endpoints.

8.1 The Anosov Dichotomy In the 1960s Anosov first proved that there are classes of flows on surfaces for which the lift to the universal covering space of any positive or negative semi-orbit is either bounded or goes to infinity. Thus the semi-orbits exhibit a dichotomous behavior which we call the Anosov dichotomy. It is a useful dichotomy because for such a class of flows either the Poincaré–Bendixson theory or the geometric techniques developed in this book can be used to analyze the dynamics of every positive and negative orbit closure. Although Anosov reports speaking about these results in Russian conferences and seminars, they were essentially unknown in the West until he published his first paper [3] on them in 1988. This section focuses on proving that a flow with a finite number of fixed points on a compact connected surface exhibits the Anosov dichotomy. The proof makes use of the fact that the torus has finite Euclidean area and compact connected orientable surfaces have finite hyperbolic area and then depends upon carefully controlling the diameters of many different sets. We begin with a few pages of necessary general concepts and results to set the stage for the proof of the main theorem, beginning with a simple lemma about the diameters of sets. Lemma 8.1.1 Let U and V be sets of finite diameters a and b, respectively, in a metric space X with metric d. If x and y are elements of U and V , respectively, such that .d(x, y) > a + b, then .U ∩ V is empty. Exercise 8.1.2 Prove Lemma 8.1.1. It is now convenient to have definitions of a positive semi-orbit of a flow going to infinity or being unbounded that do not depend on the choice of a metric. Given a flow .(Y, R) on a locally compact second-countable metric space which obviously includes .C and .B2 , the positive semi-orbit .O+ (y) goes to infinity if for every compact subset C of Y there exists .T > 0 such that yt is not in C for all .t ≥ T . This definition is clearly equivalent to .limt→∞ |yt| = ∞ or .limt→∞ |yt| = 1 for a flow on .C or .B2 , respectively, but has the advantage of being an isomorphic invariant of the flow. Similarly, the positive semi-orbit .O+ (y) is unbounded if for every compact subset C of Y and .T > 0 there exists .t > T such that yt is not in C. For examples of flows on compact surfaces that have lifts of positive semi-orbits that are unbounded and do not go to infinity, see [3].

8.1 The Anosov Dichotomy

219

A (left) transformation group .(X, ) has an invariant metric provided that the topology on X is given by a metric d satisfying .d(hx, hy) = d(x, y) for all .h ∈  and .x, y ∈ X. In other words, for all .h ∈ , the function .x → hx is an isometry of X onto itself. For example, the hyperbolic metric .dh is an invariant metric for .(B2 , G  ) and the Euclidean metric is an invariant metric for the Euclidean rigid motions. Proposition 8.1.3 Let .(X, ) be a transformation group on a metric space with an invariant metric d. If the orbits .a are closed subsets of X for all .a ∈ X, then ρ(a, b) = inf{d(ha, h b) : h, h ∈ }

.

(8.1)

is a metric for the orbit space .X/ . Exercise 8.1.4 Prove Proposition 8.1.3. A special case of this proposition is a closed subgroup H of a metric group G with a left invariant metric and the continuous action of H on G is defined by .(h, g) → hg. The proof in this case (Theorem 1.4.7 in [59]) can easily be modified to prove the proposition. The metric .ρ given by Proposition 8.1.3 will be called a quotient metric. Corollary 8.1.5 Let .(X, ) be a transformation group with . a discrete group and X a metric space such that the action of . on X is both proper and free. If there exists an invariant metric d for .(X, ), then there exists a quotient metric .ρ for the orbit space .X/  given by (8.1). Proof By Proposition 1.1.19, the orbits of . are closed sets of X. So the proposition implies there exists a quotient metric .ρ given by (8.1).

Proposition 8.1.6 Let .(X, ) be a transformation group with . a discrete group and X a metric space having an invariant metric d. If the action of . on X is both proper and free, then the following hold for the quotient metric .ρ given by Corollary 8.1.5: (a) Given v in .X/ , there exists .αv > 0 such that the restriction of the natural projection .π : X → X/  to .{x : d(x, y) < αv } is an isometry onto {u ∈ X/  : ρ(u, π(y)) < αv }

.

for each .y ∈ π −1 (v). (b) If the orbit space .X/  is compact, then there exists .α > 0 such that .π |{x : d(x, y) < α} is an isometry of .{x : d(x, y) < α} onto {u ∈ X/  : ρ(u, π(y)) < α}

.

for all .y ∈ X. Moreover, h{x : d(x, y) < α} ∩ {x : d(x, y) < α} = φ

.

220

8 Lifts and Limits

for all .h ∈  \ {ι}. Proof To prove part (a) let y be an element of .π −1 (v). There exists an open neighborhood U of y such that .gU ∩ U = φ for all .g = e (Proposition 1.1.20) and .π |gU is a homeomorphism of gU onto .π(U ) for every .g ∈  (Corollary 1.1.21). Without loss of generality, .U = {x : d(x, y) < r} for some .r > 0. Set .V = {x : d(x, y) < r/3} and suppose that .ρ(π(w), π(x)) = d(w, x) for some .w, x ∈ V . Then there exist .h, h ∈  such that .d(hw, h x) < d(w, x). Setting −1 h , we have .g = h d(w, gx) = d(w, h−1 h x) = d(hw, h x) < d(w, x)

.

and .g = h−1 h = e. Consequently, d(y, gx) ≤ d(y, w) + d(w, gx) < d(y, w) + d(w, x) < r/3 + 2r/3 = r,

.

and x is in .U ∩g −1 U , a contradiction. Therefore, .π |V is an isometry of V onto .π(V ) such that .π(V ) ⊂ {u ∈ π(U ) : ρ(u, v) < r/3}, which is an open neighborhood of v. Given .u in .{u ∈ π(U ) : ρ(u, v) < r/3}, suppose that .u is not in .π(V ). Then there exists .w ∈ U \ V such that .d(w, y) > r/3. As in the previous paragraph, there exists .g ∈  with .π(w) = u such that .g = e and .d(gw, y) < r/3. This again produces the contradiction that .gU ∩ U = φ. Therefore, .π({x : d(x, y) < r/3}) = {u : ρ(u, v) < r/3} to complete the proof of part (a). The proof of part (b) builds on part (a). Since .X/  is compact, there exist .v1 , . . . , vm such that X/  =

m 

.

{u : ρ(u, vi ) < αvi /2},

i=1

where .αvi is given by part (a). Set .α = min{αv1 /2, . . . , αvm /2}, the first important constant in the proof of Anosov’s theorem. Given .w ∈ X/ , there exists i such that w is in .{u : ρ(u, vi ) < αvi /2} and hence .{u : ρ(u, w) < α} is contained in .{u : ρ(u, vi ) < αvi }. The remaining details follow easily from the properties of .{u : ρ(u, vi ) < αvi }.

Using the Euclidean metric, Propositions 8.1.5 and 8.1.6 apply to the torus and Klein bottle with their usual universal covers and covering groups. Using the hyperbolic metric on .B2 , they apply to .B2 /  for all covering subgroups of .G  . From Section 7.2, we know that for every compact connected orientable surface of genus at least 2 and every compact connected nonorientable surface of genus at least 3, there exists a covering subgroup . of .G  such that X is homeomorphic to .B2 / . Thus Propositions 8.1.5 and 8.1.6 apply to all the compact connected surfaces of interest here. In particular, they provide a metric and a base of evenly covered

8.1 The Anosov Dichotomy

221

open connected sets for the surface consisting of disks isometric to Euclidean or hyperbolic disks in the universal covering space of the surface. The hyperbolic metric can also be obtained from the Riemannian metric ds 2 =

.

4(dx 2 + dy 2 ) 4 dz dz = . (1 − x 2 − y 2 )2 (1 − zz)2

Using this Riemannian metric, the length of a differentiable curve .f : [a, b] → B2 given by .f (t) = (x(t), y(t)) = x(t) + iy(t) is 

b

.

a

 2 x˙ 2 + y˙ 2 dt. 1 − x2 − y2

A simple integration shows that the arc length of .f (t) = (t, 0) on the closed interval [0, r] is .ln[(1 + r)/(1 − r)] which agrees with the formula for .dh [(0, 0), (r, 0)] in (6.15). Given T in .G, a more difficult calculation shows that the arc length of .Tf (t) equals the arc length of .f (t). In particular, the arc length of a hyperbolic line segment is the hyperbolic distance between the endpoints of the segment. The hyperbolic area of E, a hyperbolic circle, triangle, polygon, and so forth, is given by .

 .

E

4dx dy (1 − x 2 − y 2 )2

and is invariant under transformation by elements of .G. (A proof of the invariance of area for the upper half-plane model can be found in Section 1.4 of [38].) The hyperbolic circumference and the hyperbolic area of a hyperbolic circle of radius r are .2π sinh r and .2π(cosh r − 1), respectively. Moreover, the hyperbolic area of a compact connected orientable surface X of genus .γ (X) > 1 is .4π(γ (X) − 1). This formula, which will be used in the main theorems of this section, can be obtained from the Gauss–Bonnet theorem for hyperbolic triangles (Theorem 1.4.2, [38]) and the Dirichlet region constructed in Section 7.2. Recall that when .xσ and .xτ are distinct consecutive crossings of a local section .λ of a planar flow, one component of the complement of .J = [xσ, xτ ]ϕ ∪ [xσ, xτ ]λ is bounded and hence has a compact closure. The loop J is called a Bendixson sack because it traps either the positive or the negative orbit in a compact subset of the plane. Proposition 8.1.7 Let .λ and .λ be local sections for a flow .ϕ on an open connected subset Y of .R2 . Let x be a point of Y such that .xσ and .xτ are distinct consecutive crossings of .λ with .0 < σ < τ . If .λ ∩ λ = φ, then .(xσ, xτ )ϕ ∩ λ contains at most one point. Proof If .(xσ, xτ )ϕ ∩ λ contains more than one point, there exist times s and .s  ,     .σ < s < s < τ such that xs and .xs are consecutive crossings of .λ . Let .J be the    Bendixson sack defined by .J = [xs, xs ]ϕ ∪ [xs, xs ]λ and note that .xσ and .xτ

222

8 Lifts and Limits

are in .JN and .JP , respectively. Since .xσ and .xτ are consecutive crossings of .λ and by hypothesis .λ ∩ λ = φ, it follows that .λ ∩ J  = φ. Therefore, either .λ ⊂ JN or  .λ ⊂ J P by the Jordan curve theorem because .λ is connected, contradicting either   .xτ ∈ J or .xσ ∈ J .

P N Proposition 8.1.8 Let  .λ be a lift of a local section .λ for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1. Let x be a point in X and let . x ∈ {π −1 (x)}. If .O+ ( x ) is unbounded and . x σ and . x τ are distinct consecutive .λ with .0 < σ < τ , then the following hold: crossings of  (a) .J = [ x σ,  x τ ] x σ,  x τ ]λ defines a Bendixson sack such that .O+ ( x τ ) is in the ϕ ∪ [ − x σ ) is in the interior of J . In particular, .JP is the exterior exterior of J and .O ( of J and .JN is the interior of J . (b) Neither . x nor x is periodic. (c) .( x σ,  x τ ) λ = φ for all .T ∈ \{ι}. In particular, .xσ and .xτ are consecutive ϕ ∩T  crossings of .λ. (d) .T J ∩ J = φ for all .T ∈  \ {ι}. (e) .T (J ∪ JN ) ∩ (J ∪ JN ) = φ for all .T ∈  \ {ι}. Proof Starting with part (a), the simple closed curve J defines a Bendixson sack by definition. By Proposition 4.3.15 and Corollary 4.3.2, .O+ ( x τ ) must be in the positively invariant component of .B2 \ J {.C \ J }, which must be the exterior x ) is unbounded by hypothesis. Hence .O− ( x σ ) is in the interior of because .O+ ( J . By definition, .JP is the exterior and .JN is the interior of J . Part (a) implies that .O− ( x ) is bounded. It follows from .O+ ( x ) unbounded and − x ) bounded that neither . .O ( x nor x is periodic. Moving on to (c), first notice that Proposition 8.1.7 applies with .λ = T  λ to show  that .( x σ,  x τ ) ∩ T λ contains at most one point for all . T ∈ . ϕ Suppose that it is not true that .( x σ,  x τ ) λ = φ for all .T ∈  \ {ι}. Clearly, ϕ ∩ T there exist at most a finite number of .T ∈  \ {ι} such that .( x σ,  x τ ) λ = φ. ϕ ∩ T Thus there exists a smallest .s ∈ (σ, τ ) such that . x s ∩ S λ = φ for some .S ∈  \ {ι}. n λ = φ for all .n ∈ Z, which is the weak control curve x σ,  x s) In particular, .( ϕ ∩S   by setting condition (7.13). Hence we can construct a weak control curve .K = K



.

  x σ,  x s] S n [ x σ,  x s]Sλ . ϕ ∪ [S

n∈Z

P , it follows that .J ⊂ K ∪K P by Proposition 7.3.28. (See Since .( x s,  x τ ] ϕ ⊂ K   Figure 8.1.) Consequently, .KN ∩ J = φ and .KN must be in the exterior of J , N , it follows that N ⊂ JP because .K N is unbounded. Since .O− ( that is, .K x) ⊂ K − − .O ( x ) ⊂ JP , contradicting .O ( x ) ⊂ JN . Part (d) follows from parts (b) and (c) and Standing Assumption 2. For the last part, suppose .T (J ∪ JN ) ∩ (J ∪ JN ) = φ for some .T ∈  \ {ι}. By part (d), Proposition 4.2.3 applies to J and T J . Thus either .T J ∪ T JN ⊂ JN or .J ∪ JN ⊂ T JN .

8.1 The Anosov Dichotomy

223

J

x τ

 K xσ

x s

P K

Sxσ

∪K P Fig. 8.1 The Jordan curve J is a subset of .K

By the Schöenflies theorem, the closed set .J ∪ JN is homeomorphic to .D2 , and the Brouwer fixed point theorem applies to T or .T −1 restricted to .J ∪ JN . Thus T or −1 has a fixed point in .J ∪ J . This is impossible because the elements of . \ {ι} .T N have no fixed points in .B2 {.C}, completing the proof.

We are now ready to prove that Anosov’s dichotomy occurs when the fixed point set is finite. The proof will follow the main ideas of Anosov’s proof except for the last paragraph where there is a small gap in the proof. He provided a bridge for this gap using Lemma 15 in [7], while we use Proposition 8.1.8 to finish the proof. The contradiction, however, is the same in both cases. Theorem 8.1.9 (Anosov) Let .(X, R) be a flow on a compact connected but not simply connected surface. If the set of fixed points of .(X, R) is finite, then the lift of every positive {negative} semi-orbit is either bounded or goes to infinity. Proof When .X = P2 , the lift of every orbit is bounded because its universal covering space is the compact space .S2 . The nonorientable case follows trivially from the orientable case because the lift to the double cover would also have a finite number of fixed points. So it will be assumed that X is orientable. It will also be assumed that .γ (X) ≥ 2. The proof for .γ (X) = 1, which is left for the reader, is essentially the same, but the geometry is simpler. By Proposition 7.3.11, the lifted flow on .B2 extends to .D2 by fixing the points in .S1 . So if the theorem is false, there must exist .x ∈ X and a lift . x of x such that .ω( x ) ∩ S1 = φ and .ω( x ) ∩ B2 = φ. The proof is completed by showing that such behavior leads to a contradiction of part(e) of Proposition 8.1.8 and depends on a careful construction using the hyperbolic quotient metric .ρh . The first step is to isolate the finite set F of fixed points. The first important constant is .α given by part (b) of Proposition 8.1.6. Set ε = inf{ρh (x, y) : x, y ∈ F and x = y}

.

and .ε = min{α, ε /3, 1/3}, the second important constant in the proof. Then .Uε = {x ∈ X : ρh (x, F ) < ε} consists of disjoint open connected sets each containing exactly one fixed point and each isometric to a hyperbolic disk of radius .ε such that ε = {z ∈ the closures of these disks are disjoint because .ε ≤ ε /3. It follows that .U

224

8 Lifts and Limits

B2 : dh (z, π −1 (F )) < ε} is the disjoint union of hyperbolic disks of radius .ε < 1/3 centered at the points of .π −1 (F ), which is the fixed point set of the lifted flow on 2 .B . The second step is to construct a sequence of Bendixson sacks in .B2 . The .ωlimit set of . x for the lifted flow extended to .D2 is connected (Proposition 1.2.7). It ε because .ω( cannot be contained in one of the components of .U x ) ∩ S1 = φ and 2 2 ε ) ∩ B . Clearly, .w .ω( x ) ∩ B = φ. Therefore, there exists .w  ∈ (ω( x) \ U  is not a fixed point of the lifted flow. Let .λ be a local section at .w = π( w) such that the diameter of .λ is less than .ε, and let  .λ be its lift containing .w . So .O+ ( x ) must cross  .λ infinitely often. Let + x ). . x a1 , . . . ,  x ai , . . . be the infinite sequence of consecutive crossings of  .λ by .O ( Theorem 4.3.9 implies that . x ai is a strictly monotonic sequence converging to .w . Then Ji = [ x ai ,  x ai+1 ] x ai ,  x ai+1 ]λ ϕ ∪ [

.

is a sequence of Bendixson sacks. Because .O+ ( x ) is unbounded, .O+ ( x ai+1 ) must − x ai ) must lie in the interior of .Ji for all i. lie in the exterior of .Ji and .O ( When working with sequences of Bendixson sacks, it will also be convenient to avoid double subscripts by using an alternative notation for the positively and negatively invariant regions determined by a Bendixson sack, namely, Ji+ = (Ji )P and Ji− = (Ji )N .

(8.2)

.

Thus the exterior, denoted by .Ji+ , is positively invariant and the interior, denoted by − .J i is negatively invariant. (See Figure 8.2.) Consequently, (a) .w  must lie in .Ji+ for all i. (b) . x ai+1 must lie between .w  and . x ai for all .i > 1. (c) .sup{dh ( x t,  x ai ) : ai ≤ t ≤ ai+1 } → ∞ as .i → ∞. ε . Set .β = The third step is to construct a countable open cover of .B2 \ U min{α, 1/5}, the third important constant. For each point .y ∈ X \ Uε , there exists a local section .λy of length .4δy at y such that the flow box satisfies: λy [−2δy , 2δy ] ⊂ {x : ρh (x, y) < β}.

.

w  x ai+1

Fig. 8.2 A Bendixson sack .Ji

Ji− Ji

x ai  λ

Ji+

8.1 The Anosov Dichotomy

225

Note that in the present context, the set .{x : ρh (x, y) < β} is an open connected evenly covered neighborhood of .y ∈ X and the components of .π −1 ({x : ρh (x, y) < β}) are open hyperbolic disks of radius .β by Proposition 8.1.6. Observe that the diameter of .λy [−2δy , 2δy ] is less than or equal to the diameter of .{x : ρh (x, y) < β} which is at most 2/5. It follows that the diameters of the components of .π −1 (λy [−2δy , 2δy ]) are also at most 2/5 because .π is an isometry of each component of .π −1 ({x : ρh (x, y) < β}) onto .{x : ρh (x, y) < β}. Let .fy : [−1, 1] → λy parameterize .λy such that .fy (0) = y, and set .λcy = fy ([−1/3, 1/3]). Then .λcy is also a local section at y, and .Wy = λcy [−δy , δy ] is a neighborhood of y (Proposition 4.1.8). Thus the collection of sets .{Wyo : y ∈ X \ Uε }, where .Wyo is the interior of .Wy , is an open covering of the compact set .X \ Uε , and a finite subset of them cover .X \ Uε . Rather than using .yn , .n = 1, . . . , m to index the finite open cover, we will simply use n, .n = 1, . . . , m and write .Wno , .Wn , .λn , .δn , and .fn . Of course, .fn (0) = yn , if needed. + Observe that .λ− n = fn ([−1, −1/3]), shown in Figure 8.3, and .λn = f ([1/3, 1]) − + are local sections such that .λn [−2δn , 2δn ] and .λn [−2δn , 2δn ] are neighborhoods of the points .fn (−2/3) and .fn (2/3), respectively. Since λn [−2δn , 2δn ] ⊂ {x : ρh (x, yn ) < β},

.

there exist open sets in both .λ− n [−2δn , 2δn ] (Picture a small disk in the flow box in Figure 8.3) and .λ+ n [−2δn , 2δn ] that are isometric to hyperbolic open disks of 2 − + .B . So there exists .η > 0 such that every .λn [−2δn , 2δn ] and .λn [−2δn , 2δn ] for .n = 1, . . . , m contains an open set isometric to a hyperbolic open disk of radius .η, the fourth important constant. Clearly, .η < β < α. The fourth step in the proof is to use the flow boxes constructed in step 3 to obtain a contradictory estimate of the hyperbolic area of the interior of one of the Bendixson sacks constructed in step 2. Choose a positive integer k such that .k2π(cosh η − 1) > 4π(γ (X) − 1), where .2π(cosh η − 1) is the area of a hyperbolic disk of radius .η. Then there exists a positive integer i such that x t,  x ai ) : ai ≤ t ≤ ai+1 } > 2(k + 1). dˆ = sup{dh (

.

For the rest of the proof i will be a fixed positive integer satisfying the above condition. Let Fig. 8.3 The flow box − [−2δ , 2δ ] has diameter n n at most 2/5

.λn

yn λ− n [ 2δn , 2δn ]

226

8 Lifts and Limits

ˆ x t,  x ai ) = d}. tˆ = inf{t : ai < t < ai+1 and dh (

.

Then set σ2j = sup{t : ai < t < tˆ and dh ( x t,  x ai ) = 2j }

.

and σ2j +1 = inf{t : σ2j < t < tˆ and dh ( x t,  x ai ) = 2j + 1}

.

for .j = 1, . . . , k. x σ2j ,  x σ2j +1 ] It follows that .Lj = [ ϕ is an arc starting on the hyperbolic circle with center at . x ai of radius 2j and ending on the hyperbolic circle with center at . x ai of radius .2j + 1. Moreover, it never leaves the hyperbolic annular region between these circles as shown in Figure 8.4. Thus the diameter of .Lj is at least one, and ε because the diameter of a component of .U ε is .Lj cannot lie in a component of .U ε . Then at most 2/3. Therefore, there exists a point .zj in .Lj such that .zj is not in .U  .dh (zj , zj  ) ≥ 1 for .j < j because the annular region between the circles of radius .2j + 1 and .2(j + 1) lies between .zj and .zj  . no of .π −1 (Wno ). It Now there exists a .Wnoj such that .zj lies in a component .W j j follows from the construction of .Wnj that there exists a lift  .λnj of .λnj and .τj such that .|τj | < δnj and .zj τj ∈  λcnj . It was pointed out in step 3 that the diameter of the .λnj [−2δnj , 2δnj ] is at most 2/5. Therefore, flow box   .λnj [−2δnj , 2δnj ] ∩  λnj  [−2δnj  , 2δnj  ] = φ Fig. 8.4 The arcs .L1 and .L2 are dashed and the annular regions between them are shaded

x σ3 z1 x σ2 x σ4 z2 x σ5

x ai  λ

8.1 The Anosov Dichotomy

227

for .j = j  by Lemma 8.1.1 because .dh (zj , zj  ) ≥ 1. Likewise,  .λnj [−2δnj , 2δnj ] ∩   λ = φ because .dh (ai , zj ) ≥ 2 for .j ≥ 1 and the diameter of .λ is at most .ε < 1/3. Proposition 8.1.7 implies that .[ x ai ,  x ai+1 ] λnj = {zj τj }. Hence either ϕ ∩ − − + .λn [−2δnj , 2δnj ] or .λn [−2δnj , 2δnj ] is contained in .J , the interior of .Ji , for i j j .j = 1, . . . , k. Thus the interior of .Ji contains disjoint hyperbolic disks .Dj of radius  .η for .j = 1, . . . , k. If .π(Dj ) ∩ π(Dj  ) = φ for .j = j , then the hyperbolic area of k  .

π(Dj )

j =1

equals .k2π(cosh η − 1) which is greater than .4π(γ (X) − 1) by the choice of k. This is impossible because .4π(γ (X) − 1) is the hyperbolic area of X. Thus there exist  .Dj and .Dj  with .j = j and .T ∈  with .T = ι such that .T Dj ∩ Dj  = φ. Hence − − .T J ∩ J =  φ, contradicting part (e) of Proposition 8.1.8.

i i The following corollary overlaps with Theorem 12 in [7]. Corollary 8.1.10 Let .(T2 , R) be a flow on the torus such that the fixed point set is finite. If .O+ ( x ), the lift of the positive semi-orbit of .x ∈ T2 , is unbounded, then + .O ( x ) lies between two parallel lines and .limt→∞ | x t| = ∞. Proof The theorem implies that .limt→∞ | x t| = ∞. Because .ω(x) contains a minimal set, .ω(x) contains a fixed point, a non-fixed periodic point, or a nontrivial minimal set. In the latter two cases, .ω(x) contains moving points and Theorem 5.3.14 applies. In the first case, .ω(x) contains one or more than one fixed point. If .ω(x) is a single fixed point, then .O+ ( x ) is bounded and this case is ruled out by the hypothesis. In the second case, .ω(x) must contain moving points because .ω(x) is connected, and again Theorem 5.3.14 applies.

Anosov’s proof of Theorem 8.1.9 (Theorem 2 in [3]) is in the context of smooth flows. However, he points out that the proof works for flows in general because of Whitney’s theorem and then describes how to modify the proof of Theorem 8.1.9 when the set of fixed points is a contractable subspace of X (Theorem 1 in [3]). A closed set F of a topological space X is a contractible subspace in X if there exists a continuous homotopy function .g : F × [0, 1] → X such that .g(x, 0) = x for all .x ∈ F and .g(F, 1) is a point .q ∈ X. Notice that the above definition of a contractible subspace is more general than the definition of a contractible space (Lee, [42, p. 161]) because q need not be in F . For example, .S1 is not a contractible space because all contractible spaces are simply connected. But an embedded circle in a surface X is contractible in X if and only if it is null-homotopic. (The extension of the embedding to .D2 (p. 124) provides the function g in Anosov’s definition by composing with .(z, s) → (1−s)z, mapping 1 2 .S × [0, 1] onto .D .)

228

8 Lifts and Limits

We will prove the following theorem which is not as general as Anosov’s stated theorem but fits more comfortably into the context of this book: Theorem 8.1.11 Let .(X, R) be a flow on a compact connected orientable surface with .γ (X) ≥ 1. If the set F of fixed points of .(X, R) is contained in a closed connected locally path connected set E that is contractible to a point in X, then the lift of every positive {negative} semi-orbit is either bounded or goes to infinity. The key to Anosov’s proof of Theorem 8.1.9 was isolating the fixed points of the lifted flow in a countable .-invariant family of disjoint open disks in step 1. The same strategy works when the set of fixed points is contained in a connected locally path connected set E that is contractible to a point in X. Most of the effort in the proof will go into a new first step followed by simple instructions on reusing the remaining steps of the proof of Theorem 8.1.9. We begin with a general proposition about proper free continuous actions. Proposition 8.1.12 Let .(X, ) be a proper free continuous action of a discrete group . on a metric space X. If C is a compact subset of X, then the set {γ ∈  : C ∩ γ C = φ}

.

is finite and there exists an open set U containing C such that {γ ∈  : U ∩ γ U = φ} = {γ ∈  : C ∩ γ C = φ}.

.

Proof Suppose .{γ ∈  : C ∩ γ C = φ} is not finite. So there exist sequences .xk and yk in C and a sequence .γk of distinct elements of . such that .xk = γk yk . Because C is compact and X is metric, we can assume without loss of generality that .xk and .yk converge, respectively, to x and y in C. Because the action is proper there exist open neighborhoods U and V of x and y, respectively, such that .{γ ∈  : U ∩ γ V = φ} is a finite set. And then there exists .K ∈ Z+ such that .xk ∈ U and .yk ∈ V for .k > K. Consequently, the infinite set .{γk : k > K} is contained in the finite set .{γ ∈  : U ∩ γ V = φ}, a contradiction. For the second part let d be a metric for X and set .U k = {x : d(x, C) < 1/k}. Note that every .Uk is an open set containing C and .C = ∞ k=1 Uk . If no .Uk satisfies the required condition, then there exist sequences .xk and .yk in .Uk and a sequence .γk of elements of . such that .xk = γk yk and .γk is not in the finite set .{γ : C ∩ γ C = φ}. For each .xk and .yk there exist .xk and .yk in C such that .d(xk , xk ) < 1/k and    .d(yk , y ) < 1/k. Because C is compact, we can assume that .x and .y converge to k k k x and y in C. Then .xk and .yk also converge to x and y, respectively. As in the first part of the proof there exist open neighborhoods U and V of x and y, respectively, such that .{γ ∈  : U ∩ γ V = φ} is a finite set. By convergence, there exists K such that .γk is in .{γ : U ∩ γ V = φ} when .k > K. Thus there exists a subsequence .γki of .γk such that .γki = γ  for all i and some .γ  in the finite set .{γ ∈  : U ∩ γ V = φ} and .xki = γ  yki . Therefore, .x = γ  y and .γ  is in .{γ ∈  : C ∩ γ C = φ}, contradicting the choice of .γk .

.

8.1 The Anosov Dichotomy

229

Proof (Theorem 8.1.11) Let F denote the set of fixed points of .(X, R). By hypothesis, there exists a closed connected locally path connected set E containing F and a homotopy function .g : E × [0, 1] → X such that .g(x, 0) = x for all .x ∈ E and .g(E, 1) is a point .q ∈ X. Form the quotient space .Y = (E × [0, 1])/ ∼ with .(x, s) ∼ (y, t) if and only if .(x, s) = (y, t) or .s = t = 1. (The space Y is called the cone over E in [72] and is also the mapping cylinder for the constant function on E in both [42] and [72].)  2   The set .{ (x, s), (y, t) : (x, s) ∼ (y, t)} is a closed subset of . E × [0, 1] , so Y is a compact Hausdorff space. It is readily shown that Y is connected, locally path connected, and hence path connected. Letting f denote the natural  projection  of .E × [0, 1] onto Y , the function .H : Y × [0, 1] → Y defined by .H f (x, s), t = f (x, t + (1 − t)s) shows that Y is a contractible space and thus simply connected [42, Theorem 7.24 and Exercise 7.7]. Clearly g is a quotient map such that .(x, s) ∼ (y, t) implies that .g(x, s) = g(y, t). Consequently, the function g passes to the quotient, that is, there exists a continuous function .g  : Y → X such that .g = g  ◦ f . Now the lifting criterion (p. 45) applies to the function .g  because Y is connected and locally path connected. Because Y is simply connected, for each . x ∈ π −1 (q), there exists a unique lift . g    of .g such that . g (f (E × {1})) =  x . Setting . g = g ◦ f , we have the following commutative diagram:

   g  g   

  f - Y  E × [0, 1] H H @ HH @  H g H @g HH @ HH @ H H @ H

 X

π

X

To complete the new step 1, it is necessary to show the dependence of . g on . x by writing . g g g  ◦ f is the unique lift of g such that . g (E × {1}) =  x for x , that is, . x = . x in .π −1 (q). The function . g satisfies the following equation: π( g g  (f (x, 0))) = g  (f (x, 0)) = g(x, 0) = x. x (x, 0)) = π(

.

 Set .E g g x = x (E × {0}). It follows that the function .x →  x (x, 0) is an injective map −1 −1    of E to .E x and .E x ∩ π (x) = φ for .x ∈ E. If we can show that .E x ∩ π (x) contains exactly one point, then .x →  g x (x, 0) is also surjective.

230

8 Lifts and Limits

If . g g x (x, 0) =  x  (y, 0), then the above equation again implies that .x = y. Since the paths .t → g x (x, t) and .t → g x  (x, t) are both lifts of .t → g(x, t), they are equal by the unique lifting property because E is connected. Therefore, . x = g x (x, 1) = −1   g x  and .E g g (x, 0) is x ∩ π (x) = { x (x, 0)}, implying that .x →  x  (x, 1) =    surjective and .E x ∩E x  = φ.  Consequently, .x →  g x ∈ π −1 (q), x (x, 0) is a homeomorphism of E onto .E x for . −1 and the sets .E x ∈ π (q), have the following properties: x for . (a) (b) (c) (d)

  x =  x . E x ∩E x  = φ when . .T E x = ET  x for .T ∈ .   .{T : T E ∩ E x in .π −1 (q). x x = φ} = {ι} for any given . −1 x . T  .π (E) = T ∈ E x = T ∈ T E .

  Clearly, Proposition 8.1.12 applies to .E x and there exists .ε > 0 such that 2    V x = {z ∈ B : dh (z, E x) < ε }

.

satisfies   {T ∈  : V x ∩TV x = φ} = {ι}

.

   or, equivalently, .V x ∩TV x = φ for .T = ι. At this point, it appears that .ε depends on . x , but it does not. Because T is an isometry, 2  2    T  T  TV x = T {z ∈ B : dh (z, E x ) < ε } = {z ∈ B : dh (z, E x) < ε } = V x.

.

  It is an exercise to verify that .V x ∩TV x = φ for .T = ι if and only if −1  S V x ∩ ST S V S x =φ

.

  for .T = ι. Since .T → ST S −1 T is an isomorphism of . onto itself, .V x ∩TV x =φ    for .T = ι if and only if .VS x. x ∩ T VS x = φ for .T = ι. Thus .ε is independent of .  Let .α be given by Proposition 8.1.6. Set .ε = min{ε /3, α, 1/3}, and set .U x =  {z ∈ B2 : dh (z, E x ) < ε}. Then − ∩ T U − = φ} = {ι} {T ∈  : U  x  x

(8.3)

− = φ − ∩ T U U  x  x

(8.4)

.

or, equivalently, .

− is compact and connected because .E  for all .T = ι and . x ∈ π −1 (q). Note that .U x  x  is compact and connected. Furthermore, .π |U x is a homeomorphism onto the open set .Uε = {x ∈ X : ρ(x, E) < ε} of X. The sets .Uε and

8.1 The Anosov Dichotomy

231

ε = π −1 (Uε ) = U



.

T  U x

T ∈

suitably isolate the fixed points of the flow on X and its lift to .B2 to implement the last 3 steps of the proof of Theorem 8.1.9 without changing steps 2 and 3. The fourth part requires little more than a change of scale. Again choose k such  that .k2π(cosh η − 1) > 4π(γ (X) − 1). The diameter .μ of .U x x is independent of . −1 in .π q. Setting .ν = μ + 1, there exists a positive integer i such that x t,  x ai ) : ai ≤ t ≤ ai+1 } > 2(k + 1)ν. dˆ = sup{dh (

.

For the rest of the proof i will be a fixed positive integer satisfying the above condition. Let ˆ x t,  x ai ) = d}. tˆ = inf{t : ai < t < ai+1 and dh (

.

Then set σ2j = sup{t : ai < t < tˆ and dh ( x t,  x ai ) = 2j ν}

.

and σ2j +1 = inf{t : σ2j < t < tˆ and dh ( x t,  x ai ) = 2(j + 1)ν}

.

for .j = 1, . . . , k. It follows that .Lj = [ x σ2j −1 ,  x σ2j ] ϕ is an arc starting on the hyperbolic circle with center at . x ai of radius .(2j )ν and ending on the hyperbolic circle with center x ai of radius .(2j + 1)ν. Thus the diameter of .Lj is at least .ν, and .Lj cannot lie at . ε because the diameter of a component of .U ε is .μ. Therefore, in a component of .U ε and .dh (zj , zj  ) ≥ ν for .j < j  there exists a point .zj in .Lj such that .zj is not in .U because the annular region between the circles of radius .2j + 1 and .2(j + 1) lies between them. The contradiction is obtained as it was in the proof of Theorem 8.1.9.

Corollary 8.1.13 Let .(X, R) be a flow on a compact connected but not simply connected orientable surface. If the set F of fixed points of .(X, R) is contained in a regular Euclidean ball of X, then the lift of every positive {negative} semi-orbit is either bounded or goes to infinity. Anosov’s example (Example 2 in [3]) of an unbounded lifted orbit that does not go to infinity depends on the construction of a simple closed curve of fixed points that is not null-homotopic. The presence of a simple closed curve of fixed points that is not null-homotopic does not rule out the Anosov dichotomy. Observe that applying Theorem 1.3 to a flow satisfying the Anosov dichotomy preserves the Anosov dichotomy. Now it is not difficult to construct flows with

232

8 Lifts and Limits

simple closed curves of fixed points that are not null-homotopic by applying Theorem 1.3 to periodic orbits that are not null-homotopic in flows having the Anosov dichotomy property.

8.2 Winding Anosov’s dichotomy theorem (Theorem 8.1.9) was the first theorem where the proofs for .γ (X) = 1 and .γ (X) ≥ 2 were essentially the same. This was possible because we now have similar geometric structures that can be used to study flows on compact connected surfaces of genus at least one. Specifically, given a flow .(X, R) on a compact connected surface with .γ (X) ≥ 1,  R) such that the flow .(X/  , R) is isomorphic there exists a bitransformation .(, X,  to .(X, R) and the action of . on .X is both proper and free. It follows that the natural  → X is a universal covering. The key to this approach is that . projection .π : X is either a group of rigid motions of the Euclidean plane or of the hyperbolic plane depending on whether .γ (X) = 1 or .γ (X) ≥ 2. This common structure allows us to take a more unified approach to the study of flows on compact connected surfaces of genus at least one. When an orbit crosses a local section and recrosses it at a later time to form a loop that is not null-homotopic, it is winding in some fashion around the surface. This is a phenomenon that does not occur on the sphere or on the projective plane. The consecutive crossings that give rise to a control curve or a weak control curve are the most familiar examples. A positive orbit that winds without unwinding is forced to enter many different fundamental regions and wander far from its starting point. In this section we explore connections between winding and the behavior of positive orbits as time approaches infinity. Proposition 8.2.1 Let .λ be a local section at a point x for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, and let  .λ be a lift of .λ containing . x ∈ π −1 (x). If .O+ ( x ) is unbounded and .O+ ( x) ∩  λ = { x }, then + x )∩T  .O ( λ contains at most one point for all T in .. Furthermore, if .xσ and .xτ are crossings of .λ with .0 ≤ σ < τ , then the loop J given by .J = [xσ, xτ ]ϕ ∪ (xσ, xτ )λ is not null-homotopic. Proof Suppose the conclusion is false. Then there exists .T ∈  \{ι} and .0 < σ < τ , such that . x σ and . x τ are consecutive crossings of .T  λ. Let J be the Bendixson sack defined by .J = [ x σ,  x τ ] x σ,  x τ ]T λ . Since .O+ ( x ) is unbounded, .O+ ( x τ ) is in ϕ ∪ [ x is in .JN , the interior of J . the exterior of J and thus .JP is the exterior of J . Clearly . −1 (J ∪J ) = φ, Now .O+ ( x )∩ λ = { x } implies that  .λ ⊂ JN . Therefore, .(J ∪JN )∩T N contradicting part (e) of Proposition 8.1.8. x σ and . x τ cannot be in If .xσ and .xτ are crossings of .λ with .0 ≤ σ < τ , then . the same lift of .λ and hence J is not null-homotopic.



8.2 Winding

233

The local sections used in proofs can often be modified without disturbing the hypotheses. For example, replacing the local section .λ with .[a, b]λ containing x is an easy way to modify .λ. Or replace x with .xμ where .μ = max{t :  xt ∈  λ}. These are the ways that the previous proposition will be used to study the behavior of the lifts of positive orbits. Proposition 8.2.2 Let x be a non-periodic point of a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, and let .xσ and .xτ with .0 ≤ σ < τ be crossings of a local section .λ for .(X, R). If there exists a lift  .λ of .λ containing . x σ where . x ∈ π −1 (x), a primitive .T ∈ , and .κ ∈ Z+ such that J =



.

mκ [T mκ  x σ, T mκ  x τ ]  x τ, T (m+1)κ  x σ ]T (m+1)κλ ϕ ∪ [T

(8.5)

m∈Z

is weak control curve of degree .κ, then (a) .T j J ∩ J = φ when .1 ≤ j ≤ κ − 1. (b) For each j , with .1 ≤ j ≤ κ − 1, there exists s, .σ < s < τ , such that . x s is in j ∩ J  and . .T J x s ∈ T k ( x τ, T κ  x σ )T κλ , where

k ≡ ±j (mod κ) ≡

.

j (mod κ)

or

(κ − j ) (mod κ).

m x τ, T κ  (c) If .( x σ,  x τ ) x σ ]T κλ = φ for all .m ∈ Z, then .κ = 1. ϕ ∩ T [ (d) Without assuming that equation (8.5) is a weak control curve, if .O+ ( x ) is n ( κ x σ,  x τ ) ∩ T x τ, T x σ ) =  φ for some . κ ≥ 2, and unbounded, .( κ ϕ T  λ .O( xσ ) ∩  λ = { x σ }, then .−κ < n < 0 or .0 < j < κ with .n + κ = j . (e) If .O+ ( x ) is unbounded, .O( xσ ) ∩  λ = { x σ }, and .j ∈ Z such that .0 < x sj is in j < κ, then there exists a unique .sj , .σ < sj < τ , such that . j x σ, T j −κ  .[T  x τ ]T jλ . nκ x τ, T κ  Proof By definition of a weak control curve, .( x σ,  x τ ) x σ ]T κλ = φ ϕ ∩ T [  for all .n ∈ Z. In the hyperbolic case, .J is the type of the axis .LT of T . Consequently, there exist equidistant curves .E1 and .E2 of .LT such that .J and .LT lie between .E1 and .E2 . Moreover, .J divides .B2 into 2 open connected sets U and V containing .E1 and .E2 , respectively. Since .T n Ei = Ei for all .n ∈ Z and i equal 1 or 2, it follows that .T κ U = U and .T κ V = V by connectivity. Suppose there exists j , .1 ≤ j ≤ κ − 1, such that .T j J∩ J = φ. Now we use the same technique used to finish the proof of Proposition 7.3.7. Clearly, .T j U ∩ U = φ and .T j V ∩ V = φ. Since .B2 = T j J ∪ T j U ∪ T j V and .J is connected, either  ⊂ T j U or .J ⊂ T j V . We will show that both cases lead to a contradiction. If .J  ⊂ T j U , then .J ∩ T j V = φ from which it follows that .T j V ⊂ V because .J j j .T V ∩ V = φ and .T V is connected. Since

T j J ⊂ T j V − ⊂ V − = V ∪ J

.

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8 Lifts and Limits

and .T j J ∩ J = φ, it follows that .T j J ⊂ V and .T j V = V . Therefore, V  T j V  · · ·  T j (κ−1) V  T j κ V = V ,

.

which is impossible. The same argument works when .J ⊂ T j V , and .T j J∩ J = φ when .1 ≤ j ≤ κ − 1. Therefore, .T j J ∩ J = φ when .1 ≤ j ≤ κ − 1, completing the proof of part (a). Because x is not periodic, .T n O( x ) ∩ O( x ) = φ for all .n = 0 (Corollary 2.2.9). λ ∩ λ = φ for all .n = 0. Therefore, .T j J ∩ J = φ By standing assumption 2, .T n when .1 ≤ j ≤ κ − 1 implies j +nκ T mκ ( x σ,  x τ ) ( x τ, T κ  x σ )T κλ = φ ϕ ∩T

.

or mκ T j +nκ ( x σ,  x τ ) ( x τ, T κ  x σ )T κλ = φ ϕ ∩T

.

for some m and n, both of which simplify to the conclusion of part (b). Since the proof of parts (a) and (b) is based entirely on the powers of a single primitive covering transformation, the proof for .γ (X) = 1 or .X = T2 is just a matter of replacing the equidistant curves with parallel lines. Part (c) is an immediate consequence of the proof of part (b). The proof of part (d) starts by applying Proposition 8.2.1 to assert that .O+ ( x) ∩ T λ contains at most one point for all T in .. λ. Without x σ  ∈ T j Suppose there exist .σ  , .σ < σ  < τ , and .j < 0 such that .  λ=φ loss of generality, .σ is the largest such time. Consequently, .( xσ ,  x τ ) ∩ T m ϕ when .m < 0. x ) crosses a copy of  .λ. Then Let .τ  be the first time greater than .σ  such that .O+ (   λ. Let . be the integer such that . λ. .σ < σ < τ ≤ τ because . x τ is in .T κ x τ  ∈ T  xσ ) ∩  λ = { x σ } and .τ  > σ . Then . − j ≥ 2 and Note that . > 0 because .O+ ( m λ = φ for .m ∈ Z. Therefore, with .κ  =  − j ≥ 2, .( xσ ,  x τ  ) ϕ ∩T  J =



.





T mκ ([ xσ ,  x τ  ] xτ , T κ  x σ  ]T λ ) ϕ ∪ [

m∈Z mλ = φ is a weak control curve of degree .κ  ≥ 2. By construction, .( xσ ,  x τ  ) ϕ ∩T   for .m ∈ Z and .κ = 1. This contradiction proves that .( x σ,  x τ ) ϕ does not intersect j λ for .j < 0. A similar argument shows that .( j λ for .T  x σ,  x τ ) ϕ does not intersect .T  .j > κ. Thus .0 < j < κ. Observe that .n + κ = j if and only if

λ. T n ( x τ, T κ  x σ )T κλ = (T n+κ  x σ, T n x τ )T n+κλ ⊂ T j

.

It follows that .0 < n + κ < κ if and only if .−κ < n < 0.

8.2 Winding

235

Part (e) follows from part (b) and the previous paragraph.



We now turn our attention to homotopic properties of control curves. Two closed curves or loops f and g are homotopic loops provided there exists a homotopy .H : [0, 1] × [0, 1] → X such that H (s, 0) = f (s);

.

H (s, 1) = g(s); and H (0, t) = H (1, t) for all s and t in .[0, 1]. The function .ξ × ι(s, t) = (e2π is , t) is a quotient map of 2 1 1 .[0, 1] onto .S × [0, 1] and so there exists a continuous function .Ho : S × [0, 1] → X such that .H = Ho ◦ (ξ × ι). Clearly, .Ho is just an ordinary homotopy between the circle representatives .fo and .go (p. 123). This is not the same as path homotopy. Although the loop homotopy H will be used with disjoint simple closed curves, it will not be required that .s → H (s, τ ) is simple for .0 < τ < 1. The function H has the advantage over .Ho in that it can always be lifted to the universal covering because .[0, 1]2 is simply connected. Proposition 8.2.3 Let .λ be a local section for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, let .x ∈ X be a non-periodic point of the flow .(X, R), let .xτ0 , xτ1 and .xτ2 , xτ3 be two pairs of consecutive crossings of .λ such that .τ0 < τ1 < τ2 < τ3 , let . x ∈ π −1 (x), and let  .λ0 and  .λ2 be the lifts of .λ 2   to .(B , R) such that . x τ0 is in .λ0 and . x τ2 is in .λ2 . (a) If J = [xτ0 , xτ1 ]ϕ ∪ [xτ1 , xτ0 ]λ

.

and J  = [xτ2 , xτ3 ]ϕ ∪ [xτ3 , xτ2 ]λ

.

are disjoint simple closed curves in X which are not null-homotopic, then the control curves .J and .J constructed using .O+ ( x ) satisfy .JP ⊂ JP .  (b) If J and .J are homotopic simple closed curves in X, then there exists .T ∈  such that .S2 = T −1 S0 T , where .S0 and .S2 are the elements of . satisfying λi =  . x τi+1 ∈ Si λi+1 for i equals 0 and 2. Proof Clearly, .J and .J are disjoint because J and .J  are disjoint, and .J ⊂ JN because .τ1 < τ2 . It follows that .J∩ JP = φ. Then .JP ⊂ JP because . x (τ3 + 1) is in P ∩ J . .J P Next, let .H : [0, 1] × [0, 1] → X be a homotopy between J and .J  (with their standard parameterizations). Since .[0, 1] × [0, 1] is simply connected, there exists (s, 0)  : [0, 1] × [0, 1] → B2 of H such that .H (0, 0) =  a lift .H x τ0 . Then .s → H   is the lift of J starting at .H (0, 0) =  x τ0 by the x τ0 and ending at .H (1, 0) = S0

236

8 Lifts and Limits

(s, 1) is a lift of .J  starting at .H (0, 1), which definition of .S0 . Furthermore, .s → H (1, 1). Thus there exists .T ∈  such that is not necessarily . x τ2 , and ending at .H (s, 1) is the lift of .J  starting at .T  .s → H x τ2 and ending at (1, 1) = T S2 x τ2 H

.

because the lift of .J  starting at . x τ2 ends at .S2 x τ2 by the definition of .S2 . (0, t) and .t → H (1, t) are lifts of the curve .t → H (0, t) = H (1, t) Both .t → H (1, t) = S0 H (1, 0) = S0 (0, t) from .xτ0 to .xτ2 . Now .H x τ0 implies that .t → H (1, t) starts at .H (1, 0) = S0 H (0, 0). In particular, because .t → H (1, 1) = S0 H (0, 1) = S0 T  x τ2 H

.

(0, t) starts at . because .H x τ0 and ends at .T  x τ2 . Putting the pieces together, we have (1, 1) = T S2 S0 T  x τ2 = H x τ2 .

.

Since a covering transformation is completely determined by its value at one point, S0 T = T S2 or .S2 = T −1 S0 T .



.

Given a local section .λ of a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1 and point .x ∈ X, we extend the hypotheses used in Proposition 8.2.3 to define a sequence of pairs of consecutive crossing times as a sequence .τi of real numbers satisfying the following conditions for .i ≥ 0: (a) .τi < τi+1 . (b) .xτi ∈ λ. (c) .(xτ2i , xτ2i+1 )ϕ ∩ λ = φ. It follows from (a) and (b) that .τi + 2α < τi+1 , where .2α is the length of the section λ. Thus .τi → ∞ as .i → ∞. In this context,

.

J2i = [xτ2i , xτ2i+1 ]ϕ ∪ [xτ2i+1 , xτ2i ]λ

.

(8.6)

is a sequence of simple closed curves on the surface indexed by the even nonnegative integers. Given . x ∈ π −1 (x), let  .λm be the lift of .λ containing . x τm and let .S2i be the element of . such that λ2i =  S2i λ2i+1 .

.

Then .S2i  x τ2i is in  .λ2i+1 and .J2i is null-homotopic if and only if .S2i = ι. So

(8.7)

8.2 Winding

J2i =

237



.

  n [ x τ2i ,  S2i x τ2i+1 ] x τ2i+1 , S2i  x τ2i ]S2iλ2i ϕ ∪ [

(8.8)

n∈Z

is sequence of control curves when .S2i = ι for all .i ≥ 0. We will use the following less cumbersome notation for the positively and negatively invariant regions for sequences control curves: J2i+ = (J2i )P and J2i− = (J2i )N .

.

Proposition 8.2.4 Let .λ be a local section for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, let .τi be a sequence of consecutive pairs of crossing times for a non-periodic point .x ∈ X, and let . x ∈ π −1 (x). If the following hold for .J2i defined by equation (8.6): (a) (b) (c) (d)

The simple closed curves .J2i are not null-homotopic for all i, J2i ∩ J2i  = φ for .i = i  ,  .J2i is homotopic to .J2i  for .i = i , .[xτ2i , xτ2i+1 ]λ ∩ ω(x) = φ for at least one i, .

then there exists .S ∈  such that .S2i defined by equation (8.7) satisfies .S2i = S for infinitely many values of i. Proof Since .J2i is homotopic to .J2i  , by Proposition 8.2.3 each .S2i = T2i S0 T2i−1 for some .T2i in .. When .γ (X) = 1 and X is the torus, . is abelian, .S2i = S0 for all i, and the result is trivial. So, for the rest of the proof, .γ (X) ≥ 2. By deleting a finite number of terms from the sequence, we can assume without loss of generality that .[xτ0 , xτ1 ]λ ∩ ω(x) = φ. If the conclusion of the proposition is false, we can also assume without loss of generality that .S2i = S2i  for .i = i  by passing to a subsequence. The .S2i must be primitive because the .J2i are simple closed curves that are not null-homotopic (Proposition 7.3.7). Since there are only two primitive elements of . with the same axis, it can also be assumed that .L2i = L2i  for .i = i  , where .L2i is the axis of .S2i . It follows from .S2i = T2i S0 T2i−1 that .L2i = T2i L0 . Let .α2i and .β2i denote the endpoints of .L2i in .S1 . They divide .S1 into two open intervals. Let .I2i be the one + bounding .J2i+ . Since .J2(i+1) ⊂ J2i+ for all i (Proposition 8.2.3), ∞  .

I2i− = {a} ∈ S1

i=1

follows from Proposition 7.1.22, and thus .limt→∞  x t = a. Because distinct axes of elements in . have no endpoints in common (Proposition 6.3.7), .I2(i+1) is contained in the interior of .I2i , and a is not an endpoint of .I2i for all i. (See Figure 8.5.) Now consider .T2i J0 , noting that .T0 = ι. It is a control curve with axis .L2i , and it has the same endpoints as .J2i . Hypothesis (b) implies that .T2i J0 ∩ J2i  = φ for  + x ) must cross .T J .i = i . Because a is in .I2i , the positive orbit .O ( 2i 0 for all i.

238

8 Lifts and Limits

a

J2( i+2)

J2( i+1)

J2 i x˜ Fig. 8.5 The orbit of .x˜ is used to construct .J2i

There are now two possibilities. Either .O+ ( x ) crosses .T2i J0 as part of the orbit used to construct .T2i J0 , or it crosses in a copy of the portion of  .λ0 used to construct 0 . In other words, for some .n ∈ Z either . .J x σ = T2i S0n x τ0 for some .σ > τ0 and .n ∈ Z or O+ ( x ) ∩ T2i S0n ( x τ1 , S0 x τ0 )S0λ0 = φ.

.

The first case would imply that . x is periodic, which it is not. Thus, for all i, the second case holds and there exists .si such that .xsi ∈ (xτ0 , xτ1 )λ . Then .si  ∞ because .si+1 − si is greater than the length of the local section .λ. Therefore, .ω(x) ∩ [xτ0 , xτ1 ]λ = φ, a contradiction.

The next relatively simple lemma provides critical contradictions for several proofs in this section and the next section. Figure 8.6 illustrates the first type of region to which it will be applied. Lemma 8.2.5 (Rational Boundary Point Lemma) Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 2, and let U be a positively invariant open connected set of .(B2 , R) such that .U − ∩ S1 = {a}. Suppose .T a = a for some primitive .T ∈  and .limt→∞ zt = a. If .Rzτ ∈ U for some .R ∈  and n for some .n ∈ Z. .τ ∈ R, then .limt→∞ Rzt = a, .Ra = a, and .R = T Proof Clearly .limt→∞ Rzt = Ra is in .U − ∩ S1 = {a}. It follows that .Ra = a and .limt→∞ Rzt = a. Therefore, a is the endpoint of the axis of R, and R and T have the same axis (Theorem 7.1.6). Since T is primitive, .R = T n for some .n ∈ Z (Proposition 7.1.15).



8.2 Winding

239

Fig. 8.6 Lemma 8.2.5 applies to the open set U

x τ

T −2 x τ T −1 x 

Tx 

x ˜

 T −1 λ

 λ

U a

 Tλ

R x

Recall that a point w of .S1 is a rational point for .γ (X) = 1 provided .{sw : s ∈ R} ∩ Z[i] is not the trivial subgroup of .C and that a point w of .S1 is rational point for .γ (X) ≥ 2 provided w is a fixed point of some .T = ι in .. In this terminology, the rational points of .S1 depend on the covering group. Using the two Weil theorems (Theorems 5.1.4 and 7.3.15), the positive orbit + x ) that goes to infinity has by definition a rational limit provided that .O ( .

lim

t→∞

 xt | x t|

is a rational point of .S1 when .γ (X) = 1 (p. 121) and .

lim  xt

t→∞

is a rational point of .S1 with respect to . when .γ (X) ≥ 2. The limit of a positive orbit that goes to infinity is said to be irrational when it is not rational. If .O+ ( x ) has a rational limit, then there exists a unique primitive covering transformation T , called the asymptotic covering map of .O+ ( x ) such that .

lim  x t/| x t| =

t→∞

m + in = lim T k x /|T k x| |m + in| k→∞

for some .m + in ∈ Z[i] when .γ (X) = 1, and when .γ (X) ≥ 2 .

lim  x t = lim T k x.

t→∞

k→∞

The author showed in [47] that if x was positively recurrent and not periodic, x t = a ∈ S1 . Aranson and Grines [10] were the first to show that then .limt→∞  a was irrational. Their paper, however, assumes the flow is differentiable and the proof uses this assumption when the authors refer to a lemma [8, Lemma 3] in an earlier paper on structural instability. The next theorem comes from [61] and is the

240

8 Lifts and Limits

companion of Theorem 5.3.10 for genus 1. Its proof that .limt→∞  x t = a is irrational when x is positively recurrent and not periodic does not use differentiability. Theorem 8.2.6 Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 1. If x is positively recurrent and not periodic, then .limt→∞  x t is irrational for all . x in .π −1 (x). Proof The .γ (X) = 1 case follows from Theorem 5.3.10. So, for the remainder of the proof, .γ (X) ≥ 2. Given . x in .π −1 (x), Theorem 7.3.21 implies that . x t → a ∈ S1 as .t → ∞. Then a is rational if and only if there exists a primitive T in . such that .T a = a by definition and Proposition 7.1.15. Let .λ be a local section at x of length .2α and let  .λ be the lift of .λ containing . x. Because . x t → a ∈ S1 as .t → ∞, the positive orbit of . x can cross  .λ at most a finite number of times, and by making .λ smaller, we can assume that .O+ ( x) ∩  λ = { x }. +  Then Proposition 8.2.1 implies that .O ( x ) crosses .S λ at most once for any S in .. Because x is positively recurrent and not periodic, there exist infinitely many .S ∈  for each of which there exists a unique .t > 0 such that . x t ∈ S λ and . x t = S x. In the context of this proof, the following lemma will be useful:

Lemma 8.2.7 If a is a fixed point of a primitive transformation .T ∈  and . x τ is in S λ for some .τ > 0 and S in ., then .S = T n for all .n ∈ Z.

.

Proof Suppose .S = T n , noting that .S = ι because .O+ ( x) ∩  λ = { x } and .τ > 0. Without loss of generality, we can assume .τ is the smallest positive real number such λ for some .κ ∈ Z. Replacing T with .T −1 if necessary, we can assume x τ ∈ T κ that . n λ = φ for all .n ∈ Z, and a weak control that .κ > 0. It follows that .( x,  x τ ) ϕ ∩T  curve of degree .κ can be constructed using equation (8.5) and .( x,  x τ ) ϕ . Then part (c) of Proposition 8.2.2 implies that .κ = 1. Thus I =



.

  x,  x τ ] T m [ x τ, T  x ]T λ ϕ ∪ [

m∈Z

is a weak control curve of degree 1. Now consider the open set U (shown in Figure 8.6) bounded by .O+ (T −2 x τ ), + .O ( x ), and −1 [T −2 x τ, T −1 x ]T −1λ ∪ [T −1 x , T −1 x τ ]  x τ,  x ]λ . ϕ ∪ [T

.

Clearly, .U ⊂ IP . Using the positive invariance of .IP , it is easy to see that U is positively invariant. Moreover, a is the only boundary point of U in .S1 . −1 λ satisfying the following conditions: Choose . u and  .v in .T (a) .T −1 x is in .( u, v )T −1λ . (b) .T −2 x τ is not in .[ u, v ]T −1λ . (c) .( u, v )T −1λ (τ + α) is contained in U . The first two are trivial and the third follows from the continuity of the lifted flow.

8.2 Winding

241

Because x is positively recurrent, there exists .R ∈  such that .R x σ is in the flow box .( u, v )T −1λ (−α, α) and .σ > 4α. So .R x s is in .( u, v )T −1λ for a unique s such that .σ − α < s < σ + α. Then .3α < s implies that .R x =  x because . x s must be in .IN  and .R x is in .IN . And it is obvious that .R x = ι. It follows that .R x (s + τ + α) is in U . Figure 8.6 shows .O+ (R x ) crossing  .λ −1  to enter U , but it can also enter U immediately by crossing .T λ. In either case, + .O (R x ) crosses .I and .R x is in .IN . Since .T n x is in .I, it follows that .R = T n for .n ∈ Z contradicting Lemma 8.2.5 because .R x t is in U for large t.

Continuing to assume that T is a primitive element of . such that .T a = a, we refine our construction. Suppose .xτ is the first crossing of .λ, that is, 0 and .τ are consecutive crossings and .J = [x, xτ ]ϕ ∪ [xτ, x]λ is a simple closed curve. Then J x τ is is not null-homotopic by Proposition 8.2.1, and there exists .S ∈  such that . in .S λ and S is primitive. It follows that 

J =

.

  x,  x τ ] S m [ x τ, S x ]Sλ ϕ ∪ [

m∈Z



=



S [ x,  x τ ] ϕ m

m∈Z







S [ x τ, S x ]Sλ m

m∈Z

is a control curve, as is .R J for all .R ∈ . Either .R J ∩ J = φ or .R J = J (Proposition 7.3.1). Moreover, .R J = J if and only if .R = S n for some .n ∈ Z. If .S m = T n , then they have the same fixed points. Obviously .m = 0 if and only if .n = 0 and we can assume .m = 0 = n. Then .LS = LT and .S = T or .S = T −1 by Proposition 7.1.15. By Lemma 8.2.7, both possibilities are impossible, and hence n ∩ J  = φ for all .n = 0. .T J Let A denote the closed arc of .S1 containing a and with the endpoints of .J, which equal those of .LS . Then a cannot be an endpoint of A because .LS and .LT have no common endpoints (Theorem 7.1.6). So either .LS ∩ LT = φ or .LS ∩ LT = φ. The proofs in both cases will proceed by contradiction. First assume .LS ∩ LT = φ. By replacing T by .T −1 , if necessary, we can assume that a is the attracting fixed point of T . Because T is hyperbolic and a is in A, it follows that T k JP ⊂ T k−1 JP and T k J ⊂ T k−1 JP ,

.

for .k ≥ 1 and that .O+ ( x ) intersects .T k J for all .k > 0. In particular, for each .k > 0, there exists a unique .mk ∈ Z and .σk > τ such that  x σk ∈ T k S mk ( x τ, S x )Sλ .

.

The sequence .σk is clearly increasing to infinity. Suppose .mj = mk for some  j m +1 λ which is another lift of .λ. Then . 1 ≤ j < k, and let  .λ = T S j x σj is in   λ , contradicting Lemma 8.2.7 applied to .λ and .( x σj )(σk − σj ) =  x σk is in .T k−j .

242

8 Lifts and Limits

a

T k J U

x ˜τ J

x ˜

T −k x˜σk

Sx ˜ LT

Fig. 8.7 The open set U when .LS ∩ LT = φ

 x σk ∈  λ . Thus .mk is a sequence of distinct integers for .k > 0. So there exists k such that .|mk | ≥ 3. For convenience, suppose .mk ≥ 3. x τ ), .O+ (T −k x σk ) and Consider the open set U (see Figure 8.7) bounded by .O+ (

.

⎛

⎞

m k −1

.⎝

S j [ x,  x τ ] ϕ⎠ ∪ ⎝

j =1

⎛

m k −1

⎝

⎞

j

j =1

m k −2

⎞ S j ( x τ, S x )Sλ ⎠ ∪ (S mk −1 x τ, T −k x σk ]S mk λ =

j =0

⎛

m k −2

[S  x, S  x τ ] ϕ⎠ ∪ ⎝ j

⎛

⎞

(S  x τ, S j

 x )S j +1λ ⎠ ∪ (S mk −1 x τ, T −k x σk ]S mk λ ,

j +1

j =0

which is the piece of .J joining . x τ and .T −k x σk in .S mk λ. As before, U is positively invariant and a is the only boundary point on .S1 because .

lim (T −k x σk )t = T −k lim  x (σk + t) = T −k a = a.

t→∞

t→∞

Then .O+ (S x ) enters U at .S x τ and contradicts the rational boundary point lemma because .S = T n for all .n ∈ Z by Lemma 8.2.7. Next suppose .LS ∩LT = φ . After modifying this situation with S and .Jto .RSR −1 and .R J such that .LRSR−1 ∩ LT = RLS ∩ LT = φ , a modification of the previous argument will be used to complete the proof. Observe that .limt→∞ S k ( x τ )t = S k a and .S j a = S k a when .j = k because .S m = n x τ ) partition .JP into disjoint positively T unless .m = n = 0. Thus the orbits .O+ (S k invariant h-congruent regions. If .O+ ( y ) crosses .J and .limt→∞  y t = b ∈ S1 , then b is not an endpoint of the axis .LS .

8.2 Winding

243

Given R in . , the control curve .R J is the type of the h-line .RLS which is the axis of .RSR −1 . Because R is an automorphism of the lifted flow extended to .D2 , the  P into disjoint positively invariant horbits .R O+ (S k x τ ) = O+ (RS k x τ ) partition .RJ congruent regions. Consequently, if .O+ ( x ) crosses .R J, then a is not an endpoint of .RLS . Since Corollary 4.2.9 can be applied to .(x, xτ )λ with .x = y because x is positively recurrent, there exists an increasing sequence .τn of positive consecutive crossing times .xτn of .(x, xτ )λ . In the third paragraph of the proof, .λ was chosen so that .O+ ( x ) crosses .S λ at most once for any S in . . So the simple closed curves .(x, xτn )ϕ ∪ [xτn , x]λ are not null-homotopic. For each .τn , there exists .Rn ∈  \ {ι} such that . x τn ∈ Rn J. Moreover, .O+ ( x ) can cross .Rn J exactly once and must go from .Rn JN to .Rn JP . Consequently, .Rn+1 J ⊂ Rn JP because J is a simple closed curve in X . It follows that ∪ .Rn+1 (J

JP ) ⊂ Rn (J ∪ JP )

for .n ≥ 1, and .a ∈ Rn+1 A ⊂ Rn A for all .n ∈ Z+ . It is now routine to show that ∞  .

Rn A = {a}

n=1

using Proposition 7.1.22. Since a is never an endpoint of the interval .Rn A by Lemma 8.2.7 and Theorem 7.1.6 again, there exists n such that the fixed points of T lie on opposite sides of .Rn LS , that is, .LRn SRn−1 ∩ LT = Rn LS ∩ LT = φ . Having chosen such an n, set .R = Rn and .τ  = τn . (See Figure 8.8.) From the previous paragraph, we know that .RLS ∩ LT = φ . It follows that

a T kR

Sa J

RJ LT

x ˜

Sx ˜ LS

Fig. 8.8 Modification of the .LS ∩ LT = φ case

J

244

8 Lifts and Limits . xτ



∈ RS μ ( x τ, S x )Sλ ⊂ R J

for some .μ ∈ Z and that k

.T

RLS ∩ LT = φ

for .k > 0. Thus .O+ ( x ) intersects .T k R J for all .k > 0, and the situation is now very similar to the case .LS ∩ LT = φ , but somewhat more complicated. For each .k > 0, there exists a unique .mk ∈ Z and .σk > τ  such that . x σk

∈ T k RS mk ( x τ, S x )Sλ .

The same argument used in the first case shows that .mk is again a sequence of distinct integers and there exists k such that .|mk − μ| ≥ 3. For convenience, suppose .mk − μ ≥ 3. Consider the region U bounded by + x τ  ), .O + (T −k λ. It .O ( x σk ) and the piece of .R J joining . x τ  and .T −k x σk ∈ RS mk consists of a beginning piece .[ xτ



, RS μ+1 x )RS μ+1λ ,

a middle piece ⎛ .⎝

m k −1

⎞

⎛

RS j [ x,  x τ ] ϕ⎠ ∪ ⎝

j =μ+1

m k −1

⎞ RS j ( x τ, S x )Sλ ⎠ ,

j =μ+1

and an end piece .[RS

mk

 x τ, T −k x σk ]RS mk +1λ .

As before, U is positively invariant and a is the only boundary point on .S1 because .

lim (T −k x σk )t = T −k lim  x (σk + t) = T −k a = a.

t→∞

t→∞

Finally, .mk − μ ≥ 3 implies that .O+ (RS μ x ) enters U at .RS μ x τ and that μ+1 enters U at .RS  x τ . (See Figure 8.9 for a depiction of the region when .μ = 1 and .mk = 4.) The rational boundary point lemma implies that μ p and .RS μ+1 = T q . Hence .RS = T + μ+1 .O (RS x)

.S

= (RS μ )−1 RS μ+1 = T q−p ,

a contradiction to Lemma 8.2.7. Therefore, .limt→∞  x t cannot be rational, when x is positively recurrent and not periodic.

8.3 Omega Limit Points at Infinity

245

a

U

˜ RS x ˜ RS 2 x x) O+ (˜

RJ

O+ (T −k x ˜ σk )

Fig. 8.9 A depiction of region U when .μ = 1 and .mk = 4

8.3 Omega Limit Points at Infinity Let .(X, R) be a flow on a compact connected orientable surface of genus at least 2, and let x be a point in X. The .ω-limit sets of positive semi-orbits whose lifts go to infinity will be the focus of this section. We first consider flows having a positive semi-orbit whose lift limits to a rational point. Because the covering transformations are flow automorphisms, .limt→∞  x t is a rational point in .S1 (p. 206) for some . x ∈ −1 π (x) if and only if .limt→∞  x t is a rational point in .S1 for all . x ∈ π −1 (x). Consider a point . x ∈ π −1 (x) such that .limt→∞  x t is a rational point a of .S1 . Then there exists .T ∈  with axis .LT such that a is one endpoint of .LT . Suppose y is in .ω(x) and .tn is a sequence of real numbers such that .tn → ∞ and .xtn → y as .n → ∞. Given . x ∈ π −1 (x), there are two interesting possibilities for the sequence .dh ( x tn , LT ) of nonnegative real numbers. Either it is bounded or unbounded. In the first case, y is said to be a bounded .ω-limit point. In the latter case, it can be assumed without loss of generality that .dh ( x tn , LT ) → ∞ as .n → ∞ by passing to a subsequence, and y is said to be a remote .ω-limit point. More formally, define .ωR (x), the remote .ω-limit set of x and .ωB (x), the bounded .ω-limit set of x , by .ωR (x)

= {y ∈ ω(x) : ∃ tn → ∞ as n → ∞  xtn → y and dh ( x tn , LT ) → ∞}

and .ωB (x) = {y

∈ ω(x) : ∃ tn → ∞ as n → ∞  xtn → y and dh ( x tn , LT ) is bounded},

where .∃ means “there exists” and . means “such that.” It is easily seen that .ωR (x) and .ωB (x) do not depend on the choice of . x ∈ π −1 (x). Clearly, .ωR (x) ∪ ωB (x) = ω(x), but .ωB (x) ∩ ωR (x) = φ is not precluded. It is not hard to verify that .ωR (x) and .ωB (x) are invariant and that .ωR (x) is a closed set.

246

8 Lifts and Limits

Furthermore, .ωR (x) = φ if and only if .{dh ( x t, LT ) : t ≥ 0} is a bounded set of real numbers. Exercise 8.3.1 Verify that .ωR (x) and .ωB (x) are invariant and that .ωR (x) is a closed set.

Although the definition of rational limits of orbits (p. 121) could be used to define the set of remote .ω-limit points for any orbit on the torus such that .limt→∞ | x t| = ∞, there would be no remote limit points unless .ω(x) was contained in the fixed point set by Theorem 5.3.14. For the torus, .ωR (x) is empty, when .ω(x) contains a moving point. Moving points in .ω(x) also play a critical role when .γ (X) > 1, but not as decisively as when .γ (X) = 1. Theorem 8.3.2 Let .(X, R) be a flow on a compact connected orientable surface X with ≥ 2, and let x be a point of X such that .limt→∞  x t exists and is a rational point of 1 .S for . x ∈ π −1 (x). If .ω(x) contains a non-fixed periodic point y, then .ω(x) = ωB (x) = O(y). .γ (X)

Proof It follows from Theorem 4.2.10 that .ω(x) = O(y). Then use part (b) of x t, LT ) : t ≥ 0} Proposition 8.1.6 and the compactness of .O+ (y) to show that .{dh ( is a bounded set of real numbers.



There are two well-known homotopy results for simple closed curves on a compact connected orientable surface X. (A reference for the second homotopy theorem is [34].) • Homotopy Theorem A There are at most .3γ (X) − 3 disjoint simple closed curves on .X that are neither null-homotopic nor are any pairs homotopic. • Homotopy Theorem B There are at most .6γ (X) − 3 simple closed curves on .X ,

intersecting pairwise only at exactly one common point that are neither null nor are any pairs homotopic. Theorem 8.3.3 Let .(X, R) be a flow on a compact connected orientable surface X with ≥ 2, and let x be a point of X such that .limt→∞  x t exists and is a rational point of .S1 for . x ∈ π −1 (x). If .ω(x) contains a moving point, then .ωB (x) contains a moving point and is not empty.

.γ (X)

Proof By Theorem 8.3.2, it can be assumed that x is not periodic. From here the proof proceeds by assuming .ωB (x) ⊂ F , dividing it into three cases, and deriving a contradiction in each of them. Let y be a moving point in .ω(x) and let .λ be a local section at y. Since we are assuming .ωB (x) ⊂ F , it follows that .y ∈ ωR (x) and .λ ∩ ω(x) = λ ∩ ωR (x). The first case is when .λ ∩ ω(x) has an interior in .λ. Then x is positively recurrent because xt is in .ω(x) = ω(xt) for some t. By Theorem 8.2.6, x cannot be positively recurrent because .limt→∞  x t exists and is a rational point of .S1 . Thus either .λ ∩ ω(x) contains an isolated point or it is a perfect nowhere dense set. (These three cases will also be used in the study of .ω-limit points at infinity.) 2 If  .λ is a lift of .λ in .B , then .{S λ : S ∈ } is the set of all lifts of .λ to .B2 . Let . x be a −1 x t = a, and let T be the primitive transformation in specific point in .π (x), let .limt→∞ 

8.3 Omega Limit Points at Infinity

247

that has a as its attracting fixed point. As usual, by making .λ shorter, it can be assumed that .O+ ( x ) ∩ S λ contains at most one point for all .S ∈ . This will be the context for the rest of the proof. x ) can intersect only Because it is assumed that .ωB (x) ⊂ F , it follows that .O+ ( a finite number of the sections which are a specified bounded distance from .LT . In x ) can intersect only a finite number of .T n S λ for fixed .S ∈  because particular, .O+ ( n  .T S λ are equidistant from .LT . This is a key observation. For the second case, assume that .λ ∩ ω(x) contains an isolated point y. By shortening   .λ, it can be assumed that .λ ∩ ωR (x) = {y}, which preserves the context because .λ ⊂ λ.  Again y cannot be in .O(x) because a is rational. Since .λ \ {y} consists of two open connected subsets of .λ , the positive orbit of x must cross one of them infinitely often. So removing one of the components of .λ \ {y} from .λ produces a local section .λ with y as an endpoint and with a sequence of consecutive crossings .xtk of .λ such that .xtk converges to y. .λk be the lift For convenience, we set .λ = λ for the rest of the second case, and let  x tk . The property that .O+ ( x ) ∩ S λ contains at most one point for all of .λ containing . + x) ∩  .S ∈  has not been lost. So .O ( λk =  x tk . It follows that the simple closed curves .Jk = [xtk , xtk+1 ]ϕ ∪ [xtk+1 , xtk ]λ are not null-homotopic. λk , LT ) goes to infinity as k goes to infinity. By the key observation, the sequence .dh ( So there exists a subsequence such that .dh ( λki , LT ) < dh ( λki +1 , LT ) for all .i ≥ 0.  Given .k ∈ Z, observe that .Jk ∩ Jk  = φ if .k − k  is large enough, because .Jk can only intersect .Jk  in .(xtk  , xtk  +1 )λ at .xtk  +1 which converges to y. Hence, it can also be assumed that .Jki ∩ Jkj = φ for all .i = j . Since there are at most a finite number of non-homotopic disjoint simple closed curves on X by Homotopy Theorem A, we can also assume the .Jki are all homotopic. We have now constructed a sequence of pairs of consecutive crossing times, .τ2i = tki and .τ2i+1 = tki +1 , that satisfies the hypothesis of Proposition 8.2.4. Continuing this .λ2i =  λki and  .λ2i+1 =  λki +1 , we can assume by passing to a subsequence that notation,  there exists .S ∈  such that . x τ2i+1 ∈ S λ2i =  λ2i+1 for all .i ≥ 0, Then .S = T n for all n   because .dh (λ2i , LT ) < dh (λ2i+1 , LT ). Now consider the sequence of control curves .

2i .J

=



  x τ2i ,  S n [ x τ2i+1 ] x τ2i+1 , S x τ2i ]Sλ2i , ϕ ∪ [

n∈Z

noting that each .J2i is the type of .LS . Because .S = T n for all .n ∈ Z, the axes .LT and .LS have no points at infinity in common. Since . x t → a, which is not on the axis .LS of x τ2i ,  x τ2i+1 ] λ2i , LS ) go to infinity as S, the Euclidean limits of both .dh ([ ϕ , LS ) and .dh (  i goes to infinity by Corollary 6.1.30. So .dh (J2i , LS ) → ∞ as .i → ∞ because S is an isometry. (See Figure 8.10.) Recalling that each .Jk is the universal lift of a curve .Jk in X that is not nullhomotopic, Proposition 7.3.10 applies to the original family of curves .{Jk = [xtk , xtk+1 ]ϕ ∪ [xtk+1 , xtk ]λ : k ∈ Z+ }. Thus for large i, the curves .J2i in our subsequence of .Jk are not simple, contradicting the construction of the sequence of simple closed curves .Jk in X and eliminating the second case.

248

8 Lifts and Limits

x˜τ i x˜τ4

LT

a

x˜τ 2 d

LS

x˜

c

Fig. 8.10 .O+ ( x ) crossing control curves (dashed) increasingly far from .LS

For the third case, suppose that .λ ∩ ωR (x) is a Cantor set. As before x cannot be positively recurrent. So .O+ (x) can cross .λ only in the complementary intervals of .λ ∩ ωR (x). Hence there exist two consecutive crossings, say .xt1 and .xt2 of .λ by + .O (x) such that .(xt1 , xt2 )λ ∩ ωR (x) = φ and is, therefore, also a Cantor set. Of course, .J = [xt1 , xt2 ]ϕ ∪ [xt1 , xt2 ]λ is a simple closed curve in X because the crossings are consecutive. Since .O+ ( x ) ∩ S λ contains at most one point for all .S ∈ , J is not nullhomotopic. .λ be the lift of .λ containing . x t1 and let Q be the primitive element of . such that Let  . x t2 is in .Q λ. Then let .J be the control curve constructed using . x t1 , . x t2 ,  .λ, and Q in the usual way. Of course, T continues to denote the primitive transformation in . that has a as its attracting fixed point. Because .(xt1 , xt2 )λ ∩ ωR (x) = (xt1 , xt2 )λ ∩ ω(x), the construction of .J guarantees x σi ∈ Si J and .xσi converges that there exists a sequence .σi ∈ R+ and .Si ∈  such that . x σi cannot remain to .y ∈ ωR (x) by assumption. Obviously, .σi  ∞. Consequently, . between any 2 equidistant curves of .LT . If . x σi is between the equidistant curves .E1 and .E2 and .Si maps . x σi outside of that region, then .Si = T n for all .n ∈ Z. By passing to a subsequence, it can thus be assumed that .Si = T n for all .n ∈ Z and all .i ∈ Z+ . Moreover, .Si+1 J ⊂ Si J+ and .Si+1 J+  Si J+ . Letting .Ii be the closed interval of .S1 bounding . Si J+ , it follows that a is not an endpoint of .Ii for all i. Then, as in the proof of Proposition 8.2.4, ∞  .

Ii = {a} ∈ S1 ,

i=1

and there exists i such that the other fixed point of T is in .S1 \ Ii . Consequently the axis −1 2 .LT and the axis .Si LQ of .Si QS i intersect at a point in .B . Fix an i with this intersection property.

8.3 Omega Limit Points at Infinity

249

a

a1

Fig. 8.11 The positively invariant region .U

a2

U

λ1

 which .O+ ( Let  .λ1 be the lift of .λ in .Si J x ) crosses, say at . x t3 . Let . x1 t1 and . x2 t2 be .λ1 . Also let .a1 and .a2 be the limits as .t → ∞ the lifts of .xt1 and .xt2 , respectively, in  of . xi ti , .i = 1, 2. (See Figure 8.11.) Clearly, .a1 and .a2 are on opposite sides of a in .Ii . x1 t1 ), .O+ ( x2 t2 ), .( x1 t1 ,  x2 t2 )λ1 and the arc of Now form the region U bounded by .O+ ( 1 .S from .a1 to .a2 containing a, depicted in Figure 8.11. Note that U is connected and positively invariant. Now it is useful to consider .T k Si J for .k ∈ Z+ which form a nested sequence of copies of .Si J in .Si J+ . It is clear from Figure 8.11 that .T k+1 Si U ⊂ T k Si U for .k ≥ 0. λ1 in a positive Furthermore, the only way an orbit can enter .T k Si U is to cross .T k x t3 ) must do because .limt→∞ ( x t3 )t = a for all .k ∈ Z+ . direction. This is what .O+ ( λ1 , LT ) is constant. This It follows that .λ contains a point in .ωB (x) because .dh (T k

contradicts the assumption that .ωB (x) ⊂ F and completes the proof.

Let . be a Fuchsian covering group of a compact connected orientable surface X, and let T be a primitive element of . . Proposition 7.1.15 shows that .{S ∈  : SLT = LT } is the cyclic subgroup .[T ] generated by T . Recall from Proposition 7.3.17 and the discussion in the paragraph preceding it that the index of .[T ] in . is countably infinite and so there exists a sequence of coset representatives .Qk in . such that ∞ . = k=1 [T ]Qk . When .γ (X) = 1, the covering group . = Z2 has the same structure given a primitive element T but in a simpler form. Specifically, there exists .S ∈  such that the sequence .Qk can be replaced with .S k as described in the proof of ∞ k ∞ k Proposition 5.2.8 and . = k=1 [T ]S = k=1 S [T ]. Let .λ be a local section of a flow .(X, R) on X, and let .λ be a lift of .λ. Setting k λ, the collection of all lifts of .λ is given by  .λk = Qk .λk = S  λ or  . λ

=

∞   k=1 m∈Z

T m Qk λ=

∞  

T m λk

k=1 m∈Z

for a primitive .T ∈  . From Proposition 7.3.17, we know that .

lim dh ( λk , LT ) = ∞.

k→∞

When .γ (X) = 1, it is obvious that .limk→∞ d(S k λ, LT ) = ∞.

(8.9)

250

8 Lifts and Limits

Proposition 8.3.4 Let x be a non-periodic point of a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, and let . x be a lift of x such that .limt→∞  x t is rational. If y is a moving point in .ωB (x) and .λ is a local section at y, then the following hold: (a) There exist a lift  .λ of .λ, a primitive .T ∈ , and an infinite set .E = {s ≥ 0 : λ for some m ∈ Z} containing a sequence of nonnegative real numbers .tk  x s ∈ T m converging to infinity such that .xtk converges to y. (b) There exist an increasing sequence .si of nonnegative real numbers converging to infinity such that .{si : i ≥ 0} = E. λ. (c) For each .si , there exists a unique .mi ∈ Z such that . x si ∈ T mi + m (d) There exists .μ ≥ 0 such that .O ( x sμ ) ∩ T i λ = { x si } for all .i ≥ μ. (e) If .μ ≤ i < j , then .mi = mj and .limi→∞ mi = ∞. (f) The loops .[xsi , xsj ]ϕ ∪ (xsj , xsi )λ are not null-homotopic when .μ ≤ i < j . (g) The sequence .mi satisfies .mi+1 − mi = 1 for .i ≥ μ and can be written .mi = mμ + i − μ. (h) The sequence .T −mi  x si is a strictly monotonic sequence in  .λ such that .xsi converges to y monotonically. Proof The .γ (X) ≥ 2 case will be done first, and then the .γ (X) = 1 case will be reduced to the arguments already used for (b) through (h). For part (a), first let .a = limt→∞  x t and let T be the primitive element of . such that the rational point a is the attracting fixed point of T . It is clear from the definition of .ωB (x) that there exists a bounded sequence .dh ( x tn , LT ) such that .tn goes to infinity and .xtn converges to y. By Proposition 7.3.17, there exist  .λ and a subsequence of . x tn such λ for some .m ∈ Z. Then set .E = {s ≥ 0 : that every term of the subsequence is in .T m  x s ∈ T m λ for some m ∈ Z}. For part (b), note that the distance between two positive real numbers in E must be at least the length of the local section .λ. So E is countable and .E − = E. It follows that E has the smallest element, a second smallest, etcetera. Thus E can and will be written as an increasing sequence .si of nonnegative real numbers converging to infinity. Continuing with part (c), the definition of E implies that there exists .mi such that . x si ∈ T mi λ. And .mi is unique because .T m λ ∩ T n λ = φ when .m = n. x si ∈ T m0 x sμ ) ∩ T m0 λ}. Then .O+ ( λ = { x sμ } and To prove (d), just let .μ = max{i :  Proposition 8.2.1 applies. λ is at most one point when .μ ≤ i < j by part (d), and For part (e), .O+ ( x ) ∩ T m the function .i → mi is injective when .μ ≤ i < j . It follows from part (b) that .a = m λ (Corollary 6.1.30), x si and from the compactness of  .λ that .a = limi→∞ T i  limi→∞  which is only possible if .limi→∞ mi = ∞. Part (f) is an immediate consequence of part (e). Given .mi , there exists .mk such that .i < k and .mi + 2 < mk by part (d). The first step in proving part (g) is to observe that part (d) of Proposition 8.2.2 implies that .mi < mj < mk for .i < j < k. Since there exists arbitrarily large k satisfying the condition that .mi + 2 < mk , it follows that .mi < mj for all .j > i. Because i is arbitrary, .mi is a strictly increasing sequence.

8.3 Omega Limit Points at Infinity

251

Ji+1

Ji+1

Ji+1

Ji

Ji

Ji

 T −1 λ

 Tλ

 λ Key

Ji

 T 2λ

Ends of local sections Ji ∩ Ji+1

Ji+1

Fig. 8.12 The control curves .Ji and .Ji+1 of part (e)

By the definition of E, . x t is in .T m λ for .t ≥ 0 if and only if .t = si for some i. It n λ = φ for all .n ∈ Z and follows from part (b) that .( x si ,  x si+1 ) ϕ ∩T  = .J



  T nκ [ x si+1 ] x si+1 , T κ  x si ]T κλ x si ,  ϕ ∪ [

n∈Z nλ = φ is a weak control curve of degree .κ = mi+1 − mi = 1 because .( x si ,  x si+1 ) ϕ ∩T  for all .n ∈ Z. Therefore, .mi+1 = mi + 1, and .mi = mμ + i − μ for .i > μ. .λ Finally, for part (h), we can assume that .s0 = 0, .μ = 0, and .m0 = 0 by replacing  λ. Then .mi = i and with .T mμ

i .J

=



  x si ,  T n [ x si+1 ] x si+1 , T  x si ]T i+1λ ϕ ∪ [

(8.10)

n∈Z

are weak control curves of degree 1 for .i ≥ 1 by the proof of part (d). Clearly, .Ji+1 ⊂ + (Ji+ )− , where .(Ji+ )− is the closure of .Ji+ , and .Ji+1 ⊂ Ji+ . (See Figure 8.12.) Since the weak control curves given by equation (8.10) are T invariant because their degree is 1, applying .T −i to .Ji does not change the curve but it does alter its formula in a useful way. The result is i .J

=



  −i T n [T −i  x si , T −i  x si+1 ] x si+1 , T −i+1 x si ]T λ , ϕ ∪ [T 

(8.11)

n∈Z

i ⊂ (J+ )− ∩ (J− )− and now the initial points .T −i  x si are all in  .λ. Moreover, .J i−1 i+1 forces .T −i  x si to lie between .T −i+1 x si−1 and .T −i−1 x si+1 in  .λ. Therefore, the sequence −mi  .T x si = T −i  x si is strictly monotonic and converges to a point in  .λ because  .λ is homeomorphic to .[0, 1].

252

8 Lifts and Limits

It follows that .π(T −i  x si ) = xsi is a convergent sequence in X. Since E contains a subsequence .sij such that .xsij converges to y, the sequence .xsi must also converge λ is a homeomorphism onto .λ, finishing the proof for monotonically to y because .π| .γ (X) ≥ 2. x ) lies between 2 parallel If .γ (X) = 1, then Theorem 5.3.14 implies that .O+ ( x ) is rational (p. 121). Using lines in .R2 of rational slope because the limit of .O+ ( Proposition 5.2.7, there exist  .λ and a subsequence of . x tn such that every term of the λ for some .m ∈ Z. From this point on the proof follows the proof subsequence is in .T m

of the .γ (X) ≥ 2.

An earlier version of the next result appeared in [73]. Theorem 8.3.5 Let .(X, R) be a flow on a compact connected orientable surface X with ≥ 2. If x is a point of X such that .limt→∞  x t is a rational point .a ∈ S1 for −1 . x ∈ π (x), then every point of .ωR (x) is a fixed point. .γ (X)

Proof If .ω(x) ⊂ F or .ωR (x) = φ, there is nothing to prove. If .ω(x) ⊂ F , then .ωB (x) contains a moving point y by Theorem 8.3.3. Let q be a point in .ωR (x). Then y is not periodic by Theorem 8.3.2. Since Proposition 8.3.4 applies to points in .ωB (x) as explained prior to its statement, we will just continue to use all the same notation. In particular, there is a natural sequence .si  ∞ such that .xsi converges to y satisfying conditions (a) through (g) in Proposition 8.3.4. It also provides a sequence of weak control curves of degree 1 given by equation (8.10). The first one is 1 .J

=



  T n [ x s1 ] x s1 , T  x s0 ]T λ1 . x s0 ,  ϕ ∪ [

n∈Z

The curves .O+ ( x s1 ), .[ x s1 , T  x s0 )T λ , and ∞  .

T m ([ x s0 ,  x s1 ] x s1 , T  x s0 ]T λ ) ϕ ∪ [

m=1

with the point .a ∈ S1 (depicted in Figure 8.13) form an embedded circle K in .C with interior U , so that the rational boundary point lemma applies to U . q is a lift of q such that . q ∈ U . Then there exist an evenly covered open Suppose .  ⊂  of .π −1 (V ) such that . connected neighborhood V of q and a component .V q ∈ V o x s, LT ) > U . Because q is in .ωR (x), there exists .s > 0 such that .xs ∈ V and .dh ( . There exists another component q , LT )+δ, where .δ is the hyperbolic diameter of .V 2dh ( 1 of .π −1 (V ) and . 1 . There exists .S ∈  .V q1 ∈ π −1 (q) such that both . x s and . q1 are in .V   such that .S q1 =  q and hence .S V1 = V . q1 , LT ) = dh ( q1 , ζ ) ≤ dh ( q1 , w) for all Recall that there exists .ζ ∈ LT such that .dh ( .w ∈ LT . Then, from .dh ( x s, LT ) > 2dh ( q , LT ) + δ, we obtain .2dh ( q , LT

) + δ < dh ( x s, ζ ) ≤ dh ( x s,  q1 ) + dh ( q1 , ζ ) < δ + dh ( q1 , LT ).

8.3 Omega Limit Points at Infinity

253

U

x ˜s1 x ˜s0 T x ˜s0

J1

a

Fig. 8.13 The region U is positively invariant

Hence .2dh (S q1 , LT ) = 2dh ( q , LT ) < dh ( q1 , LT ) and .S = T n for all .n ∈ Z. But .S x s is o n  in .V ⊂ U and .S = T for some integer n by the rational boundary point lemma. This contradiction proves that .π −1 (q) ∩ U = φ. / F , we will obtain a similar contradiction using .π −1 (q) ∩ U = φ. Assuming .q ∈ Clearly, q is not in .O(x) by Theorem 8.2.6. There exists a local section .λ at q in X of length .2α such that .λ [−α, α] ∩ π(J) = φ. Suppose xs is the first time .O+ (x) crosses    .λ . There exists a lift  .λ of .λ such that . xs ∈  λ . By making .λ smaller, we can assume +   that .O ( x ) ∩ λ = { x s} and make use of Proposition 8.2.1. Because q is in .ωR (x), there exist .τ1 and .τ2 satisfying the following: (a) (b) (c) (d)

.max{s, s1 }

+ 4α < τ1 < τ2 . ∈ λ for .i = 1, 2. .xτ2 ∈ (q, xτ1 )λ .  .dh ( x τ2 , LT ) > 2dh ( x τ1 , LT ) + 2δ, where .δ is the hyperbolic diameter of  .λ . .xτi

 There exist lifts  .λ such that .{ x τi } = O+ ( x) ∩  λi . Let . qi = π −1 (q) ∩  λi . Again there i   n   q1 =  q2 and .S λ1 = λ2 . Then .S = T for all .n ∈ Z because exists .S ∈  such that .S .dh (S q1 , LT ) = dh ( q2 , LT ) > 2dh ( q1 , LT ). x τ1 lies between . x s1 and . x τ2 on .O+ ( x ) and thus . x τ1 is an element of the Notice that . embedded circle K. The construction of the local section .λ and the choice of .τ1 and .τ2 guarantee that .K

∩ λi [−α, α] = { x t : τi − α ≤ t ≤ τi + α}.

The set .( λ [−α, α])



o

\ { x t : τi − α < t < τi + α}

is the disjoint union of 2 open connected sets on opposite sides of K. Then, by the first part of the proof, the . qi are in the exterior of K for .i = 1, 2. Since .S λ1 =  λ2 , the

254

8 Lifts and Limits

 λ 1 q1 x τ1

 λ 2

q2

x˜τ2

S xτ1 U

Fig. 8.14 The region .U contains the point .S x τ1

point . x τ2 must lie between .S x τ1 and . q2 by the choice of .τ1 and .τ2 . (See Figure 8.14.) x τ1 must be in U , but .S = T n for .n ∈ Z, contradicting the rational boundary Therefore, .S point lemma.



A concrete example of a flow on a compact connected orientable surface of genus 2 with remote limit points can be found in [60]. This example is constructed by modifying the system of differential equations .x˙ = 1, y˙ = 1 − y 2 to obtain a flow on a cylinder with 2 holes (a sphere with 4 holes) and attaching 2 handles with simple flows on them. Other examples of flows with remote limit points are known. Anosov discusses such examples in [4] and [7]. Nikolaev and Zhuzhoma devote a portion of Chapter 10 of [68] to this phenomenon and examples of it. These examples are obtained using existence theorems for flows and generally do not limit to a fixed point of a covering transformation. The rest of this section will be devoted to the moving points in the .ω-limit sets of positive orbits of flows on compact connected orientable surfaces whose lifts go to infinity as time goes to infinity. So the setup is a point x of a flow on a compact connected orientable surface X such that .limt→∞  x t is a point in .S1 for a lift . x of x such that .ω(x) contains a moving point. Let y be an element of .ω(x)\F where, as usual, F is the fixed point set of .(X, R). Observe that given .O+ ( x ) and . y , a lift of .y ∈ ω(x)\F , there must exist a sequence .Tn of . such that . x t passes closer and closer to .Tn y as t goes to infinity. This property will indirectly affect the possible behaviors of .ω(x). The next definition is the key to the rest of the section. Specifically, the point y in .ω(x) is said to be an isolated .ω-limit point at infinity provided .y ∈/ F and there exists a local section .λ at y such that y is an isolated point of .ω(x) ∩ λ. In fact, y is an isolated .ω-limit point at infinity if and only if there exists a local section .λ at y such that .ω(x) ∩ λ = {y}. Alternatively, a point .y ∈/ F is an isolated .ω-limit point at infinity of x if and only if .{yt : |t| < ε} is an open subset of .ω(x) for small .ε > 0.

8.3 Omega Limit Points at Infinity

255

Proposition 8.3.6 Let x be a point of a flow on a compact connected orientable surface X with .γ (X) ≥ 0 such that . x t → ∞ as .t → ∞ for a lift . x of x. If p is a periodic point in .ω(x) that is not a fixed point, then p is an isolated .ω-limit point at infinity of x. Proof First .ω(x) = O(p) by Theorem 4.2.10. Since a local section can intersect the orbit of p at most a finite number of times, .ω(x) intersects a local section at p a finite

number of times, and p is an isolated .ω-limit point at infinity.

Proposition 2.2.5 can be used to extend Proposition 8.3.6 to nonorientable compact connected orientable surfaces. The original version of the next theorem about isolated .ω-limit set at infinity appeared in [62]. Theorem 8.3.7 Let .(X, R) be a flow on a compact connected orientable surface X with ≥ 1. If x is a point of X such that .ω(x) contains at least one moving point and the x ) of the positive semi-orbit of x goes to infinity, then, with F denoting the fixed lift .O+ ( points of .(X, R), the following are equivalent:

.γ (X)

(a) Every .y ∈ ω(x) \ F is an isolated .ω-limit point at infinity of x. (b) Some .y ∈ ω(x) \ F is an isolated .ω-limit point at infinity of x. (c) .limt→∞  x t is a rational point of .S1 for . x ∈ π −1 (x). Proof By Corollary 5.1.6 and Proposition 7.3.15, the limits .

lim

t→∞

 xt | x t|

and

xt lim 

t→∞

exist and are in .S1 for .γ (X) = 1 and .γ (X) ≥ 2, respectively. Since (a) obviously implies (b), we begin with (b) implies (c). To this end, suppose y is an isolated .ω-limit point at infinity of x. The .γ (X) = 1 case follows from Theorem 5.3.15, and we assume 1 .γ (X) ≥ 2 for the rest of the proof of (b) implies (c). Let a denote the point in .S such x t = a. that .limt→∞  If x is periodic, then .O(x) = ω(x) and every y in .ω(x) is an isolated limit point at infinity of x. Since . x t → a ∈ S1 , the simple closed curve .O(x) is not null-homotopic. x ) is the type of a rational h-line and a is rational. Thus .O( If y is in .O(x) and x is non-periodic, then .y ∈ ω(x) ∩ O(x) = φ. Thus x is positively recurrent and .O(x) ⊂ ω(x). Therefore, y is not an isolated .ω-limit point at infinity. Thus / O(x). the proof is reduced to the case that x is not periodic and that .y ∈ Since y is an isolated .ω-limit point at infinity of x, there exists a local section .λ at y such that .ω(x) ∩ λ = {y}. Any infinite sequence of crossings of .λ by .O+ (x) must converge to y as the crossing time goes to infinity because .λ ∩ ω(x) = {y}. The point y divides the local section .λ into two components and .O+ (x) must intersect one of them infinitely often. So, by deleting one of the 2 components of .λ \ {y} from .λ, it can be assumed that y is an endpoint of .λ and retain the properties that .λ ∩ ω(x) = {y} and that any infinite sequence of crossings of .λ by .O+ (x) must converge to y as the crossing time goes to infinity because .λ ∩ ω(x) = {y} still holds.

256

8 Lifts and Limits

+ x ) crosses first. Then . Let . x be a lift of x and let  .λ be the lift of .λ that .O ( x t → a ∈ S1 + implies that .O ( x ) can intersect  .λ only a finite number of times. By replacing .λ with + x ) in exactly .[y, w]λ for a suitable .w ∈ λ, it can be assumed that  .λ intersects .O ( one point, . x σ , while retaining the property that there exists an infinite sequence of consecutive crossings of .λ by .O+ (x). By replacing x with .xσ , it can be assumed without + x ) = { .λ ∩ O ( x }. loss of generality that  In this context, it follows from Proposition 8.2.1 that there exists an increasing sequence of nonnegative real numbers .tk  ∞ starting with .t0 = 0 such that .xtk ∈ λ and such that .xtk and .xtk+1 are all the consecutive crossings of .λ by .O+ (x). It follows that .xtk → y as .k → ∞, and .Jk defined by .Jk

= [xtk , xtk+1 ]ϕ ∪ [xtk+1 , xtk ]λ

is a simple closed curve on X for all k. Let  .λk be the lift of .λ containing . x tk , making  .λ0 =  λ. The next step is to construct a subsequence .Jki of .Jk that satisfies the hypotheses of Proposition 8.2.4. Condition (d) will always be satisfied because .ω(x) ∩ λ = {y}. Of course, Proposition 8.2.1 already applies and each .Jk is not null-homotopic. Given .Jk , it follows that .Jk ∩ Jk  = φ for .k  sufficiently large, because .xtk → y as .k → ∞. Hence there exists a subsequence .Jki such that .Jki ∩ Jkj = φ when .i = j . By Homotopy Theorem A, it can also be assumed that .Jki is homotopic to .Jkj for all i and j . We have now constructed a sequence of pairs of consecutive crossing times, namely, .τ2i = tki and .τ2i+1 = tki +1 that satisfies the hypotheses of Proposition 8.2.4. Continuing .λ2i =  λki ,  .λ2i+1 =  λki +1 , and .J2i = Jki . Therefore, by passing to a this notation,  subsequence one more time, there exists .R ∈  such that . x τ2i+1 ∈ R λ2i =  λ2i+1 for all .i ≥ 0. In particular, 2i .J

=



  x τ2i ,  R n [ x τ2i+1 ] x τ2i+1 , R x τ2i ]Rλ2i . ϕ ∪ [

n∈Z

Let .α and .β be the endpoints of the axis .LR of R. It suffices to show that .a = limt→∞  x t equals .α or .β. Assume this is not true. Then, for every .c > 0, there exists a x t, LR ) > c when .t ≥ ρc because . x t → a ∈ S1 as .t → ∞ and constant .ρc such that .dh ( .a ∈ / {α, β}. So when .τ2i ≥ ρc , it follows that .[ x τ2i ,  x τ2i+1 ] ϕ

 : dh (w, LR ) > c}. ⊂ {w ∈ X

Let .δ be the hyperbolic diameter of  .λ. Since . is a group of isometries for the hyperbolic metric .dh , 2i .J

 : dh (w, LR ) > c − δ} ⊂ {w ∈ X

when .τ2i ≥ ρc . Because .δ is fixed and c can be arbitrarily large, .dh (J2i , LR ) → ∞ as .i → ∞. Therefore, .J2i is not a simple closed curve for sufficiently large i by Proposition 7.3.10, a contradiction that completes the proof of (b) implies (c).

8.3 Omega Limit Points at Infinity

257

To prove (c) implies (a), assume .O+ ( x ) has a rational limit a and y is a moving point in .ω(x). It must be shown that y is an isolated .ω-limit point at infinity of x. If x is periodic, then y is in .O(x) and an isolated .ω-limit point at infinity of x by Proposition 8.3.6. If y is in .O(x) and x is not periodic, then x is positively recurrent and .O+ ( x ) has an irrational limit by Corollary 5.3.13 or Theorem 8.2.6 depending on whether .γ (X) = 1 or .γ (X) ≥ 2, respectively. Thus it can again be assumed that x is not / O(x). periodic and .y ∈ When .γ (X) = 1, Theorem 5.3.14 implies that .ωR (x) = φ because y is a moving point in .ω(x). When .γ (X) ≥ 2, Theorem 8.3.5 implies that y is not in .ωR (x) because y is not a fixed point. So in either case y is not in .ωR (x). Let .λ be a local section at y of length .2α. It must be shown that there cannot be points of .ω(x) arbitrarily close to y in the flow box .λ[−α, α]. In this context, it suffices to show that there cannot be points of .ω(x) arbitrarily close to y in .λ because .ω(x) is invariant. Since we already know that y must be in .ωB (x), Proposition 8.3.4 applies and the conclusions (a) through (h) hold. Moreover, the notation here matches that of Proposition 8.3.4, and we can continue the proof at the end of part (h), having assumed that .s0 = 0, and .mi = i. In particular, .T −i  x si is a strictly monotonic sequence in  .λ and .xsi is a monotonic sequence in .λ converging to y. Starting with .γ (X) ≥ 2, suppose .xσ is in .λ and lies between two terms of the sequence .xsi for some .σ > 0. There exists  .λ1 , a lift of .λ such that . x σ is in  .λ1 , so .λ1 = R λ. If .R = T m , then .σ is in .{s ≥ 0 :  xs ∈ there exists .R ∈  such that  λ for some m ∈ Z} = {si : i ≥ 0} and .σ = si for some i, contrary to the assumption T m that .xσ is in .λ and lies between two terms of the sequence .xsi . Thus R is not in .[T ], and x σ lies between .T −i  x si and .T −i−1 x si+1 in  .λ. there exists .i > 0 such that .R −1 Let U be the open set bounded by .O

+

(T −i  x si ), O+ (T −(i+1) x si+1 ), (T −i  x si , T −(i+1) x si+1 )λ

and the point .a = limt→∞  x t, the attracting fixed point of T . Clearly the rational x σ ) is in U , producing boundary point lemma (Lemma 8.2.5) applies to U and .O+ (R −1 the contradiction that R is in .[T ]. It follows that there cannot exist .xσ with .σ > 0 in .λ that lies between two terms of the sequence .xsi for some .σ > 0. Let u and v be the endpoints of .λ. Then the monotone sequence .xsi is contained in either .(u, y)λ or .(y, v)λ . Assume the former. It follows that xt is in .(u, y)λ for .t > 0 if and only if .t = si for some .i ≥ 0. Therefore, .ω(x) ∩ (u, y)λ = φ. If there exists a sequence of points in .ω(x) ∩ (y, v)λ converging to y, then there exists an increasing sequence .tj of positive real numbers such that .xtj is in .(y, v)λ and .xtj converges to y. Because y is in .ωB (x), Proposition 8.3.4 applies, as before, but to  k λ, .k ∈ Z. The proof that .ω(x) ∩ (u, y) = φ can then be .λ not of the form .T  some  λ repeated to also prove that .ω(x) ∩ (y, v)λ = φ. x si is Turning to .γ (X) = 1, the starting point is again a primitive T such that .T −i  .λ and .xsi is a monotonic sequence in .λ converging to a strictly monotonic sequence in  y obtained from Proposition 8.3.4. There now exists a primitive element S in . such that for all .R ∈  there exist unique integers m and n satisfying .R = S m T n = T n S m

258

8 Lifts and Limits

λ = S m T n λ with (Proposition 5.2.7). Recall that every lift of .λ can be written as .T n S m m and n unique (equation (8.9)). x ) lies between 2 lines .L1 and .L2 parallel to the Theorem 5.3.14 implies that .O+ ( axis of T . It can be assumed that  .λ is contained in the region U between .L1 and .L2 . λ ∩ U = φ and n ∈ Z} is finite. Then .{m : S m T n x ) crosses .S m T n λ for infinitely many A priori, there can exist .m = 0 such that .O+ ( n, and Proposition 8.3.4 can be applied to produce another monotonic sequence of crossings of .λ converging to y. There exist a finite number of such sequences because m n λ ∩ U = φ when .O + ( .S T  x ) crosses .S m T n λ. This accounts for all the convergent + sequences in .O (x) ∩ λ. Thus .ω(x) ∩ λ = {y}.

Corollary 8.3.8 Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 1, and let x be a point of X such that .O+ ( x ) has a rational limit. If .λ is a local section at a point in .ω(x), then .E = ω(x) ∩ λi is a finite set of the .ω-limit set at infinity of x. Corollary 8.3.9 Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 1. Let x be a point of X such that .ω(x) contains at least one moving point x ), the lift of .O+ (x), has a rational limit. If .y ∈ ω(x) is not periodic, then y and .O+ ( .ω(y) ∪ α(y) ⊂ F . Proof Suppose that a moving point z is in .ω(y) ∪ α(y). Because .ω(x) is a closed invariant set, it contains .O(y)− and the moving point z. Hence there is a local section .λ at z and a sequence .yn in .O(y) converging to z. It can be assumed that .yn is in .λ for all .n ∈ Z. Therefore, z cannot be an isolated point, contradicting Theorem 8.3.7.



If .y ∈ ω(x) is periodic in the context of Corollary 8.3.9, then .O(y) = ω(y) = α(y) by Theorem 4.2.10 The idea of other types of .ω-limit sets at infinity can be developed in the same context that x be a point of a flow on a compact connected orientable surface X such x t is a point in .S1 for a lift . x of x and .ω(x) contains at least one moving that .limt→∞  point. Of course, it will be required that they be closed and distinct from other types of .ω-limit set at infinity. It must be noted, however, that Theorem 8.3.7 itself imposes some conditions on new types of .ω-limit sets at infinity. First, it follows from Theorem 8.3.7 that every point in a specific new type of .ω-limit set at infinity must be an accumulation of other points in the same one. Second it must be a perfect set because the closure of a subset A of X that has no isolated points is a perfect set [81, p. 218, Exercise 30B]. Since the following theorem, which will be used to prove that there are exactly two additional types of .ω-limit sets at infinity, may be a somewhat obscure reference, we quote it in full from [66]: Theorem 8.3.10 (Natansan) Every non-void bounded perfect set P is either a closed interval or is obtained from a closed interval by removing a finite or denumerable number of disjoint open intervals which have no common endpoints with each other or with the original interval. Conversely, every set obtained in this manner is perfect.

8.3 Omega Limit Points at Infinity

259

Starting as above with a point x of a flow on a compact connected orientable surface X such that . x t → a ∈ S1 as .t → ∞ and .ω(x) contains at least one moving point, a closed interval .[w1 , w2 ]λ of a local section .λ is said to be an open recurrent .ω-limit set at infinity provided .[w1 , w2 ]λ is contained in .ω(x). In this situation, .U = (w1 , w2 )λ (−α, α) is an open set of X, and .U ⊂ ω(x). Since there must exist .τ > 0 such that .xτ is in U , the point x is positively recurrent. Because .ω(x) o is invariant, .O(x) ⊂ ω(x) o . It follows that .ω(x)

⊂ O+ (x)− ⊂ O(x)− ⊂ (ω(x) o )− ⊂ ω(x),

.ω(x)

= O+ (x)− = O(x)− = (ω(x) o )− = ω(x).

and therefore,

Observe, however, that .ω(x) o and .ω(x) \ ω(x) o are disjoint invariant sets of .ω(x) from which it follows that the orbits of .ω(x) \ ω(x) o are not positively recurrent orbits of .ω(x) o and .O+ (x) is dense in .ω(x) o . Therefore, x uniquely determines a recurrent orbit closure. Then .limt→∞  x t must be irrational, which follows from either Theorem 8.3.7 or Theorem 8.2.6. We now have two distinct types of .ω-limit sets at infinity, namely, the isolated and the open recurrent .ω-limit sets at infinity. Although Theorem 8.3.10 includes a finite union of closed intervals, only one is needed to determine a unique positively recurrent orbit closure .O+ (x)− . Thus there is no interest in perfect sets consisting of more than one closed interval, that is, made by removing a finite number of open intervals greater than 1. The remaining case determined by Theorem 8.3.10 is removing a countable number of open intervals from a closed interval which have no common endpoints with each other or with the original interval. Again there is an additional restriction from the open recurrent type of .ω-limit sets at infinity, namely, having no interior which is the same as a nowhere dense set. Recall that a Cantor set is, by definition, a perfect nowhere dense set. Given a local section .λ of the flow, a Cantor subset C of .λ with endpoints .w1 and .w2 is a Cantor .ω-limit set at infinity provided C is contained in .[w1 , w2 ]λ and no points in the complementary intervals of C are contained in .ω(x). Theorem 8.3.11 If x is a point of a flow on a compact connected orientable surface X x t → ∞ as .t → ∞ and .ω(x) contains at least one moving point, then the such that . moving points of .ω(x) are of the same type of .ω-limit sets at infinity. Proof It follows from Theorem 8.3.7 that every isolated .ω-limit set at infinity satisfies x t → ∞ as .t → ∞ must be irrational for both the desired condition. Furthermore, . open recurrent and Cantor .ω-limit sets at infinity. So an isolated .ω-limited set at infinity cannot occur in either open recurrent .ω-limit sets at infinity or Cantor .ω-limit sets at infinity for a specific x, that is, the isolated type of the .ω-limit sets at infinity cannot intermingle with either of the other 2 types for a specific x.

260

8 Lifts and Limits

Since a single open recurrent .ω-limit set at infinity for a particular x implies that .ω(x) is the closure of an open positively recurrent set of X, every local section contained in the interior of .ω(x) intersects .O+ (x) in the interior of the local section. It follows that open recurrent and Cantor types of .ω-limit sets at infinity cannot intermingle for a specific x with an irrational limit as .t → ∞ and a moving point in .ω(x).

Corollary 8.3.12 Let .(X, R) be a flow on a compact orientable surface with .γ (X) ≥ 1. If x is a point of X such that .ω(x) contains at least one moving point and .O+ ( x ) limits x ∈ π −1 (x), then either to an irrational point of .S1 for . (a) Every .y ∈ ω(x) \ F is an open recurrent .ω-limit point at infinity of x or (b) Every .y ∈ ω(x) \ F is a Cantor .ω-limit point at infinity of x.

Part (a) implies that x is positively recurrent and not periodic, but part (b) by itself does not imply anything about the recurrent behavior of x .

8.4 Geometry of Almost Periodic Orbits Let X be a compact connected orientable surface of genus at least two. Because periodic points, which includes fixed points, have no role in this section, it will be convenient to refer to almost periodic points that are not periodic as strictly almost periodic. Since the closure of a periodic orbit is obviously just a periodic orbit and a minimal set, every point in the closure of a strictly almost periodic orbit is strictly almost periodic. Since strictly almost periodic orbits are recurrent and not periodic, the limits .u

= lim  x t and v = lim  xt t→∞

t→−∞

(8.12)

exist and are distinct irrational points of .S1 , when . x is a lift of a strictly almost periodic point x (Theorems 7.3.21 and 8.2.6). The main result of this section (Theorem 8.4.7) is that the lift of the orbit of a strictly almost periodic point is the type of an irrational h-line. To get started, we need a little more hyperbolic geometry; specifically we need to generalize equation (6.25) to equidistant curves and apply it. Proposition 8.4.1 Let L be an h-line, E an equidistant curve to L, and .ζ a point in .B2 . If .ζ is on the same side of L as E and .L⊥ , the h-line perpendicular to L passing through .ζ , then .dh (ζ, E)

≡ inf{dh (ζ, z) : z ∈ E} = dh (ζ, E ∩ L⊥ ).

(8.13)

Proof Suppose .ζ is on the side of E that does not contain L and .z ∈ E. Let .K ⊥ be the perpendicular to L passing through z and let .w = K ⊥ ∩ L as shown in Figure 8.15.

8.4 Geometry of Almost Periodic Orbits

261

Fig. 8.15 The point .ζ is on the side of E that does not contain L and .z ∈ E

ζ z

E

w

L L⊥

.dh (ζ, z)

K⊥

+ dh (z, w) ≥ dh (ζ, w) ≥ dh (ζ, L) =

dh (ζ, E ∩ L⊥ ) + dh (E ∩ L⊥ , L ∩ L⊥ ), because the hyperbolic metric is additive on h-lines (Proposition 6.1.23). Notice that and .dh (E ∩ L⊥ , L ∩ L⊥ ) are both equal to the distance from the equidistant curve E to L and can be subtracted to obtain

.dh (z, w)

.dh (ζ, z)

≥ dh (ζ, E ∩ L⊥ )

from which equation (8.13) follows. When .ζ is in the region between E and L, start with .dh (z, ζ ) + dh (ζ, L ∩ L⊥ ) and

eventually subtract .dh (ζ, L ∩ L⊥ ). Corollary 8.4.2 Let L be an h-line, let E and .E  be equidistant curves of L on the same side of L, let .ζ be a point between E and .E  , and let r be a positive real number. If .r < min{dh (ζ, E), dh (ζ, E  )}, then the disk of radius r centered at .ζ is contained in the region between E and .E  . Exercise 8.4.3 Prove Corollary 8.4.2.

From Chapter 7, we know that for every compact connected orientable surface X with .γ (X) ≥ 2 there exists a Fuchsian covering group . containing only hyperbolic elements such that .B2 /  is isomorphic to X, making the natural projection of 2 2 2 .B onto .B /  a universal cover of X . For every .ζ ∈ B , there exists a compact fundamental region .Dζ () for . , called the Dirichlet region. Moreover, by Proposition 7.1.14, there exists a radius .ν(ζ ) in .R+ such that

262

8 Lifts and Limits .S Dζ ()

⊂ {z : dh (z, Sζ ) < ν(ζ )}

for all .S ∈  , and for every w in .B2 , there exists .S ∈  such that .w

∈ S Dζ () ⊂ {z : dh (z, w) < 2ν(ζ )}.

This result plays a key role in the proof of the main theorem in this section. Lemma 8.4.4 Let .β : R → X be an injective curve on a compact connected orientable  : R → B2 be a lift of .β. If there exist irrational points surface with .γ (X) ≥ 2, and let .β 1 u and v of .S (with respect to .) satisfying .

(t) = u = v = lim β (t), lim β

t→∞

t→−∞

(8.14)

then the following hold:  is injective. (a) The lift .β (R) ∩ T β (R) = φ when .T = ι. (b) Given .T ∈ , .β (c) Letting L be the h-line determined by u and v, .L ∩ T L = φ for .T = ι. (s) = β (t), then .β(s) = π(β (s)) = π(β (t)) = β(t), proving part (a). Proof If .β   If .β (R) ∩ T β (R) = φ for some .T ∈ , then there exist real numbers s and t such that (s) = T β (t) from which it follows that .β(s) = β(t). Therefore, .s = t and .β = Tβ  .β   (unique lifting property). In particular, .β (0) = T β (0) and T has a fixed point. Since .ι is (R) ∩ T β (R) = φ when .T = ι to complete the only element of . having a fixed point, .β the proof of part (b). (R) ∩ T β (R) = φ and If .L ∩ T L = φ, then, using the Jordan separation theorem, .β part (c) follows from part (b).



Let .λ be a local section of length .2α at . y ∈ O( x ) where . x is a lift of a strictly almost periodic point x and satisfies equation (8.12) with u and v irrational points of .S1 . So the function .h : λ × [−α, α] → B2 defined by .h(x, t) = xt is a homeomorphism of .λ × [−α, α] onto the flow box .B = λ[−α, α]. In the present context, .λ is an arc with endpoints p and q , and .(a, b)λ (−α, α) is an open connected set of .B2 for a and b in .λ. In particular, .U = (p,  y )λ (−α, α) and .V = ( y , q)λ (−α, α) are open connected subsets of .B2 such that .U

∪ V = (p, q)λ (−α, α) \ { y t : −α < t < α}.

By the Jordan separation theorem, .B2 \ O( x ) has exactly 2 components, and U and V are in different components. Moving forward along .O( y ) in the flow box, U is on the left and V is on the right or vice versa. The one on the right will be called the right component of .λ and the other one will be the left component of .λ. Let .DR and .DL denote components of .B2 \ O( x ) containing the right and left components of .U ∪ V , respectively. Given a local section .λ of length .2α  at . y ∈ O( x ), it is an exercise to show that .DR = DR and .DL = DL . Therefore, a local

8.4 Geometry of Almost Periodic Orbits

263

section at any point of .O( x ) can be used to consistently identify the components of 2 \ O ( x ).

.B

Exercise 8.4.5 Prove that .DR = DR and .DL = DL . Theorem 8.4.6 Let .(X, R) be a flow on a compact connected orientable surface with ≥ 2 and with a Fuchsian covering group .. If . x in .B2 satisfies equation (8.12) 1 with .u = v irrational points of .S and L is the h-line determined by u and v, then there exists .μ > 0 with the property that for any .τ > 0 there exists .t > τ and .t  < −τ < 0 such that .dh ( x t, L) < μ and .dh ( x t  , L) < μ.

.γ (X)

Proof If .O( x ) is the type of the h-line L, there is nothing to prove so we will assume is not the type of the h-line L. Since the Dirichlet region .D x () is a compact () = D () : S ∈ } give a tessellation of fundamental region for ., the sets .{S D x S x 2 .B by fundamental regions. By Proposition 7.1.14, there exist a radius .ν( x ) in .R+ such 2 that given .w ∈ B there exists .S ∈  satisfying the following: .O ( x)

.w

∈ S D x )}. x () ⊂ {z : dh (z, w) < 2ν(

Let .E1 be the equidistant curve of L containing . x . There exists an equidistant curve for L on the side of .E1 not containing L such that .dh (E1 , E2 ) = 4ν( x ). Then there is a disk centered at every point midway between .E1 and .E2 containing an .S D x () for some .S ∈ . Using .E3 and .E4 , repeat the construction on the other side of L, and let .D1 and .D3 be the regions between .E1 and .E2 and between .E3 and .E4 , respectively. (See Figure 8.16.) It is sufficient to prove the .t > τ > 0 case because the proof of the .t  < −τ case will be essentially the same. If .max{s :  x s ∈ E1 ∪ E3 } is not finite, then the result holds for x s ∈ E1 ∪ E3 } is finite, then it suffices to prove the some .μ > dh (E1 , L). If .max{s :  x is in .E1 and . x t is in the open set bounded by .E1 and containing .E2 . theorem when . In this context it will be shown that the result holds for some .μ > dh (E2 , L). Suppose .μ does not satisfy the conclusion of the theorem. Then there exists .τ > 0 such that .dh ( x t, L) > μ for all .t > τ , and there exists an h-line .L⊥ perpendicular to L such that − .O ( x τ ) ∩ L⊥ = φ. Then .L⊥ divides .D1 into 2 parts one of which does not intersect − .O ( x ), denoted by .D  (shaded region in Figure 8.16). Make the crucial observations x is not in .D  . that .D  does not intersect the region between L and .E1 and that . The construction in the first 2 paragraphs of the proof now guarantees that there exists .T ∈  such that .T  x is in .D  . Clearly, .T = ι because . x = T  x . If .T L = L, then either L is the axis of a hyperbolic element of . and u and v are rational, which is impossible because they are irrational, or T interchanges u and v and has a fixed point in L, which is impossible because . does not contain any elliptical elements. Thus .T L = L. x must be in a component of .B2 \ O( x ), as shown in Figure 8.16. Part (c) The point .T  of Lemma 8.4.4 implies that .L ∩ T L = φ. It follows from the hypothesis that .E2

.T

u = lim T  x t and T v = lim T  xt t→∞

t→−∞

264

8 Lifts and Limits

Fig. 8.16 Case when > μ for all .t > τ

.dh ( x t, L)

L⊥

u

Tx  x τ

TL x 

E2

E1

E3

L

E4

v

with u and v distinct irrational elements of .S1 . Thus for .|t| large, .T  x t and T L are in x ). A perpendicular h-line segment from .T  x to T L the same component of .B2 \ O( measuring .dh (T  x , T L) must start in .D  , cross the region between .E2 and .E1 , and x , L) < dh (T  x , T L) producing a enter the region between L and T L. Therefore, .dh (

contradiction.

One way of viewing Theorem 8.4.6 is that when .O( x ) satisfies equation (8.12), cannot approach its limiting points “tangentially” to .S1 because the previous theorem shows that there must be sequences that violate such an approach. It applies to all recurrent orbits, that is, to all orbits that are both positively and negatively recurrent. In particular, it applies to strictly almost periodic orbits and will provide a foundation for proving the main theorem about them next.

.O ( x)

Theorem 8.4.7 Let .(X, R) be a flow on a compact connected surface with .χ(X) = 0. x ) is the type of an h-line with If x is a strictly almost periodic point of .(X, R), then .O( irrational endpoints for all . x in .π −1 (x). Proof By part (b) of Theorem 2.2.6, the nonorientable case follows from the orientable case using the orientable double covering. So we can assume that X is orientable and .γ (X) > 1. Since a strictly almost periodic point of .(X, R) is a recurrent point that is not periodic, x t and .limt→−∞  x t exist and that they are Theorem 8.2.6 implies that the limits .limt→∞  distinct irrational elements, which we will denote as u and v, respectively, of .S1 . Let L x ). be the h-line determined by the limit points u and v in equation (8.12) for the orbit .O( x ) is the type of an h-line with irrational endpoints, it now suffices to To prove that .O( prove the existence of .D > 0 such that .dh ( x t, L) < D for all .t ∈ R. Suppose no such D exists. Then there exists a sequence .τn of real numbers such x τn , L) > n + μ + 4ν(0) where .μ is obtained from Theorem 8.4.6 and .ν(0) is that .dh ( given by Proposition 7.1.14. Clearly .|τn | → ∞ as .n → ∞. It can be assumed that .τn converges to infinity and . x τn is on the same side of L for all n.

8.4 Geometry of Almost Periodic Orbits

265

Let E be the equidistant curve of L such that .dh (E, L) = μ and E is on the same side of L as . x τn is for all n. Then E divides .B2 into 2 open connected sets. Let .E + be the region determined by E that does not contain L. Now using Theorem 8.4.6 and passing to a subsequence of .τn , we can construct 2 sequences .tn and .tn such that for all .n > 0: .tn

< τn

< tn ,

tn

tn+1 ,


μ when tn < t < tn .

Setting .Jn

= [ x tn ,  x tn ] x tn ,  x tn ]E ϕ ∪ [

produces a sequence of disjoint simple closed curves in .B2 that will be used to obtain a contradiction of the strict almost periodicity of .x ∈ X. Note that the sequence of simple x ) for closed curves .Jn is constructed so that the interior of .Jn is on the same side of .O( all n, namely, the right for the particular configuration being used in the proof. The same ideas apply to the other configurations. Let .E 1 be the equidistant curve to L in .E + at a distance of .μ + 4ν(0) from L. Let  1 2 + .E be the closed region between E and .E . Then the equidistant curve .E to L in .E at 1  1 a distance of .μ + 2ν(0) lies halfway between E and .E in .E . Note that L, E, .E , .E 2 , and .E  are independent of n. (See Figure 8.17.) x sn , L) = max{dh ( x t, L) : tn ≤ t ≤ Fix .n > 4 and let .sn be the first time that .dh ( tn+1 }. Let .Ln be the perpendicular h-line to L passing though . x tn , and let .Hn be the half plane determined by .Ln and u. There exists an h-line .Ln in H and perpendicular to L with the following properties: (a) Letting .Hn be the open half plane determined by .Ln and u, the set 

.Hn

∩ E  ∩ [ x tn ,  x sn ] ϕ

is empty. (b) The set .n

= E  ∩ Ln ∩ [ x tn ,  x sn ] ϕ

is not empty. Define 2 positive real numbers by .σn = max{t :  x t ∈ n } and by .σn is the unique positive real number such that

266

8 Lifts and Limits 



.dh ( x σn , Ln

∩ L) = max{dh ( x t, Ln ∩ L) :  x t ∈ n }.



Lemma 8.4.8 The following hold: (a) .σn = σn . (b) If . x s lies between .E ∩ Ln and . x t in .n , then .s < t. Proof Clearly, .dh ( x σn , Ln ∩ L) ≤ dh ( x σn , Ln ∩ L) and .σn ≤ σn . So it suffices to show x σn , Ln ∩ L) < dh ( x σn , Ln ∩ L) leads to a contradiction. that .dh ( Observe that . x σn lies on .Ln and between .E ∩ Ln and . x σn . Now a routine orbit x σn ) cannot go to infinity without intersecting .( x tn ,  x σn ) analysis shows that .O+ ( ϕ to

complete the proof.

Let .Ln be the h-line in .Hn and perpendicular to L such that .dh (L ∩ Ln , L ∩ = 4ν(0). Then the equidistant curves E , .E 1 to L and the h-lines .Ln and .Ln perpendicular to L form a quadrilateral denoted by .Bn . (See Figure 8.17.) Ln )

Lemma 8.4.9 For every m and n in .Z+ , there exists a hyperbolic transformation T with fixed points with u and v (T is not in .) that is an h-isometry of .Bm onto .Bn . Proof The proof of this lemma uses Proposition 6.2.4 and the following pages. Recall the group .H

= {Tρ : ρ ∈ R and Tρ 0 = tanh(ρ/2)}

of all hyperbolic transformations with fixed points 1 and .−1. Since .LR is the orbit of the contains a unique hyperbolic transformation that maps any specified point of .LR onto another specified of .LR . Then a suitable conjugate of .H does the same for L. So there exists a hyperbolic transformation T of L onto itself such that .T (L ∩ Lm ) = L ∩ Ln . Since T is distance and direction preserving, .T (L ∩ Lm ) = L ∩ Ln . Moreover,     .T Lm = Ln and .T Lm = Ln . It follows from Proposition 8.4.1 that .dh (E ∩Lm , E 1 ∩Lm ) and .dh (E ∩Lm , E 1 ∩Lm ) are equal to .dh (E, E 1 ), and likewise for n. Thus all 4 of these terms are equal. Now the T -invariance of E and .E  implies that the 4 edges of .Bm match properly with the 4 edges of .Bn . In particular, the diameters of .Bm and .Bn are equal for all m and n, and denoted

by .δ. .Tρ -flow, .H

Lemma 8.4.10 .T  x tn is not contained in the interior of .Jn for all .T ∈  \ {i}. Proof Suppose there exists .T ∈  such that .T  x tn is in the interior of .Jn . Since, by x ) it follows that .T  x tn is on construction, the interior of .Jn is on the right side of .O( the right side of .O( x ). Note that .(T  x tn )t → T u ∈ S1 as .t → ∞ and .(T  x tn )t → x ) ∩ O(T  x ) = φ and .T L ∩ L = φ it follows that T L T v ∈ S1 as .t → −∞. Since .O( x tn , T L) > dh ( x tn , L), a is on the opposite side of L from E. This implies that .dh (T  contradiction.



8.4 Geometry of Almost Periodic Orbits

267

Fig. 8.17 The interior of .Bn is shaded

E1

bn bn βn E

x tn

x tn

L Ln

Ln

Let .bn =  x σn =  x σn , which is on the left boundary of .Bn (see Figure 8.17). Note that, by construction, .Bn contains a disc of radius .2ν(0) and hence .Bn contains a fundamental region. If .( x sn ,  x tn ) ϕ does not intersect the interior of .Bn , then .Bn is contained in .Jn by connectivity and hence .T  x tn is in the interior of .Bn for some  .T ∈  , which is impossible by Lemma 8.4.10. Thus for all .n > 4 there exists .sn ,    .sn < sn ≤ tn such that . x sn is in the interior of .Bn . That is, .bn βn is in the interior of   .Bn where .βn = sn − σn . Hence, for every .n ≥ 4, there exist .Sn ∈  such that .bn βn is in .Sn D0 () for all n. Thus the sequences .cn = Sn−1 bn and .cn = Sn−1 bn βn = cn βn are both in .{z : dh (z, 0) ≤ δ}, and it can be assumed that .cn = Sn−1 bn converges to c. Now almost periodicity enters the proof for the first time. The sequence .π(cn ) = π(bn ) is a convergent sequence in .O(x) and its limit .π(c) is a strictly almost periodic point in the minimal set .O(x)− of .(X, R). It follows that there exists a control curve  constructed using .O( .J x ) such that .{z : dh (z, 0) ≤ δ} ⊂ J− and that .O(c) crosses − +  from .J to .J at some time .α > 0. By almost periodicity and the continuity .J of the lifted flow there exists N such that for all .n > N , .O(cn ) crosses the same local section of .J that .O(c) crossed at time .α at some time .αn and .αn converges x sn , L) = max{dh ( x t, L) : tn ≤ t ≤ tn } = n + μ + 4ν(0). to .α . Recall that .dh (  Also recall that .bn =  x σn and .bn βn =  x sn where .σn < sn < sn . It follows that   + when .n > N and .βn > α , but by .βn = (sn − σn ) → ∞. Therefore, .cn βn is in .J − construction .cn βn is in .J for all n.

Chapter 9

Recurrent Orbit Closures

In 1943, the Russian mathematician A. Maier [46] published a seminal paper about flows on surfaces. This chapter is devoted to presenting covering space proofs of his essential results. The Lefschetz fixed point theorem implies that every flow on a compact connected surface has at least one fixed point unless the surface is the torus or the Klein bottle (Theorem 1.2.11). Since a positively recurrent point of a flow on the Klein bottle is periodic, only the torus can support minimal flows such as the irrational straight line flows. The only other minimal sets that are not periodic orbits of a flow on a compact connected surface are the nontrivial minimal sets, and they are closed and nowhere dense sets. Maier, however, showed that the recurrent orbit closures of a flow on a compact connected surface behave more like minimal sets than recurrent orbit closures do in general. For example, Maier proved that 2 recurrent orbit closures of a flow on a compact connected surface are equal, if their intersection contains a positively or negatively recurrent point. Similarly, Maier proved that a point in the omega limit set of a positively recurrent point is positively recurrent if it contains a moving point in its omega limit set. Section 9.1 lays the covering space foundation for Maier’s theorems with a density result that follows from properties of the lifted flow on the universal covering space. From this result, we obtain existence theorems for positively recurrent orbits with the hypothesis on the behavior of orbits in the universal covering space as time goes to infinity. It follows from Maier’s first theorem that a flow on a compact connected surface does not contain a strictly decreasing sequence of recurrent orbit closures. The last section is devoted to showing that, in addition, there is a finite bound on the number of distinct recurrent non-periodic orbit closures a flow on a compact connected surface can have. This bound is a function of the genus. Here we use covering spaces to obtain some general results about simple closed curves on a compact connected

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_9

269

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9 Recurrent Orbit Closures

orientable surface and then use these results to count the maximal number of distinct recurrent orbit closures.

9.1 Covering Space Criteria The focus of this section is a density result from [62] establishing conditions on lifted orbits that imply the existence of a dense positively invariant set in .ω(x) \ F . Theorem 9.1.2 then provides the essential tool for proving Maier’s fundamental theorems. The mathematical context will be the same as described in the opening paragraphs of Section 8.2. The results in this section hold for .γ (X) ≥ 1, and the proofs for .γ (X) = 1 and .γ (X) ≥ 2 are essentially the same. The notion of a sequence of pairs of consecutive crossing times (p. 236) is again a useful tool. The first result is a lemma that shows how the control curves from a sequence of pairs of crossing times can be modified to have a common single intersection point. Lemma 9.1.1 Let .λ be a local section for a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1. If .τi is a sequence of consecutive pairs of crossing times of .λ for a non-periodic point .x ∈ X, then there exists a sequence of simple closed curves .J2i and a point p in .λ with the following properties:  = {p} whenever .i = j . (a) .J2i ∩ J2j (b) If .J2i and .J2i are lifts of .J2i = [xτ2i , xτ2i+1 ]ϕ ∪ [xτ2i , xτ2i+1 ]λ , and .J2i beginning, respectively, at . x τ2i and .p  in  .λ, a lift of .λ, then there exist .Si ∈  λ. such that .J2i and .J2i end at .Si  x τ2i and .Si p , respectively, in .Si

Proof Let the length of .λ be .4α. If .θλ is a homeomorphism of .[−1, 1] onto .λ, then h(s, t) = θλ (s)t is a homeomorphism of .[−1, 1] × [−2α, 2α] onto the flow box .B = {wt : w ∈ λ and |t| ≤ 2α} mapping vertical line segments onto pieces of orbits. In particular, we can use h to modify .J2i within B. By hypothesis, .xτi is a sequence of distinct points in .λ. Consider the two sequences of points in .[−1, 1] × [−2α, 2α] given by .

(θλ−1 (xτ2i ), α)

.

and (θλ−1 (xτ2i+1 ), −α).

.

Let . 2i be the line segment in .[−1, 1]×[−2α, 2α] from .(0, 0) to .(θλ−1 (xτ2i ), α) and −1 . 2i+1 be the line segment from .(θ λ (xτ2i+1 ), −α) to .(0, 0). Then . 2i and . 2i+1 are all distinct line segments emanating from .(0, 0), and the intersection of any two of them is the point .(0, 0). Set .p = h(0, 0), and define .J2i as follows:

9.1 Covering Space Criteria −α

271 α

p x τ2i

x (τ2i + α)

 B Si p x (τ2i+1 − α)

x τ2i+1 Si x τ2i

Si B

Fig. 9.1 The lifts .J2i and .J2i shown schematically

  J2i = h( 2i ) ∪ x(τ2i + α), x(τ2i+1 − α) ϕ ∪ h( 2i+1 ).

.

 = p = h(0, 0) when Clearly, each .J2i is a simple closed curve and .J2i ∩ J2j  .i = j . The two curves .J2i and .J 2i traverse the same path outside of B. Suppose     .J2i and .J x τ2i and .p  in  .λ, a lift of .λ. (See 2i are lifts of .J2i and .J2i beginning at .  = {  and there xi (τ2i + α) is in .B, Figure 9.1.) Let .B wt : w  ∈ λ and |t| ≤ 2α}. Then .  Since .J2i and .J  can only differ exists .Si ∈  such that . xi (τ2i+1 − α) is in .Si B. 2i  inside B, and they both include the arc . xi (τ2i + α), xi (τ2i+1 − α) ϕ , it follows that .Si satisfies the conclusion of (b), because B is evenly covered.



Theorem 9.1.2 Let .λ be a local section of a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, and let x be a point of X such that the positive x ) goes to infinity when t goes to infinity for . x ∈ π −1 (x). Suppose there orbit .O+ ( exists a subset D of .λ with the following properties: (a) .D ⊂ ω(x) ∩ λ. (b) .D − is a Cantor set in .λ. (c) When .π( u) ∈ D, the positive orbit .O+ ( u) is unbounded. If .λ is a local section at .y ∈ ω(x) \ F such that .λ ∩ λ = φ, then .O+ (u) ∩ λ = φ for some .u ∈ D. Furthermore, the set .{ut : u ∈ D and t ≥ 0} is dense in .ω(x) \ F , where F denotes the set of fixed points of the flow .(X, R). Proof Assume that .γ (X) ≥ 2. (The proof for .γ (X) = 1 is almost identical.) Note that x is not periodic because .ω(x) ∩ λ is not finite and .D − ⊂ ω(x) because .D ⊂ ω(x). The proof of the theorem will now be split into two cases. Case 1: Suppose .O(x) ∩ D − = φ. If .xτ is in .D − , then .xτ is in .ω(x) = ω(xτ ) and .xτ is positively recurrent. Since .λ is a local section at .y ∈ ω(x), there exists  .σ > 0 such that .(xτ )σ = x(τ + σ ) is in .λ and is not an endpoint. Because .xτ is − in .D , there exists a sequence .un in D converging to .xτ . Then by continuity of

272

9 Recurrent Orbit Closures

the flow, .un σ converges to .x(τ + σ ). Thus .un σ is in the .λ flow box, and .O+ (un ) crosses .λ for large n, proving the first statement. Note that .O+ (xτ ) dense in .ω(x) and the continuity of the flow can also be used to prove that .{ut : u ∈ D and t ≥ 0} is dense in .ω(x) \ F when .xτ is in .D − . This completes the proof of the theorem in the first case. Case 2: Suppose .O(x) ∩ D − = φ. By making .λ a little shorter if necessary, we can assume that the endpoints of .λ are not in .D − and that the hypotheses are still satisfied. Since y is in .ω(x), there exists .t > 0 such that .xt ∈ λ , and in fact, we can assume that x is in .λ .   Let . x be in .π −1 (x), and let  .λ be the lift of .λ containing . x . Because .O+ ( x ) goes +  to infinity when t goes to infinity by hypothesis, .O ( x ) can intersect  .λ at most a finite number of times. By letting .τ = max{t :  xt ∈  λ } and replacing x with .xτ , it +  +  can be assumed .O ( x ) ∩ λ = { x }. Thus .O (S x ) ∩ T ( λ ) is a point or the empty set for all S and T in . by Proposition 8.2.1. Let  .λ be the first lift of .λ that . x t crosses for .t > 0. Again the positive orbit .O+ ( x)  can intersect .λ at most a finite number of times, say .s1 < · · · < sk .

Lemma 9.1.3 There exist disjoint closed intervals .λj , .1 ≤ j ≤ k, of .λ with the following properties: (i) (ii) (iii) (iv)

For .j = 1, . . . k, .xsj is in .λj . The endpoints of .λj are not in .D − for .j = 1, . . . k. k − ⊂ .D j =1 λj . − is a Cantor set. .λj ∩ D = φ if and only if .λj ∩ D

Proof Each .xsi is in a complementary interval of .D − because .O(x) ∩ D − = φ, and the endpoints of .λ are not in .D − . Hence, . x s1 , . . . ,  x sk can be indexed as a x ) goes to infinity when t goes to infinity for strictly monotonic sequence since .O+ ( . x ∈ {π −1 (x)}. Starting with .xs1 and using endpoints not in .D − , it is straightforward to construct a sequence of disjoint closed intervals .λj containing .xsj and satisfying

the four properties. It follows from the lemma that .λj and .D ∩ λj satisfy conditions (a), (b), and (c) in the statement of the theorem. Thus .λ and D can be replaced with .λj and .D ∩ λj for any choice of j without loss of generality by setting .σ = sj . Whether or not  .λ is still the first lift of .λ that . x t crosses for .t > 0 will not affect the rest of the x σ ) ∩ T ( λ) is a point or the proof. The advantage of this change is that now .O+ (S empty set for all S and T in ., and .O+ ( x τ ) has the same property when .τ > σ and . x τ ∈ T ( λ) because . x τ is the only point of .O+ ( x τ ) in the local section .T ( λ) (Proposition 8.2.1). Let .(am , bm )λ , .m = 1, . . . , be the complementary intervals of .D − . Because − = φ and .D − ⊂ ω(x), the orbit of x must intersect infinitely many .O(x) ∩ D complementary intervals of .D − . Since .y ∈ ω(x) ∩ λ , there exists a sequence of consecutive crossings .xτk of .λ by .O+ (x) such that .τk > σ . Clearly, .τk goes to infinity. Although infinitely many

9.1 Covering Space Criteria

273

of the orbit segments .(xτk , xτk+1 )ϕ must cross complementary intervals of .D − , a particular .(xτk , xτk+1 )ϕ need not intersect any of them and can intersect at most a finite number of complementary intervals .(am , bm )λ because the length of the local section .λ governs the rate of consecutive crossings (Proposition 4.1.1). Two inductively defined increasing subsequences .kj and .mj for .j ≥ 0 of integers can now be constructed with the following properties: (a) .τkj < τkj +1 for .1 ≤ j . (b) .(xτkj , xτkj +1 )ϕ ∩ (amj , bmj )λ = φ. (c) .(xτk , xτk +1 )ϕ ∩ (amj , bmj )λ = φ for all . < j . As in Section 8.2, consider the sequence of pairs of consecutive crossing times τ2i = τki and .τ2i+1 = τki +1 of .λ , and set

.

J2i = [xτ2i , xτ2i+1 ]ϕ ∪ [xτ2i+1 , xτ2i ]λ .

.

Then the simple closed curves .J2i are not null-homotopic (Proposition 8.2.1). Applying Lemma 9.1.1 to the sequence of pairs of consecutive crossing times  .τ2i = τki and .τ2i+1 = τki +1 of .λ produces a sequence of simple closed curves     .J 2i and a point .p ∈ λ such that .J2i ∩ J2j = {p} for all .i = j . By passing to a subsequence, it can be assumed that .J2i has the property that the simple closed curves .J2i are all path homotopic at p because on a compact connected orientable surface of genus .γ (X) there are at most .6γ (X)−3 simple closed non-path homotopic curves that intersect only at one common point. (See page 246.)   Let .p  be the lift of p in  .λ , let .w 2i be the lift of .xτ2i such that .w 2i is in  .λ , and let  beginning .σ2i = τ2i+1 − τ2i . Then there exists .T ∈  such that the lift of any .J 2i at .p  ends at .T p  in .T  λ . It follows by Lemma 9.1.1 that .w 2i σ2i is in .T  λ for all i. Consequently, J (i, j ) = [ w2i , w 2i σ2i ] w2j , w 2j σ2j ] w2i , w 2j )λ ∪ ( w2i σ2i , w 2j σ2j )T λ ϕ ∪ [ ϕ ∪ (

.

is a simple closed curve in .B2 with nonempty interior .I (i, j ) when .i < j . See Figure 9.2. .( w2j, w 2j σ2j ) .λ2j of .λ at  .c2j in a complemenBy construction, ϕ intersects a lift   tary interval .  am2j ,  bm2j λ such that .(xτki , xτki +1 )ϕ ∩ (amj , bmj )λ = φ for .i < j . 2j Furthermore, .O+ ( w2j ) ∩  λ2j = { c2j } because .O+ ( w2j ) can intersect a lift of .λ at most once. Therefore,  .am2j is in the exterior of .J (i, j ), and . bm2j is in the interior of .J (i, j ) or vice versa. Without loss of generality,  .am2j is in .I (i, j ). − .am2j is in .D , there must exist . u in .I (i, j ) such that .π( u) ∈ D. By Since  + 1 hypothesis, .O ( u) is unbounded for all .u ∈ D, and hence, .ω( u) ∩ S = φ (in the extended flow on .D2 ). Therefore, .O+ ( u) must intersect .T  λ and .O+ (u) ∩ λ = φ for some .u ∈ D. This completes the proof of the first conclusion of the theorem. Since we already proved the entire theorem in case 1, we continue to assume that − = φ. To show that .{ut : u ∈ D and t ≥ 0} is dense in .ω(x) \ F , let .O(x) ∩ D y be a point in .ω(x) \ F , and let U be an open neighborhood of y. If y is in .D − ,

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9 Recurrent Orbit Closures

Fig. 9.2 The simple closed curve .J (i, j ) with interior .I (i, j )

c2j 

I(i, j) w 2j σ2j

w 2j  λ

 Tλ

w 2i

w 2i σ2i

then .D ∩ U = φ. If y is not in .D − , it suffices to find a local section .λ at y such that .λ ⊂ U ∩ (X \ λ) and apply the first part of the theorem. There are two cases: If y is not in .λ, then .U ∩ (X \ λ) is an open neighborhood of y, and there exists a local section .λ at y contained in .U ∩ (X \ λ) by Corollary 4.1.9. If y is in .λ, then there exists a complementary interval .(a, b)λ of .D − containing y. Let .λ be a closed interval contained in .(a, b)λ containing y. Replace .λ by one of the two components

of .λ \ (a, b)λ and apply the first part of the theorem. Although the statement of Theorem 9.1.2 gives the impression that the hypotheses are specific to special local sections, they are actually satisfied throughout .ω(x) \ F when they are satisfied for one local section intersecting .ω(x). Theorem 9.1.4 Let .λ be a local section of a flow .(X, R) on a compact connected orientable surface X with .γ (X) ≥ 1, and let x be a point of X such that the positive semi-orbit .O+ ( x ) goes to infinity when t goes to infinity for . x ∈ π −1 (x). Suppose there exists a subset D of .λ with the following properties: (a) .D ⊂ ω(x) ∩ λ. (b) .D − is a Cantor set in .λ. (c) The positive orbit .O+ ( u) is unbounded when .π( u) ∈ D. Then the following hold: (i) .limt→∞  x t is irrational for . x ∈ π −1 (x). − (ii) .{yt : y ∈ D and t ∈ R} is a recurrent orbit closure contained in .ω(x). (iii) There exist recurrent non-periodic points in .ω(x) whose orbits are dense in .ω(x) \ F . (iv) There are no periodic points in .ω(x) \ F . Proof Because .D − is a Cantor set in .λ ∩ ω(x), the points in .D − cannot be isolated .ω-limit points of x. Therefore, x has no isolated .ω-limit points, and .limt→∞  x t is irrational for . x ∈ π −1 (x) by Theorem 8.3.7 to prove (i). For (ii), set .A = {yt : y ∈ D and t ∈ R}. Obviously, .A− ⊂ ω(x) because − .A ⊂ ω(x). By Theorem 9.1.2, .ω(x) \ F ⊂ A . It follows that

9.1 Covering Space Criteria

275

A− \ F = ω(x) \ F.

.

(9.1)

Given .y ∈ D and .σ ∈ R, choose any .u1 and .u2 in .λ \ D − such that y is in .(u1 , u2 )λ . Then .[u1 , u2 ]λ σ is a local section at .yσ , and D  = A ∩ [u1 , u2 ]λ σ

.

satisfies the hypothesis for D in Theorem 9.1.2. If U is any open neighborhood of .yσ ∈ A, then we can choose .u1 and .u2 such that .[u1 , u2 ]λ σ is contained in U . The final conclusion of Theorem 9.1.2 now applies to .D  , and consequently, .D  R+ is dense in .ω(x) \ F . Since   D  R+ = A ∩ [u1 , u2 ]λ σ R+ ⊂ (A ∩ U )R+ ⊂ AR+ = A ⊂ ω(x) \ F,

.

it follows that .(A ∩ U )R+ is dense in .ω(x) \ F = A− \ F for all open sets U such that .A ∩ U = φ. Clearly, .A ∩ F = φ. If U and V are open sets of X such that .U ∩ A− = φ and − = φ, then U is an open neighborhood of some .yσ ∈ A and V is an open .V ∩ A neighborhood of some .wτ ∈ A. It follows from the previous paragraph that there exists .t > 0 such that (A ∩ U )t ∩ ((A− \ F ) ∩ V ) = φ

.

because .(A− \ F ) ∩ V is an open neighborhood of .wσ in the relative topology on − − .A \ F . Therefore, for any two open sets U and V of X such that .A ∩ U = φ and − .A ∩ V = φ, there exists .t > 0 such that (A− ∩ U )t ∩ (A− ∩ V ) = φ.

.

It follows from equation (9.1) that .A− \ F is invariant and hence .A− is invariant. By Proposition 1.1.3 applied to the flow restricted to the compact invariant set .A− , there is a dense residual subset of transitive points in .A− . If for some .n > 0, .{yt : |t| ≤ n} has interior in .A− for a moving point y in .A− , then there exists an open set U of X such that .φ = U ∩ A− ⊂ {yt : |t| ≤ n}. In particular, there exists .τ such that .|τ | < n and .yτ is in U . By Corollary 4.1.9, there exists a local section .λ at .yτ of length .α > 0 such that the flow box .λ[−α, α] is a subset of U . It follows that .λ[−α, α] ∩ A− consists of a finite number of disjoint pieces of the orbit of y each of length .2α. Therefore, .yτ is an isolated .ω-limit point of x because .A− \ F = ω(x) \ F , contradicting the fact that x has no isolated .ωlimit points because .D − is a Cantor set. Now Corollary 1.2.9 implies that .A− is a recurrent orbit closure. Let y be a recurrent point in .A− such that .O(y)− = A− . Clearly, y is not periodic. By equation (9.1), .O(y)− \ F = ω(x) \ F , proving (iii). And then Corollary 4.2.11 applies to .O(y)− to prove (iv).



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9 Recurrent Orbit Closures

The suspension of a Denjoy cascade with a minimal Cantor set is a flow on the torus with a non-periodic and nowhere dense minimal set M. Moreover, the lift of every positive and negative orbit to .R2 goes to infinity in an irrational direction. If we apply Theorem 1.3.3 to M, we obtain a flow with uncountably many orbits whose positive and negative lifts go to infinity in irrational directions, but every positively or negatively recurrent point is a fixed point. Thus the hypotheses pertaining to .ω(x) in Theorem 9.1.4 are essential. Theorem 9.1.5 Let .(X, R) be a flow on a compact connected orientable surface X with .γ (X) ≥ 1, and let x be a point of X such that the positive semi-orbit .O+ ( x) goes to infinity for . x ∈ π −1 (x) when t goes to infinity. If there exist uncountably  then .ω(x) contains a many positive orbits in .ω(x) with unbounded lifts in .X, positively recurrent orbit that is not periodic. Proof Since .ω(x) \ F is second countable, there exist a countable collection .λn of local sections of length .αn such that the sets .λn [−αn , αn ] cover .ω(x) \ F by Lindelof’s theorem ([39], Chapter 1, Theorem 15 or [59], Theorem 5.3.4). Let .A = {y ∈ ω(x) : O+ ( y ) is unbounded}. By hypothesis, A contains uncountably many orbits. Each orbit in A must intersect at least one .λn . So there exists a .λn such that .B = A ∩ λn is uncountable. Since an orbit can intersect a local section at most a countable number of times, uncountably many different orbits in A must intersect this .λn . Because .λn is homeomorphic to the closed interval .[0, 1], the set C of condensation points of B is a perfect set and .B \ C is countable or finite. Hence, C is closed, .B ∩ C = φ, and C is a subset of .ω(x), because C is a subset of − .A . (See [66], Chapter II, Section 6.) If C contains an interval, then x is positively recurrent. Otherwise, C is a Cantor set, that is, a perfect nowhere dense subset of .λn . Set .D = B ∩ C. Let u be an element of C. An interval of .λn containing u contains an uncountable number of points from B and hence must contain points of D because .B = (B ∩ C) ∪ (B \ C) and .B \ C is countable or finite. Thus .C ⊂ D − and .C = D − . Now Theorem 9.1.4 applies.



9.2 Maier’s Theorems Section 9.1 provides a new approach to Maier’s first and second theorems ([62]). Maier’s first theorem is a generalization of Theorem 5.3.4. Theorem 9.2.1 (Maier) Let x be a positively recurrent point of a flow (X, R) on a compact surface X. If y in ω(x) is a positively recurrent point, then O(x)− = O(y)− .

.

Proof Since O(x)− is connected, it can be assumed that X is connected. By Proposition 2.2.5 and Theorem 2.2.6, it suffices to prove the theorem for compact connected orientable surfaces.

9.2 Maier’s Theorems

277

When x is periodic, the result is obvious. Because every recurrent point is periodic for flows on S2 , it can also be assumed that γ (X) ≥ 1. When x is not periodic, y is not periodic unless it is fixed by Corollary 4.2.11. When the genus is 1 and x is not periodic, the result follows from Theorem 5.3.4. For the rest of the proof, it can be assumed that γ (X) ≥ 2 and that both x and y are not periodic. Clearly, O(y)− ⊂ O(x)− . So it suffices to show that x is in O+ (y)− . By Theorem 8.2.6, the lift of every positively recurrent orbit that is not periodic limits to an irrational point at infinity. Thus limt→∞  x t is an irrational point of S1 for −1  x ∈ π (x) and limt→∞  y t is an irrational point of S1 for  y ∈ π −1 (y). Since y ∈ ω(y), ω(y) contains a moving point. By Corollary 8.3.12, the moving points in ω(y) are either open recurrent or Cantor ω−limit points at infinity. In the first case, ω(y) is the closure of an open invariant set U containing y. In this case, x is in U because y ∈ ω(x), y is in U , and U is invariant. Thus x ∈ ω(y) ⊂ O+ (y)− . For the second case, let λ be a local cross section at y such that the endpoints are not in ω(y). Let D = O+ (y) ∩ λ. Since y ∈ ω(x), D ⊂ ω(x) ∩ λ. Clearly, D − = ω(y) ∩ λ is a Cantor set. Observe that the hypotheses of Theorem 9.1.2 are satisfied. The second conclusion applies to D, and O+ (y) is dense in ω(x) \ F . Since x ∈ ω(x) \ F , it follows

that x ∈ O+ (y)− . Corollary 9.2.2 Let x be a positively {negatively} recurrent moving point of a flow (X, R) on a compact surface X. If y in ω(x) is a negatively {positively} recurrent point, then O(x)− = O(y)− . Proof There is a recurrent point w in O(x)− by Theorem 1.2.8. Applying both versions of the theorem yields O(x)− = O(w)− = O(y)− .

Normally, z has been reserved for points in B2 or C. It is helpful to set this notational convention aside for the remainder of this section because it is easier to remember Maier’s second theorem as Maier’s xyz Theorem. It considers the following situation: ω(x) contains a moving point y and ω(y) contains a moving point z. The conclusion of Maier’s xyz theorem is that y is positively recurrent. The xyz theorem for the sphere is Theorem 4.3.11. We will prove a covering space xyz theorem and use it to prove Maier’s xyz theorem. Conversely, Maier’s xyz theorem implies the covering space xyz theorem, so the two results are equivalent. The covering space version, however, does not naturally include the possibility that y is periodic, which is the trivial case of Maier’s xyz theorem. Obviously, z is periodic when y is periodic. Conversely, if z is periodic, then both ω(x) and ω(y) contain a periodic point that is not a fixed point. Then Theorem 4.2.10 implies that ω(x) = ω(y) = O(z), which violates the xyz hypothesis unless y is periodic. Thus y is periodic if and only if z is periodic. We will take advantage of this observation in our statement of the covering space version of the xyz theorem. Theorem 9.2.3 Let (X, R) be a flow with fixed points F on a compact connected orientable surface X such that γ (X) ≥ 1. If x, y, and z are moving points in X such that y is in ω(x) and z is a non-periodic point in ω(y), then:

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9 Recurrent Orbit Closures

(a) The semi-orbit O+ ( x ) has an irrational limit for  x ∈ π −1 (x). + . (b) The semi-orbit O ( y ) is unbounded and ω( y ) ∩ B2 ⊂ F (c) z is not an isolated ω-limit point of y. Proof Since z is a non-periodic point in ω(y), y is not periodic. Let λ be a local section at z. Then there exist positive real numbers τ1 and τ2 such that yτ1 and yτ2 are consecutive crossings of λ with τ1 < τ2 . Since y is in ω(x), Proposition 7.3.20 implies that the semi-orbit O+ ( x ) has a limit, say a, in S1 for  x ∈ π −1 (x). Suppose a is rational. Then Corollary 8.3.9 implies that ω(y) ∪ α(y) ⊂ F . But this contradicts z being a moving point in ω(y). Thus a is irrational. Turning to part (b), suppose ω( y ) contains a moving point  z  . Let λ be a local      section at z = π( z ), and let λ be the lift of λ containing z . Since  y is not periodic, there exist distinct consecutive crossings  y τ1 and  y τ2 of  λ ; hence, the loop J = [ y τ1 ,  y τ2 ] y τ2 ,  y τ1 ]λ is null-homotopic, which implies that J = π(J) ϕ ∪ [ is also null-homotopic. The proof of part (a) of Proposition 7.3.20 applies almost verbatim to obtain a contradiction. Therefore, ω( y ) does not contain any moving points in B2 . If the positive semi-orbit O+ ( y ) is bounded for some  y ∈ π −1 (y), then some −1  z ∈ π (z) is in ω( y ), contradicting previous paragraph. Therefore, O+ ( y ) is unbounded for all  y ∈ π −1 (y), completing the proof of (b). The proof of part (c) is split into 2 cases: first γ (X) = 1 and then γ (X) ≥ 2. Assuming that γ (X) = 1 or that X is the torus, let λ be a local section at z in X. Suppose z is an isolated ω−limit point for y. Because z is in ω(y), there exist 2 distinct consecutive crossings yτ1 and yτ2 of λ by O+ (y). Then the simple closed curve J = [yτ1 , yτ2 ]ϕ ∪ [yτ2 , yτ1 ]λ is not null-homotopic by Proposition 7.3.20. It follows that a universal lift    = .J T n [ y τ1 ,  y τ2 ] y τ2 , T  y τ1 ]T λ ϕ ∪ [ n∈Z

is a control curve, and O+ ( y τ2 ) is contained in J+ . Since J is the type of a rational line, there exists a rational line L and a half plane H determined by L such that J and J+ are contained in H . x of x and Because y is in ω(x), Corollary 4.2.9 implies that there exists a lift  σ > 0 such that  x σ ∈ (T −1 y τ2 ,  y τ1 )λ . (For the remainder of the γ (X) = 1 case, refer to Figure 9.3.) Then T  x σ is in ( y τ2 , T  y τ1 )T λ . Let U be the region bounded x σ ) and T O+ ( x σ ) and the arcs [ x σ,  y τ1 ]λ , [ y τ1 ,  y τ2 ] by the positive orbits O+ ( ϕ, and [ y τ2 , T  x σ ]T λ . It is easily seen that U is positively invariant. x ) is irrational by part (a), there exist parallel irrational Since the limit of O+ ( lines L1 and L2 such that O+ ( x ) and T O+ ( x ) are between L1 and L2 by Theorem 5.3.14. Let B be the band between L1 and L2 , and set V = B ∩ H . y τ2 ) does not intersect L, L1 and L2 . Then U is contained V , and O+ ( There exists a primitive S in  such that  = [S] ⊕ [T ] because T is primitive (Proposition 5.2.7) and k ∈ Z such that J is between L and S k L. Hence, S nk J is y τ2 ) must intersect between the rational lines L and S (n+1)k L. Consequently, O+ (

9.2 Maier’s Theorems

279

U S nk J

J+

x σ yτ1  λ

J L

Tx σ

yτ2

L1

L2

Fig. 9.3 The case when γ (X) = 1

S (n+1)k L because O+ ( y τ2 ) is unbounded by part (b). To do so, it must cross S nk J. + y τ2 ) goes to infinity between the parallel irrational lines L1 and L2 . Therefore, O ( It follows that the limit of O+ ( y τ2 ) is irrational, and hence, z is not an isolated ω-limit point at infinity of y by Theorem 8.3.7, a contradiction, completing the γ (X) = 1 case. Suppose γ (X) ≥ 2 and z is an isolated ω-limit point of y. Theorem 8.3.7 cannot y ) is unbounded, but it can be be applied directly because we only know that O+ ( applied indirectly using Proposition 8.2.4. Again we use the construction from the beginning of the proof of Theorem 8.3.7. There exists a local section λ at z such that ω(y) ∩ λ = {z}. Let s = min{t ≥ 0 : xt ∈ λ}. Given  x ∈ π −1 (x), let  λ be the lift of λ such that  xs ∈  λ. Then λ and s x) ∩  λ and z is in λ. Using the group action of can be modified so that { x s} = O+ (  and Proposition 8.2.1, it follows that O+ ( x s) ∩ T  λ contains at most one point for all T ∈ . The point z divides the local section λ into two local sections, and O+ (y) must cross one of them infinitely often. So we can assume z is an endpoint of λ and still have the properties that λ ∩ ω(y) = {z} and that there is an infinite sequence of consecutive crossings of λ by O+ (y). Moreover, the property in the previous paragraph will not be lost. Replacing y with ys where s = min{t ≥ 0 : yt ∈ λ}, it can be assumed that y is in λ. (See Figure 9.4.) In this context, the set {t ≥ 0 : yt ∈ λ} is a sequence of nonnegative real numbers tk starting with t0 = 0 and increasing to infinity such that ytk and ytk+1 are consecutive crossings of λ by O+ (y) for all k ≥ 0. Then Jk = [ytk , ytk+1 ]ϕ ∪ [ytk+1 , ytk ]λ

.

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9 Recurrent Orbit Closures

z

Fig. 9.4 Consecutive crossings of λ by O+ (y)

yt2 yt1 y

is a sequence of simple closed curves that are not null-homotopic by Proposition 7.3.20 because y is in ω(x). It follows from λ ∩ ω(y) = {z} that .

lim ytk = z

k→∞

and that, given k  , Jk ∩ Jk  = φ for k sufficiently large. As in the proof of Theorem 8.3.7, there exists a subsequence Jki with i ≥ 0 such that Jki ∩ Jkj = φ when i = j and such that Jki is homotopic to Jkj for all i and j . Thus τ2i = tki , and τ2i+1 = tki +1 is a sequence of pairs of consecutive crossing times that satisfies the hypotheses of Proposition 8.2.4. (Condition (d) is trivial because λ ∩ ω(y) = {z} and z is an endpoint of λ.) Given  y ∈ π −1 (y) and a lift  λ containing  y τ0 , let  λm be the component of π −1 (λ)    λ2i = such that  y τm ∈ λm (λ0 = λ), and let S2i be the element of  such that S2i  y τ2i is in  λ2i+1 . Since J2i is not null-homotopic, S2i = ι λ2i+1 . By construction, S2i  and

 n 2i = [ y τ2i ,  .J S2i y τ2i+1 ] y τ2i+1 , S2i  y τ2i ]S2iλ2i (9.2) ϕ ∪ [ n∈Z

is a sequence of control curves. It follows from Proposition 8.2.4 that we can assume without loss of generality that S2i = S for all i by passing to a subsequence again. Letting [ y τ2i ,  y τ2i+1 ] y τ2i+1 , S y τ2i ]Sλ2i ϕ ∪ [

.

be the lift of J2i = Jki starting at  y τ2i , J2i =



.



y τ2i ,  S n [ y τ2i+1 ] y τ2i+1 , S y τ2i ]Sλ2i ϕ ∪ [

(9.3)

n∈Z

is a control curve for all i. Thus every J2i is the type of LS , the axis of S, and + J2j ⊂ J2i+ when i < j . It follows that J2j ⊂ J2i+ when i < j .

9.2 Maier’s Theorems

281

S −1 a

a Sa

S −1 yτ2i+1

yτ0 λ0

U

SO + (xσ )

O+ (x σ )

Rk J0

yτ2i

yτ1 Sλ0

Fig. 9.5 The region U is a positively invariant subset of J0+

Because y is in ω(x), it follows from Corollary 4.2.9 that there exists a lift  x of x and σ > 0 such that  x σ ∈ ( y τ0 , S −1 y τ1 )λ0 . Notice that the positive orbits of S n xσ + as n runs through Z divide J0 into disjoint regions (See Figure 9.5) because the semi-orbit O+ ( x ) has an irrational limit for  x ∈ π −1 (x) by part (a). Let U be the interior of the simple closed curve in D2 bounded by the positive orbits O+ ( x σ ) and SO+ ( x σ ); and the arcs [ x σ,  y τ0 ]λ0 , [ y τ0 ,  y τ1 ] y τ1 , S x σ ]Sλ0 , ϕ , [ and [a, Sa]S1 , where limt→∞  x t = a as shown in Figure 9.5. Clearly, SU ∩ U = φ because a is irrational. Observe that  + ⊃ .J SnU 0 n∈Z

and that S n U is a positively invariant subset of J0+ for all n ∈ Z. n y τ2i ,  y τ2i+1 ] Then  y t is in U for all t > τ1 , and S n [ ϕ is contained in S U for all + −1 −1 n ∈ Z and i ∈ Z . In particular,  y τ2i is in U and S  y τ2i+1 is in S U , but both  y τ2i and S −1 y τ2i+1 lie on  λ2i . It is now clear from the sequence of control curves + ⊂ J2i+ when i < j that there exists an increasing given by equation (9.3) and J2j sequence σ2i such that  x σ2i is the unique point satisfying  x σ2i ∈ ( y τ2i , S −1 y τ2i+1 )λ2i

.

for i ≥ 0 with σ0 = σ . It follows for n ∈ Z and i ∈ N that

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9 Recurrent Orbit Closures

S n x σ2i ∈ S n ( y τ2i , S −1 y τ2i+1 )S nλ2i .

.

As usual, σ2i → ∞ as i → ∞ because σ2i + 2α < σ2(i+1) where 2α is the length of the local section λ. Because the elements of  are h-isometries, ρ = sup{dh (z, w) : z, w ∈  λ2i }

.

does not depend on i and is finite because the lifts of λ are compact. Consequently, dh ( y τ2i ,  x σ2i ) ≤ ρ for all i and limi→∞  y τ2i = a by Corollary 6.1.30. The hypothesis y ∈ ω(x) implies that O+ (x) crosses (yτ0 , yτ1 )λ at a positive sequence of times sk  ∞ (Corollary 4.2.9) such that s1 > σ0 . Then there exists a sequence of lifts of x,  xk ∈ π −1 (x), and a sequence of covering transformations, Rk ∈ , such that  xk sk is in ( y τ0 , S −1 y τ1 )λ and Rk xk =  x . It follows that  x sk = Rk xk sk ∈ Rk ( y τ0 , S −1 y τ1 )λ ⊂ Rk J0 .

.

Since J0 is a control curve, Rk J0 is a control curve for all k ∈ Z+ and Rk J0 = R J0 when k = k  . (If Rk J0 = Rk  J0 for some k = k  , then the orbit of  x would cross the control curve Rk J0 at sk and sk , which is impossible.) In fact, k

J0+ ⊃ · · · ⊃ Rk J0+ ⊃ Rk+1 J0+ ⊃ . . . .

.

It follows from limi→∞  y τ2i = a that  y τ2i is in Rk J0+ for large i and limt→∞  yt = a by Proposition 7.3.19. Now Theorem 8.3.7 implies that limt→∞  y t = a is rational because it was assumed that z is an isolated ω−limit point of y and that contradicts the irrationality of a.

The proof of Maier’s second theorem is a good example of using geometric properties of the lifted flow to obtain dynamical results for a flow on a compact connected orientable surface. Theorem 9.2.4 (Maier) Let (X, R) be a flow on a compact surface X such that γ (X) ≥ 1. If x, y, and z are points in X such that y is in ω(x) and z is a moving point in ω(y), or equivalently z is a moving point such that z ∈ ω(y) ⊂ ω(x), then y is positively recurrent. Proof Since O(x)− is connected, it is in a compact connected component of X. So it can be assumed that X is a compact connected surface. By Proposition 2.2.5 and Theorem 2.2.6, it suffices to prove the theorem for compact connected orientable surfaces. If x or y is periodic, there is nothing to prove. If z is periodic and y is not periodic, then Theorem 4.2.10 implies that ω(x) = O(z) and O(y) = O(z). So y is periodic if and only if z is periodic, and we can assume that x, y, and z are not periodic in the rest of the proof. Having eliminated periodicity from the proof, Theorem 9.2.3, implies the following:

9.2 Maier’s Theorems

283

(a) The semi-orbit O+ ( x ) has an irrational limit for  x ∈ π −1 (x). + . (b) The semi-orbit O ( y ) is unbounded and ω( y ) ∩ B2 ⊂ F (c) z is not an isolated ω-limit point of y. We next show that O+ ( y ) limits to an irrational point of S1 . Let λ be a local cross section at z. Since z is not an isolated ω−limit point of y (and using properties of the flow box at a moving point), there exists a sequence of distinct moving points zn such that zn ∈ ω(y) ∩ λ and zn → z. Since each element of zn is in ω(y), there exist consecutive crossings yτ1 and yτ2 of λ such that (yτ1 , yτ2 )λ contains a point of zn , say z1 . Let J = [yτ1 , yτ2 ]ϕ ∪ [yτ1 , yτ2 ]λ . Since y is not periodic and y ∈ ω(x), Proposition 7.3.20 implies that J is not null-homotopic. For a lift  y of y, there exists a primitive T ∈  and a lift  λ such that J =



.

  T n [ y τ2 ] y τ2 , T  y τ1 ]T λ y τ1 ,  ϕ ∪ [

n∈Z

is a control curve. Since z1 ∈ ω(y), O+ ( y ) must cross Tm J for a sequence of distinct covering transformations Tm ∈  and an increasing sequence of crossing y ) goes to infinity. Theorem 8.3.7 times tm . Hence, by Proposition 7.3.19, O+ ( implies that O+ ( y ) limits to an irrational point, say b, because z ∈ ω(y) \ F is not an isolated ω−limit point of y. Since limt→∞  y t = b is irrational, Theorem 8.3.11 implies that all only moving points in ω(y) are one of two types: open recurrent ω-limit points at infinity or Cantor ω-limit points at infinity. In the first case, there exists an open interval (w1 , w2 )λ ⊂ ω(y), which implies that y is positively recurrent. For the remainder of the proof, we consider the second case. As before, let λ be a local cross section at z ∈ ω(y), and let D = O+ (y) ∩ λ. Since ω(y) ∩ λ is either a Cantor set or the empty set for every local section λ , it follows that D − is a Cantor set. Note that D ⊂ ω(x) ∩ λ and the hypotheses of Theorem 9.1.2 are satisfied. Choose σ < 0 such that yσ ∈ / λ. Let Un be a ball of radius n1 about yσ , and let λn be a local cross section at yσ such that λn ⊂ Un and λ ∩ λn = φ. By Theorem 9.1.2, there exists tn such that ytn = (yσ )(tn − σ ) ∈ Un . Note that tn − σ > |σ | > 0. It follows that there exists a sequence τn = tn − σ of positive real numbers bounded away from zero such that (yσ )τn → yσ . If τn contains a bounded subsequence, then yσ and y are periodic, contrary to our assumption that y is not periodic. Therefore, τn → ∞ and y is positively recurrent.

Corollary 9.2.5 If y is in a recurrent orbit closure O(x)− and z is in ω(y), then either z is fixed or y is positively recurrent. Corollary 9.2.6 Let O(x1 )− and O(x2 )− be distinct recurrent orbit closures. If y is in O(x1 )− ∩ O(x2 )− and z is in α(y) ∪ ω(y), then z is a fixed point. Proof If z is not fixed, y is positively or negatively recurrent, contradicting Theorem 9.2.1.



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9 Recurrent Orbit Closures

Corollary 9.2.7 If a recurrent orbit closure contains no fixed points, then it is a minimal set. Corollary 9.2.8 Let (X, R) be a flow on a compact connected orientable surface X such that γ (X) ≥ 1. If x, y, and z are points in X such that y is in ω(x) and z is a non-periodic point in ω(y), then both O+ ( x ) and O+ ( y ) have irrational limits for all lifts of x and y. Proof Use the theorem, Proposition 7.3.20, and Theorem 8.3.7.



The hypotheses of Theorem 9.2.3 and Corollary 9.2.8 are the same. The conclusions of Corollary 9.2.8, however, clearly imply the conclusions of Theorem 9.2.3. Thus if Corollary 9.2.8 is known to be true, then Theorem 9.2.4 is also true. Thus Corollary 9.2.8 can be thought of as an elegant lifted version of Maier’s xyz theorem for compact connected orientable surfaces. Theorem 9.2.4 can also be extended to compact bordered surfaces using Theorem 2.3.2.

9.3 Counting Recurrent Orbit Closures Determining how the maximal number of distinct recurrent orbit closures of a flow on a compact connected surface depends on the genus of the surface requires additional knowledge of classic surface topology. By restricting our attention to compact connected surfaces, we can use the universal covering spaces to prove the necessary results. The first step is to show that null-homotopic simple closed curves (path homotopic to the constant loop at the ends) bound regular Euclidean balls defined on page 35. Theorem 9.3.1 Let X be a compact connected surface. If .f : [0, 1] → X is a nullhomotopic simple closed curve, then there exists a regular Euclidean ball U in X such that .∂U − = f ([0, 1]).  → X be the universal covering of X, and let .f be a lift of f . Proof Let .π : X Because f is null-homotopic, .f(0) = f(1) and .f is a closed curve. It follows that  must also be simple because f is simple. .f Since the result holds for .S2 by Proposition 4.2.2, it can be assumed that the  → X is not genus of X is at least one. Hence, the universal covering .π : X a homeomorphism, and the covering group . is not the trivial group. Recall, however, that the universal covering space of .P2 is .S2 , the universal covering space of .T2 and the Klein bottle is .R2 , and the universal covering space of the rest of the compact connected surfaces is .B2 , which is homeomorphic to .R2 . Hence, Schoenflies theorem implies that the result holds for the universal covering spaces of all compact connected surfaces.

9.3 Counting Recurrent Orbit Closures

285

Only the universal covering of .P2 by .S2 is finite sheeted, making the proof easier. It will be left to the reader to adapt the more general proof to this simpler case. For  is either .R2 or .B2 . the rest of the proof, .X The embedded circle .J = f([0, 1]) has an interior .JI , which is a regular  containing Euclidean ball, and an exterior .JE . So there exists an open set .U  → E2r (0) for some .r > 1  ∪ JI = J− , and a surjective homeomorphism .h : U .J such that .h(JI ) = B2 and .h(J) = S1 . So .J∪ JI is homeomorphic to .D2 . ρ = h−1 (E2ρ (0)) for .1 < ρ < r, and note that .U ρ− is homeomorphic to Set .U 2 − and hence compact. Because .π is an open function, it suffices to show that .Eρ (0) ρ ) the required ρ , for some .ρ such that .1 < ρ < r, making .π(U .π is injective on .U Euclidean ball in X. It is easy to verify directly that .π is injective on .J. Because universal coverings  is equivalent are normal coverings, showing that .π is injective on any set E in .X to showing that .T E ∩ E = φ for all .T ∈  \ {ι}. Therefore, .T J ∩ J = φ for all .T ∈  \ {ι} and Proposition 4.2.3 can be used. If .T (J∪ JI ) ∩ (J∪ JI ) = φ for some T in . \ {ι}, then either .T (J∪ JI ) ⊂ J∪ JI or .T −1 (J ∪ JI ) ⊂ J ∪ JI . Since .J ∪ JI is homeomorphic to .D2 by Schoenflies theorem, the Brouwer fixed point theorem ([72] p. 194) implies that T or .T −1 has a fixed point in .J ∪ JI , which is impossible because T is in . \ {ι}. Therefore, ∪ JI ) ∩ (J∪ JI ) = φ for all .T ∈  \ {ι} or equivalently .{T ∈  : T (J∪ JI ) ∩ .T (J (J ∩ JI ) = φ} = {ι}. Now Proposition 8.1.12 implies there exists .ρ, .r > ρ > 1 such that {γ ∈  : Uρ ∩ γ Uρ = φ} = {γ ∈  : C ∩ γ C = φ} = {ι},

.

ρ . and hence, .π is injective on .U



Corollary 9.3.2 A simple closed curve .f : [0, 1] → X on a compact connected surface X is null-homotopic if and only if the circle representative .f o extends to a homeomorphism mapping .D2 into X. Proof Recall that .f : [0, 1] → X is null-homotopic if and only if the circle representative .f o extends to a continuous function on .D2 (see page 124).

Corollary 9.3.3 If .f : [0, 1] → X is a null-homotopic simple closed curve on a compact connected surface X (with or without boundary) with embedded circle .J = f ([0, 1]), then the following hold: (a) .X \ J is the disjoint union of two open sets .U1 and .U2 such that .U1 is homeomorphic to .B2 . (b) .Ui− = Ui ∪ J for .i = 1, 2. (c) .U1− is homeomorphic to .D2 . (d) .U2− is a compact connected bordered surface. Proof Use the theorem and Proposition 2.1.11.

286

9 Recurrent Orbit Closures

Given an embedded circle J in a surface X, an open set W containing J is a collar neighborhood of J provided there exists a homeomorphism of .{z ∈ C : 1/r < |z| < r} onto W mapping .S1 onto J . Schoenflies theorem implies that every embedded circle in .C and .S2 has a collar neighborhood. It follows from Theorem 9.3.1 that every null-homotopic embedded circle has a collar neighborhood. Proposition 9.3.4 Let X be a compact connected orientable surface, and let D be a compact subset of X. If .f : [0, 1] → X is a simple closed curve with embedded circle .J = f ([0, 1]) such that f is not null-homotopic and .J ∩ D = φ, then there exists a collar neighborhood W of J such that .W ∩ D = φ. Proof As usual, there are two cases: .γ (X) = 1 and .γ (X) ≥ 2. The same method works in both cases using different geometries in the universal covering spaces. Assume .γ (X) ≥ 2, leaving the easier Euclidean geometry to the reader. Using the usual universal covering space structure, let .fbe a lift of f and let T be the covering transformation such that .f(1) = T f(0). Then .fu (s) = T [s] f(s − [s]) is the universal lift determined by .f and .J = fu (R). Recall that because f is a simple closed curve, .fu is injective, T is primitive, and .S J = J or .S J ∩ J = φ depending on whether .S ∈ [T ] or .S ∈ / [T ] with .[T ] denoting the cyclic group generated by T . In this case, it is also easy to show that .fu is a homeomorphism of  and .π |J is a universal covering of the embedded circle J . .R onto .J Let .LT be the axis of T . There exist equidistant curves .E1 and .E2 on opposite , and let .L1 be sides of .LT such that .J lies between them. Let  .a be lift of .f (0) in .J hyperbolic line through  .a perpendicular to .LT , and set .L2 = T L1 . Then .E1 , .E2 , 2 .L1 , and .L2 bound a compact set C in .B .  Let .Y be the closed set consisting of .E1 and .E2 and the region between them. = Y , and the continuous action of .[T ] is both proper and free. Let .πT Then .T Y  onto .Y /[T ]. It is easily seen that C is a fundamental be the natural projection of .Y /[T ] is a compact connected bordered region for .πT . It can then be shown that .Y surface with two boundary components and zero Euler characteristic. It follows from /[T ] is homeomorphic to a sphere with two holes. Furthermore, equation (2.4) that .Y ) is an embedded circle parameterized by the .πT is a universal covering, and .πT (J   simple closed curve .f (t) = πT f (t) with .0 ≤ t ≤ 1. Since the compact annular region of A = {z ∈ C : 1/2 ≤ |z| ≤ 2}

.

 → A is is one representation of a sphere with two holes, we can assume that .πT : Y  a universal covering of A with covering group .[T ]. Clearly, .πT (J ) is in the interior of .{z ∈ C : |z| = 2}. Since .fis also the lift of .f  and .πT is a universal covering, the path class of .f  generates the fundamental group .1 (A, f  (0)). Since .f  is nullhomotopic in .C, Proposition 4.2.3 and Corollary 9.3.2 can be used to show that ). .{z ∈ C : |z| < 1/2} is contained in the interior of .πT (J Schoenflies theorem implies there exists a homeomorphism g of .C onto itself such that .g(S1 ) = πT (J). Since .J is between .E1 and .E2 and .T J = J, it is readily verified that there exists .r > 1 such that .g({z ∈ C : 1/r ≤ |z| ≤ r}) ⊂ A and a

9.3 Counting Recurrent Orbit Closures

287

collar neighborhood of .πT (J) in A. Note that .k/(k − 1 + r) < 1 < (k − 1 + r)/k for all .k ∈ Z+ , and both sequences converge to 1 as k goes to infinity. Moreover, each   Vk = g {z ∈ C : k/(k − 1 + r) < |z| < (k − 1 + r)/k}

.

is a collar neighborhood of .πT (J) such that .Vk− ⊂ Vk−1 for .k > 1. The next step is to show that .h(z) = π(πT−1 (z)) defines a homeomorphism of −1 n .Vk onto an open neighborhood .Wk of J for large k. Because .π T (z) = {T w : o  , h is well defined on .Vk for all k. The function h is open n ∈ Z} for some w in .Y because .πT is continuous and .π is open. And h is continuous because .πT and .π are coverings. So the issue is whether or not .h(z) = π(πT−1 (z)) is injective on .Vk for large k. It is clearly injective on .πT (J) and J =

.

  πT−1 πT (J)

 =

πT−1



k∈Z

Vk

=



πT−1 (Vk ).

k∈Z

Note also that .T πT−1 (Vk ) = πT−1 (Vk ). If .h(z) = h(z ) and .z = z , then .πT−1 (z) ∩ πT−1 (z ) = φ. Consequently, .h(z) = h(z ) with .z = z implies the existence of −1 −1  .S ∈  such that .Sπ T (z) ∩ πT (z ) = φ and S is not in .[T ]. If h is not injective on .Vk for large k, then, because .Vk ⊂ Vk−1 , it can be assumed that for all k there exist .wk ∈ πT−1 (Vk ) and .Sk ∈  \ [T ] such that .Sk wk ∈ πT−1 (Vk ). Replacing .wk by .T m wk and .Sk by .T n Sk T −m for suitable m and n, it can be assumed that .wk and .Sk wk are in C. By passing to a subsequence, we can assume that .wk and   . .Sk wk converge to w and .w in C. Obviously, w and .w are in .J + If there exists .S ∈  such that .{k ∈ Z : Sk = S} is infinite, then .w  = Sw and n for some .n ∈ Z, contrary to the construction of .S . Passing to subsequence .S = T k one more time, it can be assumed that .Sk = Sj when .k = j . Thus there do not exist open neighborhoods U and .U  of w and .w  such that .{S ∈  : U  ∩ SU = φ} is finite, which contradicts the proper action of . on .B2 . Therefore, there exists K such that h is injective on .Vk for .k ≥ K, and .h(Vk ) = π(πT−1 (Vk )) is a collar neighborhood of J for .k ≥ K. It follows that = π .J = π J πT−1 (Vk ) = π(πT−1 (Vk )) = h(Vk ). k≥K

k≥K

Hence, .h(Vk ) ∩ D = φ for large k because .J ∩ D = φ.

k≥K



Proposition 9.3.5 Let X be a compact connected orientable surface X. If J is an embedded circle of X, then .X \ J has one or two components. Furthermore, if .X \ J has exactly two components U and V , then .U − = U ∪ J and .V − = V ∪ J , that is, J is the common boundary of both U and V .

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9 Recurrent Orbit Closures

Proof By Proposition 9.3.4, there exists an open set W containing J and a homeomorphism h of .{z ∈ C : 1/r < |z| < r} onto W for some .r > 1 such that .h(S1 ) = J . Set .W1 = h({z ∈ C : 1/r < |z| < 1}) and .W2 = h({z ∈ C : 1 < |z| < r}). Since the components of a manifold are open sets of the manifold, the components of .X \ J are open in .X \ J and hence in X because .X \ J is open in X. There exist components .U1 and .U2 containing the connected sets .W1 and .W2 , respectively, with the possibility that .U1 = U2 . Suppose there exist additional components of .X \ J . Then .U1 ∪ U2 ∪ W is the complement of the union of these other components of .X \ J , and thus .U1 ∪ U2 ∪ W is a closed subset of X. But .U1 ∪ U2 ∪ W is certainly an open set in X and so must equal X because X is connected, contradicting the existence of additional components of .X\J . Therefore, .X \ J has one or two components. Suppose U and V are the two distinct components of .X \ J . Then U and V are open sets of .X\J and X and closed sets of .X\J . It follows that both .U − ⊂ U ∩J = X \ V and .V − ⊂ V ∩ J = X \ U . By connectedness, .Wi is contained in either U or V . If .W1 and .W2 are both in U , then as above .U ∪ J is both open and closed and .V = φ. Therefore, either .W1 ⊂ U and .W2 ⊂ V or vice versa. Obviously, J is contained in the closure of both .W1 and .W2 and hence in the closure of both U and

V. A compact connected surface can be cut along a simple closed null-homotopic curve to obtain two compact connected bordered surfaces (Corollary 9.3.3). Using Theorem 9.3.4, we can also cut along any simple closed curve on a compact connected orientable surface and obtain a compact bordered surface with either one or two components. This result was the primary topological theorem that Maier used to prove Theorems 9.2.1 and 9.2.4. Since we will only use this topological result in this section, more of the details will be left to the reader. Theorem 9.3.6 Let X be a compact connected orientable surface. If .f : [0, 1] → X is a simple closed curve with embedded circle .J = f ([0, 1]) such that f is not null-homotopic, then there exists a compact bordered surface Y with two boundary components .J1 and .J2 and a continuous surjective function .g : Y → X with the following properties: (a) The functions .g|Ji for .i = 1, 2 are homeomorphisms of .Ji onto J for .i = 1, 2. (b) The function .g|(Y \ ∂Y ) is a homeomorphism onto .X \ J . (c) Either Y is connected and .∂Y has two components or Y has two components each having one boundary component. (d) .χ (Y ) = χ (X). (e) If Y is connected, then .γ (Y ) + 1 = γ (X). (f) If Y has two components .Y1 and .Y2 , then .γ (Y1 ) + γ (Y2 ) = γ (X). Proof As in the proof of Proposition 9.3.5, W is a collar neighborhood of J and h is a homeomorphism of .{z ∈ C : 1/r < |z| < r} onto W for some .r > 1 such that 1 .h(S ) = J . Again let .W1 = h({z ∈ C : 1/r < |z| < 1}) and .W2 = h({z ∈ C : 1 < |z| < r}).

9.3 Counting Recurrent Orbit Closures

289

Slightly extending our use of .ι, let .ι0 : X \ J → X, .ι1 : W \ W1 → X, and ι : W \ W2 → X be the inclusion functions of .X \ J , .W \ W1 , and .W \ W2 , respectively, into X. Consider

. 2

Yˆ = (X \ J )  (W \ W1 )  (W \ W2 )

.

with the coherent topology and the continuous function .G : Yˆ → X defined by

G(x) =

.

⎧ ⎪ ⎪ ⎨ι0 (x) ι1 (x)

⎪ ⎪ ⎩ι (x) 2

for x ∈ X \ J for x ∈ W \ W1 for x ∈ W \ W2 .

Define an equivalence relation .∼ on .Yˆ by .x ∼ y if and only if: (a) .x = y. (b) .x =  y and .G(x) = G(y) ∈ / J. Set .Y = Yˆ / ∼ with the quotient topology, and let .π be the natural projection of .Xˆ onto the quotient space Y . Then G passes to the quotient, and there exists a continuous function .g : Y → X such that .G = g ◦ π . Let .Jˆ1 and .Jˆ2 be the subsets of .Yˆ defined by .Jˆ1 = J ∩ (W \ W2 ) and .Jˆ2 = J ∩ (W \ W1 ). Then .Jˆ1 and .Jˆ2 are saturated closed subsets of .Yˆ such that .G|Jˆi is a homeomorphism of .Jˆi onto J . Hence, .Ji = π(Jˆi ), .i = 1, 2 are disjoint sets of Y and .g|Ji is a homeomorphism of .Ji onto J , proving part (a). Observe that .Y = π(X \ J ) ∪ J1 ∪ J2 and that .π is an open function. It follows that .θ = π |X \ J is a homeomorphism of .X \ J onto .Y \ (J1 ∪ J2 ). It is an exercise to show that Y is Hausdorff by checking the different cases. Also by using the homeomorphism h of .{z ∈ C : 1/r < |z| < r} onto W , one shows that Y is the union of three compact connected sets .C0 , C1 , and .C2 such that .C0 ⊂ π(X \ J ), .Ci ⊃ Ji , and .C0 ∩ Ci = φ for .i = 1, 2. It follows that Y is a compact connected orientable bordered surface with two boundary components .J1 and .J2 . Part (b) follows from .g ◦ θ = g ◦ π |X \ J = G(x) = ι0 (x) = x for .x ∈ X \ J . For part (c), note that .π(X \ J ) is dense in Y . Therefore, Y has at most two components by Proposition 9.3.5. Suppose Y has two components .Y1 and .Y2 and .Y2 has no boundary. Thus .Y2 must be an open subset of .Y \ (J1 ∪ J2 ), making .g(Y2 ) an open subset of .X \ J and hence an open set of X. And .Y2 must be a closed and hence compact subset of Y . Putting the pieces together, .g(Y2 ) is an open compact and hence closed subset of X contained in .X \J , a proper subset of X. This is impossible because X is connected. Therefore, each component must contain at least one of the two boundary components. For part (d), recall that the bordered surface Y has a triangulation such that every edge is a face of exactly 1 or 2 triangles. (See page 52.) By adding a finite number of vertices to .J1 and .J2 and 1 edge for each added vertex, it can be assumed that −1 (g(v)) ∩ J is a vertex of .J . Now g maps .v ∈ J1 is a vertex of .J1 if and only if .g 2 2

290

9 Recurrent Orbit Closures

the triangulations of .J1 and .J2 onto the same triangulation of J , and g maps the triangulation on Y onto a triangulation of X. Since the triangulation of a simple closed curve always has the same number of edges and vertices, the triangulations of .J1 , .J2 and J can be ignored in the calculation of .χ (Y ) and .χ (X). Therefore, .χ (Y ) = χ (X). For the last 2 parts, it follows from equations (2.2) and (2.4) that χ (X) = 2 − 2γ (X),

.

and  χ (Y ) =

.

2 − 2γ (Y ) − 2

when Y is connected

2 − 2γ (Y1 ) − 1 + 2 − 2γ (Y2 ) − 1 when Y is not connected.

Substitute .χ (Y ) into .γ (X) = (2 − χ (X))/2 = (2 − χ (Y ))/2 and simplify. )−



)−

The orbit closures .O(x1 and .O(x2 of two recurrent points .x1 and .x2 that are not periodic of a flow .(X, R) on a compact surface X are said to be distinct provided that .O(x1 )− = O(x2 )− . This does not preclude the possibility that − − = φ. .O(x1 ) ∩ O(x2 ) Theorem 3.3.7 shows that there exists a flow on a compact connected orientable surface X with exactly .γ (X) distinct recurrent orbit closures. The next theorem shows .γ (X) is the maximum possible distinct recurrent orbit closures. Theorem 9.3.7 Given a flow .(X, R) on a compact connected orientable surface with or without boundary, the number of distinct recurrent orbit closures is at most .γ (X). Proof By Theorems 2.3.2 and 4.3.6, we can assume without loss of generality that X is not bordered. The proof then proceeds by induction on the genus .γ (X). The .γ (X) = 1 case is just Theorem 5.3.4. Assume the theorem holds when .γ (X) ≤ n, and let .(X, R) be a flow such that .γ (X) = n + 1. Suppose .O(x1 )− , . . . , O(xn+2 )− are distinct recurrent orbit closures of .(X, R). Then there exists an open neighborhood W of .x1 such that .W ∩ O(xj )− = φ for .j = 2, . . . , n + 2 by Theorem 9.2.1. Let .λ be a local section of length .2α at .x1 such that .λ ⊂ W . There exists .τ > 0 such that .x1 and .x1 τ are consecutive crossings of .λ. As usual .J = [x1 , x1 τ ]ϕ ∪ [x1 τ, x1 ]λ is an embedded circle. Suppose .X \ J is not connected. Then .X \ J has two components U and V such that .U − = U ∪ J and .V − = V ∪ J by Proposition 9.3.5. Setting .P = {qt : 0 < t < α and q ∈ (x1 , x1 τ )λ } and .N = {qt : −α < t < 0 and q ∈ (x1 , x1 τ )λ }, the proof of Proposition 4.3.1 can be reused to show that P and N are in different components of .X \ J . Similarly, the proofs of Corollary 4.3.2 and Proposition 4.3.4 can then be repeated to show that the component containing P is positively invariant and contains .O+ (x1 (τ + α)). This is impossible because .x1 (−α) is in .O+ (x1 (τ + α))− and is in the component that is not positively invariant. Therefore, .X\J is connected.

9.3 Counting Recurrent Orbit Closures

291

Now apply Theorem 1.3.3 to the flow .(X, R) with .E = J , observing that O(xj )− for .j = 2, . . . , n + 2 remain distinct recurrent orbit closures because − = φ for .j = 2, . . . , n + 2. Since J cannot be null-homotopic by .J ∩ O(xj ) part (b) of Theorem 7.3.21, Theorem 9.3.6 applies to produce a bordered compact surface Y . Because J consists entirely of fixed points, .X \ J is an invariant set for the modified flow on X. Using parts (a) and (b) of Theorem 9.3.6, we can define flows on .Y \ ∂Y , .J1 , and .J2 . A routine argument shows that together these flows define a flow on Y with .n + 1 distinct recurrent orbit closures. Since it was shown that Y is connected, .γ (Y ) = γ (X) − 1 = n by part (e) of Theorem 9.3.6. Therefore, the flow on Y can have at most n distinct recurrent orbit closures by the induction assumption. This contradiction completes the proof.

.

The group .c =  ∩ G was introduced and discussed in Section 7.1. Obviously, c acts freely and properly on .B2 because . acts freely and properly on .B2 . Thus .c is a Fuchsian covering group. The classification of coverings theorem ([42], Theorem 12.19) shows that each isomorphism class of the coverings of X corresponds to a unique conjugacy class of a subgroup of the universal covering group . of X and conversely. Specifically, if H is a subgroup of ., then the natural projection .π  : B2 → B2 /H is a covering and .π passes to the quotient .π  : B2 /H → B2 / , which is also a covering. (This is the way the classification theorem is proved in [42].) Note that .B2 /H is always connected. The index of H in . is the cardinality of the number of sheets of the covering .π  : B2 /H → B2 / . If H is a normal subgroup of ., then H is the only group in its conjugacy class and .π  : B2 → B2 /H is a normal covering. .

Proposition 9.3.8 Let . be a subgroup of .G  that acts freely and properly on .B2 . If 2  2 2 .B /  is a compact nonorientable surface, then .π : B / c → B /  is a compact 2 connected orientable 2-sheeted normal covering of .B / . Proof It follows from the proof of the classification of coverings theorem that .π  : B2 / c → B2 /  is a covering. It is 2-sheeted because the index of .c in . is 2, and it is orientable because .c is a Fuchsian group. Connectivity is obvious. A finite sheeted covering of a compact manifold is always compact. Since all subgroups of index 2 are normal, .π  is a normal covering.

Proposition 9.3.9 Let X be a compact connected nonorientable surface. If .π :  → X is 2-sheeted compact connected orientable covering of X, then .γ (X)  = X γ (X) − 1. Proof The key idea is to use the fact that a compact surface is a polyhedron and thus  = 2χ (X). homeomorphic to finite Euclidean simplicial complex to show that .χ (X) So without loss of generality, X will be a finite Euclidean simplicial complex such that each edge lies in exactly 2 simplices. The first step is to construct triangulations  = 2χ (X).  such that .χ (X) of X and .X There exists a finite cover .U of X by open Euclidean balls of X that are evenly covered by .π . Let .δ be the Lebesgue number of .U (see Lemma 4.21 in [42]). By applying barycentric subdivision and Lemma 5.18 in [42] to the finite Euclidean

292

9 Recurrent Orbit Closures

simplicial complex X a finite number of times, we obtain a new triangularization of X such that every 2-simplex with its edges and vertices has diameter less than .δ and is therefore contained in an element of .U by the Lebesgue lemma. This process does not change .χ (X) ([42], Theorem 5.19).  let   if and only if To construct a triangulation of .X, .s be a simplex of .X −1  .s is contained in a component of .π (U ) for some .U ∈ U and .π maps  .s homeomorphically onto a simplex s of X. The construction of the triangulation of X guarantees that .π −1 (s) of every simplex s in the triangulation consists of precisely  that do not intersect. It follows that if two simplices of .X  do not 2 simplices in .X  is in a meet properly, then the same problem occurs in X. Clearly, every point of .X    simplex of .X. Therefore, there exists a triangulation of .X satisfying .χ (X) = 2χ (X). Now we can use equations (2.2) and (2.3) to see that .χ (X) = 2 − γ (X) and  = 2 − 2γ (X).  Therefore, .χ (X)  2(2 − γ (X)) = 2 − 2γ (X)

.

 = γ (X) − 1. and .γ (X)



Using the classification of compact connected orientable surfaces, it follows from Proposition 9.3.9 that .B2 / c is homeomorphic to the orientable 2-sheeted covering of X constructed by Proposition 2.3.4 when .B2 /  = X. The companion to Theorem 9.3.7 for nonorientable surfaces appears in [47] and [53]. The proof given here follows from Theorems 9.3.7 and 2.2.6. Theorem 9.3.10 Given a flow .(X, R) on a compact connected nonorientable surface with or without boundary, the number of distinct recurrent orbit closures is at most .[(γ (X) − 1)/2]. Proof As usual we can assume that X has no boundary. If .γ (X) ≤ 2, then .[(γ (X)− 1)/2] = 0, which agrees with Corollary 4.3.7 and Theorem 5.3.16. For the rest of the proof, we can assume that .γ (X) ≥ 3. Let m be the number of distinct recurrent orbit closures of the flow .(X, R), and let .{x1 , . . . , xm } be the set of recurrent points of .(X, R) consisting of exactly one recurrent point from each of the m distinct recurrent orbit closures of .(X, R).  → X be a compact orientable 2-sheeted cover of X, it follows Letting .π : X from Theorem 2.2.6 and Proposition 2.2.5 that .π −1 (xi ) = { xi1 ,  xi2 } is a set of recurrent points such that   π O(xij )− = O(xi )− .

.

 R). We cannot, however, assert That is, there is a set of 2m recurrent points of .(X, − − − that either .O(xi1 ) = O(xi2 ) or .O(xi1 ) = O(xi2 )− .

9.3 Counting Recurrent Orbit Closures

293

 R) contains at most 2m distinct recurrent orbit closures, and since Therefore, .(X,  by Theorem 9.3.7. By Proposition 9.3.9,  is orientable, it follows that .2m ≤ γ (X) X  = γ (X) − 1. Finally, .2m ≤ γ (X)  = γ (X) − 1 reduces to .γ (X) .

m ≤ [(γ (X) − 1)/2]

.

because .γ (X) is an integer.



It follows from Theorem 3.3.7 that Theorem 9.3.10 is also sharp in the sense that given .k ∈ Z+ there exists a flow .(X, R) on a compact connected nonorientable surface such that .γ (X) = k and such that .(X, R) has precisely .[(γ (X) − 1)/2] distinct recurrent orbit closures.

Chapter 10

Existence of Transitive Flows

The existence of minimal flows on the torus was already established in Section 1.1, and then applying Beck’s theorem (Theorem 1.3.3) to a minimal flow on the torus in various ways produces a wide variety of non-minimal transitive flows on the torus. (One such example follows the proof of Beck’s theorem.) Constructing a transitive flow on a surface of genus greater than 1 is not so easy because there are no minimal flows on such a surface. This chapter is devoted to constructing a large family of transitive flows on compact connected orientable surfaces of genus greater than 1. The purpose of the first section is to construct a family of surfaces with a natural orbit structure. Specifically, we will use the theory of simplicial complexes to construct a family of bordered compact connected orientable surfaces with an internal rectangular structure of the form .[a, b] × [0, 1]. The vertical line segments of these rectangles will eventually be flow lines that will be used to construct flows in Section 10.2. These surfaces will contain a dense copy of a suspension flow that can be extended to the whole surface with a time change by modifying the proof of Beck’s theorem. The first two sections are used to construct the necessary flows without discussing their dynamical properties. Specifically, in the first section, a class of compact connected orientable bordered surfaces are constructed simultaneously with the orbits of the flows without constructing the flows. Then, in the second section, the flows with the prescribed orbits are defined using a modified version of the proof of Beck’s theorem. Section 10.3 begins with the intimate connection between a surface being a recurrent orbit closure of a flow on it and the flow being transitive. After proving that the flows constructed in Section 10.2 are transitive flows, they are used to construct transitive flows on compact connected orientable surfaces with a finite number of fixed points. In particular, they are recurrent orbit closures that satisfy the Anosov dichotomy.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5_10

295

296

10 Existence of Transitive Flows

The final section uses the ideas of the first two sections to construct a large class of flows with interesting nontrivial sets isomorphic to the suspension of a locally circular minimal set.

10.1 Constructing Rectangular Surfaces Recall that the function .ξ(s) = e2π is from p. 123 maps the closed interval .[0, 1] onto .S1 with only 0 and 1 identified and that the usual topology on .S1 is also the quotient topology of .[0, 1] determined by .ξ . Consequently, for a real number a such that .0 < a < 1, the function .h : [0, 1] → [0, 1] defined by .h(s) = s + a mod 1 is isomorphic to the rotation .h (z) = e2π ia z because .ξ ◦ h = h ◦ ξ . It will be more convenient to use h than .h  to construct transitive flows because it will allow us to work easily with irrational rotations and simplicial complexes simultaneously. The reader, however, needs to keep in mind that .h(s) = s + a mod 1 means that .[0, 1] has the quotient topology with 0 and 1 identified. Finite simplicial complexes will be used to construct the necessary compact connected bordered surfaces. The use of simplicial complexes will be based on the development of the subject in Chapter 5 of [42]. To facilitate the reader’s use of [42] as a reference, we will use the same notation as much as possible. Assuming a is irrational, let .c1 = 0 < c2 < · · · < cr < 1 be real numbers such that .ci + ma + n = cj for all .m, n ∈ Z when .i = j , or in other words, .ci + ma is not congruent to .cj mod 1 for all m when .i = j . In still other words, the orbits of .c1 , . . . , cr for the irrational rotation .h(s) = s + a mod 1 are all distinct. For example, choosing .ci to be rational numbers works because a is irrational. Then set .I1 = [c1 , c2 ] = [0, c2 ], I2 = [c2 , c3 ], . . . , Ir = [cr , 1]. The points .c1 , . . . , cr are consecutive mod 1 in the sense that .c1 precedes .c2 and follows .cr . Consecutive mod 1 will be the important ordering throughout the section because it is the same as the counterclockwise ordering in .S1 . In particular, the sets .{h−1 (c1 ), . . . , h−1 (cr )} and .{h(c1 ), . . . , h(cr )} have the same consecutive order mod 1 as .{c1 , , . . . , cr } but not necessarily the same linear order in .[0, 1]. In other words, starting at .h(ci ) and moving to the right, the next element of .{h(c1 ), . . . , h(cr )} encountered is .h(ci+1 ) with the understanding that when no element of .{h(c1 ), . . . , h(cr )} is encountered before reaching 1, the search continues by going back to 0 and then moving to the right again. The sets E = {c1 , , . . . , cr } ∪ {h−1 (c1 ), . . . , h−1 (cr )} ⊂ [0, 1)

.

and E  = {c1 , , . . . , cr } ∪ {h(c1 ), . . . , h(cr )} ⊂ [0, 1)

.

will play a key role in the construction of a surface and a flow on it.

10.1 Constructing Rectangular Surfaces

c1

c2

0

1 4

I1

h(c3 ) c2

c1

h−1 (c1 )

I2

297

h−1 (c2 )

c3 1 2

I3

h−1 (c3 )

c4 2 3

h(c4 )

I4

h(c1 ) c3

h−1 (c4 )

1

h(c2 )

c4 1

Fig. 10.1 An example of E and .E  with .r = 4 and .a =

√ 2/2

The points of E and the intervals between pairs of consecutive mod 1 points of E are the vertices and edges of a triangulation of .[0, 1] mod 1. The same holds for .E  . Note as well that .c1 = 0 ≡ 1 (mod 1) is a vertex of both these triangulations and that both the triangulations have .2 r vertices and .2 r edges. For a specific example of E and .E  , see Figure 10.1. Proposition 10.1.1 The function .h(s) = s+a mod 1 has the following properties: (a) The function h maps E bijectively to .E  . (b) If b and d are consecutive points mod 1 of E, then .h(b) and .h(d) are consecutive points mod 1 of .E  . (c) If b and d are consecutive points mod 1 of E, then h maps the closed interval .[b, d] mod 1 homeomorphically to the closed interval .[h(b), h(d)] mod 1 (which can be in 2 pieces.) and the lengths of the 2 intervals are equal. (d) The function h provides a bijective correspondence between the .2 r closed intervals whose endpoints are consecutive points mod 1 of E with the .2 r closed intervals whose endpoints are consecutive points mod 1 of .E  . Proof Exercise.



Corollary 10.1.2 The function h maps the triangulation of .[0, 1] mod 1 determined by E onto the triangulation of .[0, 1] mod 1 determined by .E  . Corollary 10.1.3 A closed interval whose endpoints are consecutive points mod 1 of E {.E  } is one of the following four types: (i) (ii) (iii) (iv)

[ci , ci+1 ] = Ii . [ci , h−1 (ck )] {[ci , h(ck )]}. −1 (c ), c .[h k i+1 ] {[h(ck ), ci+1 ]}. −1 (c ), h−1 (c .[h k k+1 )] {[h(ck ), h(ck+1 )]}. . .

Each of these intervals is contained in .Ii for a unique i, and .h(x) = x + a or h(x) = x + a − 1 on each of these intervals.

.

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10 Existence of Transitive Flows

These 4 types of intervals provide information about the correspondence from part (d) of Proposition 10.1.1. Specifically, h maps type (i) intervals of E to type (iv) of .E  and type (iv) intervals of E to type (i) of .E  . Similarly, h maps type (ii) intervals of E to type (iii) of .E  and type (iii) to type (ii). Obviously, .h−1 exhibits the same behavior in the other direction. Setting .η(k) = i if and only if .ci < h−1 (ck ) < ci+1 defines a function η : {1, . . . , r} → {1, . . . , r}.

.

Likewise, set .η(k) = i if and only if .ci < h(ck ) < ci+1 . It is easily seen that .η −1 (i) {.η−1 (i)} is either empty or a finite subset of consecutive integers mod r. If .η(k) = i, then .h−1 (ck ) is the right endpoint of a type (ii) or (iv) and the left endpoint of a type (iii) or (iv). There is a corresponding statement for .η(k) = i. Next we want disjoint copies of the edge .{ci } × [0, 1]. This can be easily accomplished using a third coordinate. Specifically, set .Ri = Ii × [0, 1] × {i}. We will use the third coordinate to indicate the rectangle to which the point belongs. For example, in this way, we can distinguish the edges .(c2 , t, 1) from .(c2 , t, 2). Note that −1 (c ), t, η(k)) is a point of .R .(h k η(k) when .0 ≤ t ≤ 1. A Euclidean simplicial complex is both a simplicial complex in .Rn and a polyhedron in .Rn . Hence, the identity map is a triangulation of the polyhedron (see the section Euclidean Simplicial Complexes beginning on p. 92 of [42]). In this context, we will use the terms edge and triangle instead of the terms 1-simplex or 2-simplex. Edges and triangles will be denoted by .v0 , v1  and .v0 , v1 , v2 , respectively. The first step is to give .RS = R1 ∪ · · · ∪ Rr the structure of a Euclidean simplicial complex. Since .RS consists of r disjoint rectangles in .R3 , this could be done most easily by triangulating all of them the same way, but this simple approach will not work for our purposes. To triangulate .Ri , we start with vertices at the corners. Then add the vertices (ci , 1/3, i), (ci , 2/3, i)

.

and (ci+1 , 1/3, i), (ci+1 , 2/3, i)

.

on the left and right sides of .Ri , respectively. Next add 2 internal vertices  .

ci + ci+1 1 , ,i 2 3



 and

 ci + ci+1 2 , ,i . 2 3

The above 6 vertices will be used to build a buffer between the top and bottom vertices so that when a top edge is identified with a bottom edge there will not be 2 edges with the same name. They will be referred to as buffer vertices.

10.1 Constructing Rectangular Surfaces Fig. 10.2 Basic triangulation of .Ri

299

(ci , 1, i)

(ci+1 , 1, i)

(ci , 23 , i)

(ci+1 , 23 , i)

(ci , 13 , i)

(ci+1 , 13 , i)

(ci , 0, i)

(ci+1 , 0, i)

Additional vertices on the top (second coordinate equals 1) and bottom (second coordinate equals 0) edges of .RS will be needed, but first the basic triangulation with the top and bottom of .Ri as edges will be completed. The 2 left corners of .Ri and the 2 buffer vertices .(ci , 1/3, i) and .(ci , 2/3, i) naturally divide the left edge into 3 edges. Similarly, there are 3 edges on the right side of .Ri . Next add edges from  ci +ci+1 1 , 3 , i to the other 5 buffer vertices. Then complete the buffer by the vertex . 2   adding edges from . ci +c2 i+1 , 23 , i to the vertices .(ci , 2/3, i) and .(ci+1 , 2/3, i). The buffer is now a sub-complex made up of 4 small triangles, 9 edges, and 6 vertices. To complete the basic triangulation, add edges from the top corners .(ci , 1, i) and  ci +ci+1 2 .(ci+1 , 1, i) to . , 3 , i , adding 2 edges and creating 3 triangles and similarly 2 at the bottom. This completes a basic triangulation of every .Ri consisting of 10 vertices, 19 edges, and 10 triangles as shown in Figure 10.2. Thus .χ (Ri ) = 1 and .χ (RS ) = r as expected. Only the triangles  .

  ci + ci+1 2 (ci , 1, i), , , i , (ci+1 , 1, i) 2 3 

and 

 (ci , 0, i),

.

  ci + ci+1 1 , , i , (ci+1 , 0, i) 2 3

will be subdivided by adding vertices to the top and bottom edges of .Ri . The last step in triangulating .RS is to add vertices at the 2r points (h−1 (ck ), 1, η(k)) and (h(ck ), 0, η(k)).

.

Notice that these modifications are governed by the functions .η(k) and .η(k) and thus depend on the choice of a and .c1 , . . . , cr . Each of these changes increases

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10 Existence of Transitive Flows

Fig. 10.3 One possible triangulation of an .Ri

(ci , 1, i)

(ci+1 , 1, i)

(ci , 23 , i)

(ci+1 , 23 , i)

(ci , 13 , i)

(ci+1 , 13 , i)

(ci , 0, i)

(ci+1 , 0, i)

the vertices by 1, the edges by 2, and the triangles by 1. For future reference, the triangulation of .RS contains 12r vertices, 23r edges, and 12r triangles (Figure 10.3). Let q be a positive integer, and set .R = RS × {1, . . . , q}. Letting .Rij = Ri × {j }, R=

q  r 

.

Rij .

j =1 i=1

Each .Rij is a subset of .R4 and is given the obvious triangulation from .Ri . Note that an edge cannot join 2 points that are not in the same .Rij . The edge structure of R has been set up so that it links back to Corollary 10.1.3. In particular, the following hold: (i) .(ci , 1, i, j ), (ci+1 , 1, i, j ) is an edge of R if and only if .[ci , ci+1 ] is a consecutive interval mod 1 of E. (ii) .(ci , 1, i, j ), (h−1 (ck ), 1, η(k), j ) is an edge of R if and only if .[ci , h−1 (ck )] is a consecutive interval mod 1 of E. (In this case, .i = η(k).) (iii) .(h−1 (ck ), 1, η(k), j ), (ci+1 , 1, i, j ) is an edge of R if and only if −1 (c ), c .[h k i+1 ] is a consecutive interval mod 1 of E. (In this case, .i = η(k).) (iv) .(h−1 (ck ), 1, η(k), j ), (h−1 (ck+1 ), 1, η(k + 1), j ) is an edge in R if and only if .[h−1 (ck ), h−1 (ck+1 )] is a consecutive interval mod 1 of E. (In this case, .η(k) = η(k + 1).) Furthermore, if .(x, 1, i, j ), (x  , 1, i  , j ) is an edge of R, then it is of one of these same 4 types that originated in Corollary 10.1.3. In the last part of the section, they will govern how the surface is assembled by gluing top edges to bottom edges. Putting these observations together, we have the following fact: Proposition 10.1.4 For every j such that .1 ≤ j ≤ q, there is an edge connecting 2 vertices of R of the form .(x, 1, i, j ) and .(x  , 1, i, j ) {.(x, 0, i, j ) and .(x  , 0, i, j )} if and only if x and .x  are consecutive points mod 1 of E {.E  }. Alternatively, R can be thought of as an .r × q matrix of rectangles. For fixed i, the projections of the rectangles onto the first 2 coordinates of .R4 are identical,

10.1 Constructing Rectangular Surfaces

301

including the simplicial structure. For fixed j , the projections of the rectangles on the first 2 coordinates provide a triangulation of the unit square .[0, 1]2 such that 2π ix , s) is a triangulation of the cylinder .S1 × [0, 1]. .h(x, s) = (e Proposition 10.1.5 The following are equivalent: (a) (b) (c) (d)

(β1 , 1, i, j ), (β2 , 1, i, j ) is a top edge of .Rij for all j . (β1 , 1, i, j ), (β2 , 1, i, j ) is a top edge of .Rij for some i and j .    .(h(β1 ), 0, i , j ), (h(β2 ), 0, i , j ) is a bottom edge of .Ri  j for some .i and j .   .(h(β1 ), 0, i , j ), (h(β2 ), 0, i , j ) is a bottom edge of .Ri  j for all j . . .

Proof The equivalence of (a) and (b) and the equivalence of (c) and (d) follow from Proposition 10.1.4. Using part (b) of Proposition 10.1.1, the equivalence of (b) and (c) also follows from Proposition 10.1.4.  Let .Sq be the group of permutations of q symbols, and let .F : {1, . . . , r} → Sq . The action of .F (i) ∈ Sq on the symbol j will be written .F (i)j . The function F will play a critical role in the construction of the rectangular surfaces. Corollary 10.1.6 Mapping the top edge .(β1 , 1, i, j ), (β2 , 1, i, j ) to the bottom edge .(h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ) is a bijective correspondence between the top edges of R and the bottom edges of R. We are now ready to construct a vertex scheme of an abstract simplicial complex K that will turn out to be a bordered surface with negative Euler characteristic. Bold face will be used to denote the abstract vertices, and rather than using subscripts, they will be written as functions of two indices because of their complex dependency on the indices. Although now purely abstract, these vertices are motivated by the triangulation of R and our plan to identify the top edges .(β1 , 1, i, j ), (β2 , 1, i, j ) with the bottom edges

.

(h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ).

.

Abstract vertices .u(i, j ) and .v(i, j ) will play the roles of the internal vertices  .

ci + ci+1 2 , , i, j 2 3



 and,

 ci + ci+1 1 , , i, j , 2 3

respectively, so .u is the upper one. Next .p(i, j ) and .q(i, j ) will play the roles of (ci , 2/3, i, j ) and .(ci , 1/3, i, j ), respectively. Similarly, .r(i, j ) and .s(i, j ) will play the roles of .(ci+1 , 2/3, i, j ) and .(ci+1 , 1/3, i, j ), respectively. Together these first 6 abstract vertices will preserve the rectangular buffer in each .Rij . So

.

{p(i, j ), q(i, j ), r(i, j ), s(i, j ), u(i, j ), v(i, j )}

.

i = 1, . . . , r and .j = 1, . . . , q will also be referred to as buffer vertices of .K. Since we want to drop the distinction between the vertices .(ci , 0, i − 1, j ) and .(ci , 0, i, j ) of R, there will be a single .c(i, j ). When .i = 1, naturally .(ci , 0, i − 1, j ) .

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10 Existence of Transitive Flows

is the lower right corner of .Rrj = Ir × [0, 1] × {r, j }, that is, (c1 , 0, 1 − 1, j ) = (0, 0, 0, j ) ≡ (1, 0, r, j )

.

because .0 ≡ 1 mod 1 for the first coordinate and 0 is congruent to r mod r for the third coordinate. Similarly, .(c1 , 1, 1 − 1, j ) ≡ (1, 1, r, j ). The surrogate for .(h(ci ), 0, η(i), j ) will be .d(i, j ). There are 8rq abstract vertices. Next define a vertex map .θ0 mapping the vertices of R to the abstract vertices as follows: θ0 ((ci + ci+1 )/2, 2/3, i, j ) = u(i, j )

.

θ0 ((ci + ci+1 )/2, 1/3, i, j ) = v(i, j ) θ0 (ci , 2/3, i, j ) = p(i, j ) θ0 (ci , 1/3, i, j ) = q(i, j ) θ0 (ci+1 , 2/3, i, j ) = r(i, j ) θ0 (ci+1 , 1/3, i, j ) = s(i, j ) θ0 (ci , 0, i − 1, j ) = c(i, j ) θ0 (ci , 0, i, j ) = c(i, j ) θ0 (h−1 (ci ), 1, η(i), j ) = c(i, F (η(i))j ) θ0 (h(ci ), 0, η(i), j ) = d(i, j ) θ0 (ci , 1, i − 1, j ) = d(i, F (i − 1)j ) θ0 (ci , 1, i, j ) = d(i, F (i)j ).

The last 6 lines of the definition of .θ0 are the critical ones. Note that θ0 h−1 (ci ), 1, η(i), F (η(i))−1 j ) = c(i, j )

.

and 3 vertices of R map to each .c(i, j ). Likewise, .θ0 maps exactly 3 vertices of R to each .d(i, j ). It follows from the discussion preceding the definition of .θ0 that .θ0 (1, 0, r, j ) = c(1, j ) and .θ0 (1, 1, r, j ) = d(1, F (1)j ). In an abstract simplicial complex, simplices are just specified subsets of vertices with the property that all their nonempty subsets are also simplices. This condition will automatically be satisfied because R is a simplicial complex. Define a finite 2-dimensional abstract simplicial complex .K as follows: (a) The vertices of .K are .c(i, j ), .d(i, j ), .p(i, j ), .q(i, j ), .r(i, j ), .s(i, j ), .u(i, j ), and .v(i, j ) such that .i = 1, . . . , r and .j = 1, . . . , q.

10.1 Constructing Rectangular Surfaces

303

(b) .{β 0 , β 1 } is a 1-simplex of .K if and only if there exists an edge of the triangulation of R with one vertex in .θ0−1 (β 0 ) and the other in .θ0−1 (β 1 ). (c) .{β 0 , β 1 , β 2 } is a 2-simplex of .K if and only if there exists a triangle of R with one vertex in each of the sets .θ0−1 (β 0 ), .θ0−1 (β 1 ), and .θ0−1 (β 2 ). Then .θ (v0 , v1 , v2 ) = {θ0 (v0 ), θ0 (v1 ), θ0 (v2 )} and likewise for .v0 , v1  is a simplicial map of the simplices of R to those of .K. A routine check shows that there are no edges connecting vertices of R identified by .θ0 , so .θ does not collapse any edges to a vertex or triangles to an edge. The proofs of the next six lemmas describing .θ in detail are straightforward calculations and are left to the reader. Because the next 2 lemmas use only consecutive bottom and consecutive top vertices, the relationships between the indices are known and using the functions .η and .η can be avoided. Lemma 10.1.7 The simplicial map .θ maps the bottom edges of R by type as follows:

θ (ci , 0, i, j ), (ci+1 , 0, i, j )

θ (ci , 0, i, j ), (h(ck ), 0, i, j )

θ (h(ck ), 0, i, j ), (ci+1 , 0, i, j )

θ (h(ck ), 0, i, j ), (h(ck+1 ), 0, i, j ) .

= {c(i, j ), c(i + 1, j )} = {c(i, j ), d(k, j )} = {d(k, j ), c(i + 1, j )} = {d(k, j ), d(k + 1, j )}.

Lemma 10.1.8 The simplicial map .θ maps the top edges of R by type as follows:

(ci , 1, i, j ), (ci+1 , 1, i, j )

θ (cη(i) , 1, η(i), j ), (h−1 (ci ), 1, η(i), j )

θ (h−1 (ci ), 1, η(i), j ), (cη(i)+1 , 1, η(i), j )

θ (h−1 (ci ), 1, η(i), j ), (h−1 (ci+1 ), 1, η(i), j ) .θ

= {d(i, F (i)j ), d(i + 1, F (i)j )} = {d(η(i), F (η(i))j ), c(i, F (η(i))j )} = {c(i, F (η(i))j ), d(η(i), F (η(i)j ))} = {c(i, F (η(i))j ), c(i + 1, F (η(i))j )}.

Lemma 10.1.9 The simplicial map .θ maps the side edges of R as follows:  θ (ci , 1, i, j ), (ci , 2/3, i, j ) = {d(i, F (i)j ), p(i, j )}  θ (ci , 0, i, j ), (ci , 1/3, i, j ) = {c(i, j ), q(i, j )}

.

 θ (ci , 2/3, i, j ), (ci , 1/3, i, j ) = {p(i, j ), q(i, j )}

 θ (ci , 1, i − 1, j ), (ci , 2/3, i − 1, j ) = {d(i, F (i − 1)j ), r(i − 1, j )}  θ (ci , 0, i − 1, j ), (ci , 1/3, i − 1, j ) = {c(i, j ), s(i − 1, j )}

 θ (ci , 2/3, i − 1, j ), (ci , 1/3, i − 1, j ) = {r(i − 1, j ), s(i − 1, j )} .

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10 Existence of Transitive Flows

Lemma 10.1.10 If .(β1 , 1, i, j ), (β2 , 1, i, j ) is an edge of R, then



θ (β1 , 1, i, j ), (β2 , 1, i, j ) = θ (h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ) .

.

Note that .θ0 maps the buffer vertices of R bijectively onto the buffer vertices of K without altering i and j . The remaining vertices of R are mapped 3 to 1 onto the set of vertices

.

.

{c(i, j ) : i = 1, . . . , r and j = 1, . . . , q}∪{d(i, j ) : i = 1, . . . , r and j = 1, . . . , q} .

Lemma 10.1.11 The function .θ maps the edges of R that contain at least 1 buffer vertex bijectively onto the set of edges of .K that contain at least 1 buffer vertex. Since every 2-simplex of .K contains at least one buffer vertex, there is no need for that hypothesis in the next lemma. Lemma 10.1.12 The function .θ maps the triangles of R bijectively onto the 2simplices of .K. Let .L be the sub-complex of .K consisting of the vertices {c(i, j ) : i = 1, . . . , r and j = 1, . . . , q}∪{d(i, j ) : i = 1, . . . , r and j = 1, . . . , q}

.

and all simplices of .K whose vertices are all in .L. Since every triangle of R contains an internal vertex, there are no 2-simplices in .L. Proposition 10.1.13 There are exactly 2rq edges in .L, and every edge in .L is the θ -image of exactly 2 edges of R—one of the form .(β1 , 1, i, j ), (β2 , 1, i, j ) and one of the form .(h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ).

.

Proof Only top edges and bottom edges of R can be mapped by .θ to an edge of .L because the buffer vertices prevent vertices of the form .(x, 0, i, j ) forming an edge with vertices of the form .(y, 1, i  , j  ). From the definition of .θ0 , it is clear that .θ0 is injective on the vertices of the form .(x, 0, i, j ). Since there are 2rq bottom edges of R, it follows that .L contains at least 2rq edges. Now Lemma 10.1.10 implies that .L has at most 2rq edges and with Corollary 10.1.6 that every edge of .L has the required 2 pre-images.  Let .|K| be the geometric realization of .K. The vertices of .|K| are the same as the vertices of .K, but they are now a basis of a free vector space over .R. In our context, the construction of .|K| produces a compact Euclidean simplicial complex because .K is finite. The topology of the realization of a simplicial complex is the coherent topology of its simplices, that is, a set U is open if and only if the intersection of U with every simplex is open in the natural Euclidean topology on the simplex. It is an exercise to show that the relative Euclidean topology on a finite Euclidean simplicial complex coincides with coherent topology determined by its simplices. (The general construction of the geometric realization of an abstract simplicial complex can be found in Chapter 5 of [24].)

10.1 Constructing Rectangular Surfaces

305

Let .|θ | =  be the Euclidean simplicial map determined by .θ . Obviously, . is a continuous map of the compact Euclidean simplicial complex R onto the compact Euclidean simplicial complex .|K|. Thus . is a quotient map. The topology on .|K| coincides with the quotient topology determined by . because R is also compact and .|K| is Hausdorff. So we can work with .|K| as a finite simplicial complex in a large dimensional Euclidean space or a quotient space .R ∼ . Understanding the equivalence relation .(x, s, i, j ) ∼ (x  , s  , i  , j  ) if and only if     .(x, s, i, j ) = (x , s , i , j ) will be necessary in this and the next section. Proposition 10.1.14 If .(β1 , 1, i, j ), (β2 , 1, i, j ) and (h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j )

.

are the corresponding top and bottom edges, respectively, then (x, 1, i, j ) = (h(x), 0, i  , F (i)j )

(10.1)

.

for .(x, 1, i, j ) in .(β1 , 1, i, j ), (β2 , 1, i, j ). Proof It follows from Lemma 10.1.10 that



 (β1 , 1, i, j ), (β2 , 1, i, j ) =  (h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ) .

.

Let .ψ be the unique simplicial map of .(β1 , 1, i, j ), (β2 , 1, i, j ) in R onto (h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j )

.

mapping .(β1 , 1, i, j ) to .(h(β1 ), 0, i  , F (i)j ) and .(β2 , 1, i, j ) to .(h(β2 ), 0, i  , F (i)j ) in R. The uniqueness of simplicial maps implies that .(x, 1, i, j ) = ◦ψ(x, 1, i, j ) for all .(x, 1, i, j ) in .(β1 , 1, i, j ), (β2 , 1, i, j ). It is an exercise to show that the simplicial map of an edge in .Rn onto a parallel edge of the same length and orientation is a translation. It follows that ψ(x, 1, i, j ) = (x, 1, i, j ) + (a, −1, i  − i, F (i)j − j ) = (h(x), 0, i  , F (i)j )

.

or ψ(x, 1, i, j ) = (x, 1, i, j ) + (a − 1, −1, i  − i, F (i)j − j ) = (h(x), 0, i  , F (i)j )

.

depending on whether .β2 + a ≤ 1 or .β2 + a > 1. Corollary 10.1.15 The 2 points .(x, t, i, j ) ∼ an unordered pair of one of the following types:

(y, s, i  , j  )

(a) .(ci , 0, i − 1, j ) and .(ci , 0, i, j ) mod 1. (b) .(β, 1, i, j ) and .(h(β), 0, i, , F (i)j ) for some .β ∈ E.

 if and only if they are

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10 Existence of Transitive Flows

(c) .(x, 1, i, j ) ∈ (β1 , 1, i, j ), (β2 , 1, i, j ) and .(h(x), 0, i  , F (i)j ) ∈ (h(β1 ), 0, i  ,  .F (i)j ), (β2 , 0, i , F (i)j ). Recall that a point x is in the Euclidean simplex with vertices .v0 , . . . , vk if and only if x=

k

.

αj vj where 0 ≤ αj ≤ 1 and

j =0

k

αj = 1.

j =0

Furthermore, a point x is in the Euclidean open simplex with vertices .v0 , . . . , vk for k > 0 if and only if

.

x=

k

.

j =0

αj vj where 0 < αj < 1 and

k

αj = 1.

j =0

(When .k = 0, the simplex is just a vertex .v0 and its open simplex is just .v0 .) If x is in a Euclidean simplicial complex, then x is in a unique open simplex of the complex. Let .Wij = Ii × (0, 1) × {(i, j )}, which is an open subset of R. Clearly, the union of the sets .Wij for .i = 1, . . . , r and .j = 1, . . . , q is R with the top and bottom edges removed and . is injective on the vertices of .Wij by definition. Proposition 10.1.16 Each .Wij is mapped by . homeomorphically onto an open subset of .|K|. Proof Because no edges or triangles are collapsed by ., it is injective on every open simplex in R. Suppose there exist distinct z and w in .Wij such that .(z) = (w). Then neither z nor w are vertices because . is a simplicial map such that .(z) = (w). For the same reasons, z and w are either both in an open edge or both in an open triangle. Therefore, . maps both the open edges or open triangles containing z and w onto the open edge or triangle containing .(z) = (w) that contradicts Lemma 10.1.11 or Lemma 10.1.12. Because .Wij contains no points in the top or bottom edges of R, it is a saturated open set. Since . is a quotient map, it follows that .(Wij ) is an open subset of .|K| and .|Wij is a quotient map (Lemma 3.17, [42]). Because .|Wij is injective, every open subset of .Wij is saturated and .|Wij is an open map onto .(Wij ), proving that .|Wij is a homeomorphism.  Proposition 10.1.17 The geometric realization .|L| of .L is homeomorphic to .S1 × {1, . . . , q}. Proof Without loss of generality,

it can be assumed that .|L| is a sub-complex of .|K|. On the one hand, the set .Bj = ri=1 {(x, 0, i, j ) : x ∈ Ii }, which is just the bottom edges of .Rij for fixed j , is a sub-complex of R. The projection .(x, s, i, j ) → x maps .Bj onto .[0, 1] mod 1. Clearly,

10.1 Constructing Rectangular Surfaces

B=

q 

.

Bj =

j =1

307 q  r 

{(x, 0, i, j ) : x ∈ Ii }

j =1 i=1

is also a sub-complex of R. The projection .π(x, s, i, j ) = (x, j ) maps B onto [0, 1] × {1, . . . , q} mod 1. So .π(B) is homeomorphic to .S1 × {1, . . . , q}. q On the other hand, .(B) = |L| = j =1 (Bj ), which is a disjoint union. Given distinct vertices .(β1 , 0, i, j ) and .(β2 , 0, i  , j  ) of B, it follows that

.

(β1 , 0, i, j ) = (β2 , 0, i  , j  )

.

if and only if either .j = j  , .β1 = β2 = ci , and .i  = i − 1 or .j = j  , .β1 = β2 = ci , and .i = i  − 1. It follows from Proposition 10.1.13 and the definition of .θ0 that . is injective on the set B \ {(ci , 0, i  .j ) : i  = i or i  = i − 1}.

.

Now .π(x, 0, i, j ) = π(x  , 0, i  , j ) if and only if .(x, 0, i, j ) = (x  , 0, i  , j ) is easily verified. Since the spaces are all compact and Hausdorff, the conclusion  follows from the uniqueness of the quotient space. To prove that .|K| is in fact a bordered surface, it must be shown that each point of .|K| has an open neighborhood homeomorphic to .B2 or .B2 ∩ H2 . This will be accomplished by partitioning .|K| into 3 types of points and showing that each type has the required property. Set P1 =

q  r 

.

Wij

j =1 i=1

and P3 = {(x, 0, i, j ) ∈ R : x ∈ E  } ∪ {(x, 1, i, j ) ∈ R : x ∈ E}.

.

Note that .P3 is the set of vertices that are either on the top of R or the bottom of R, and .(P3 ) is precisely the vertices of .|L|. Clearly, .P1 ∩ P3 = φ. Thus .P1 , .P2 = R \ (P1 ∪ P3 ), and .P3 are a partition of R. Alternatively, P2 = {(x, 0, i, j ) ∈ R : x ∈ / E  } ∪ {(x, 1, i, j ) ∈ R : x ∈ / E}.

.

The following properties of the partition .P1 , .P2 , and .P3 of R can be easily verified: (a) The sets .(P1 ), .(P2 ), and .(P3 ) are a partition of .|K|. (b) If .(i, j ) = (i  , j  ), then .(Wij ) ∩ (Wi  j  ) = φ.

308

(c) (d) (e) (f) (g)

10 Existence of Transitive Flows

If A is a subset of .P3 , then .−1 ((A)) ⊂ P3 . −1 ((P )). .P3 =  3 If .(x, s, i, j ) is in .P3 , then the cardinality of .−1 ((x, s, i, j )) is 3. If A is a subset of .P2 , then .−1 ((A)) ⊂ P2 . −1 ((P )). .P2 =  2

Theorem 10.1.18 The compact Euclidean simplicial complex .|K| is a bordered orientable surface. Proof It must be shown that each point of .|K| has a neighborhood homeomorphic to either .B2 or .B2 ∩ H2 . This will be done separately for . of the points in each element of the partition .P1 , .P2 , and .P3 of R. Observe that .Wij is a bordered surface. Since . maps .Wij homeomorphically onto an open subset of .|K| (Proposition 10.1.16), every point in .(Wij ) has the  required property, and hence, the same holds for .(P1 ). Recall that there are 2rq top edges of R and 2rq bottom edges of R, and .(P3 ) is the set of the vertices of .|L|. The .-images of the edges of R that are neither top nor bottom edges of R cannot be edges of .|L| because they contain a vertex in .P1 . Proposition 10.1.13, however, implies that each of the 2rq edges of .|L| is the image of a pair of corresponding top and bottom edges of R. Thus exactly 2 edges of R map to each edge of .|L|. Each of the top {bottom} edges of R is an edge of a triangle of R. Although the precise definition of the vertices of R (see p. 300) specifies all the coordinates of each vertex exactly, the more general notation used in Proposition 10.1.14 is simpler and adequate here. Thus for .(P2 ), it suffices to consider a pair of corresponding edges of R, using a convenient form, such as the bottom edge .(β1 , 0, i, j ), (β2 , 0, i, j ) and the top edge .(h−1 (β1 ), 1, i  , F (i  )−1 j  ), (h−1 (β2 ), 1, i  , F (i  )−1 j  ). Since the third vertex of these triangles is  .

ci + ci+1 1 , , i, j 2 3



 or

 ci  + ci  +1 1  , , i , F (i  )−1 j  , 2 3

the .-images of these triangles are distinct triangles of .|K| even if .(i, j ) = (i  , j  ). Therefore, each edge of .|L| is the .-image of exactly 2 edges of R from 2 different triangles of .|K| without common vertices. Since each point of .(P2 ) is in an edge of .|L| but is not a vertex of .|L|, there exists an open neighborhood at each point of 2 .(P2 ) homeomorphic to .B as shown in Figure 10.4. It remains to examine .(P3 ) or equivalently the vertices in .|L|. The arguments for .c(i, j ) and .d(i, j ) are essentially the same and best understood from figures. What follows is the proof that each .c(i, j ) has an open neighborhood homeomorphic to .B2 ∩ H2 . The details are just bookkeeping and geometry. Starting with .(ci , 0, i, j ), observe that .(ci , 0, i, j ) is a vertex of 2 triangles that have a common edge. The same is true for .(ci , 0, i − 1, j ) or equivalently .(ci , 0, i + 1, j ) as is shown in Figure 10.3 on p. 300. Since

10.1 Constructing Rectangular Surfaces

(

309

ci +ci+1 1 , 3 , i, j) 2

v(i, j)

(ci , 0, i, j)

(β2 , 0, i, j)

(h−1 (ci ), 1, i , F (i )−1 j  )

Θ

c(i, j)

Θ(β2 , 0, i, j)

(h−1 (β2 ), 1, i , F (i )−1 j  )

u(i , F (i )−1 j  ) (

ci +ci+1 2  , 3 , i , F (i )j  ) 2

Fig. 10.4 How one type of top edge attaches to a bottom one

(ci , 0, i − 1, j ) = (ci , 0, i, j ),

.

the .-images of these 4 triangles all have .c(i, j ) as a common vertex. The third point of .−1 (c(i, j )) can be written as .(h−1 (ci ), 1, i  , F (i  )−1 j ). This is unambiguous because the starting point was .(ci , 0, i, j ). There are exactly 2 triangles that have .(h−1 (ci ), 1, i  , F (i  )−1 j ) as a common vertex. They also have a common edge shown in Figure 10.5, namely, the edge (h−1 (ci ), 1, i  , F (i  )−1 j ), ((ci  + ci  +1 )2, 2/3, i  , F (i  )−1 j ).

.

The top edges on the left and right sides of the vertex .(h−1 (ci ), 1, i  , F (i  )−1 j ) as shown and labeled in Figure 10.5 must necessarily be the corresponding edges to the bottom edges .(β1 , 0, i − 1, j ), (ci , 0, i − 1, j ) and .(ci , 0, i, j ), (β2 , 0, i, j ), respectively. Although we know they must be top edges of the same .Rij , we do not know which of 3 possible types they are. We can now construct a concrete picture of all the simplices of .|K| that have .c(i, j ) as a vertex by gluing the top edges of Figure 10.5 to the bottom edges .(β1 , 0, i −1, j ), (ci , 0, i −1, j ) and .(ci , 0, i, j ), (β2 , 0, i, j ). The result is shown in Figure 10.6. The open star of a vertex denoted by .St(v) in a simplicial complex is the union of the open simplices that have v as a vertex. It contains v because the open simplex of a vertex is the vertex. The open star of a vertex is open in the coherent topology determined by the simplices. Hence, for a finite Euclidean simplicial complex, the star of a vertex is open in the Euclidean topology. (See p. 304 and Problem 5–2, p. 114, [42].)

310

10 Existence of Transitive Flows (h−1 (ci ), 1, i , F (i )−1 j) (h−1 (β2 ), 1, i , F (i )−1 j)

(h−1 (β1 ), 1, i , F (i )−1 , j)

c

i +ci +1

2

, 23 , i , F (i )−1 j



Fig. 10.5 The structure of R below the vertex .(h−1 (ci ), i  , F (i  )−1 j ) Fig. 10.6 A planar view of the sub-complex .K 

s(i − 1, j) v(i − 1, j)

q(i, j) v(i, j)

c(i, j)

(β2 , 0, i, j)

(β1 , 0, i − 1, j)

u(i , F (i )−1 j)

Thus .St(c(i, j )) is an open neighborhood of .c(i, j ). It will be shown that St(c(i, j )) is homeomorphic to .B2 ∩ H2 . The simplices of .|K| that contain .c(i, j ) and their faces are a sub-complex of .|K| denoted by .K  containing .St(c(i, j )). It is homeomorphic to the planar sub-complex shown in Figure 10.6 by Exercise 5.4, p. 95 of [42]. Removing the simplices of .K  that do not contain .c(i, j ) leaves the open star  .St(c(i, j )), so .St(c(i, j )) ⊂ K . Using the rays emanating from .c(i, j ) and the  simple closed curve bounding .K , we can construct a homeomorphism of .St(c(i, j )) .

10.1 Constructing Rectangular Surfaces

311

onto .B2 with a wedge removed, which is homeomorphic to .B2 ∩H2 . Similarly, every .d(i, j ) is a boundary point. This completes the proof that .|K| is a compact bordered surface. It remains to prove that .|K| is orientable. For fixed j , the projection of the triangulations of .Rij , .1 ≤ i ≤ r, is a triangulation of the unit square that does not depend on j by construction. Putting the counterclockwise orientation on the triangles orients the unit square and .RS , which naturally extends to R. Furthermore, the orientations of the top edges of R are all from right to left, and the orientations of the bottom edges of R are all from left to right. Consequently, the identifications of the edges .(β1 , 1, i, j ), (β2 , 1, i, j ) and .(h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ) by . (Proposition 10.1.14) automatically provide an orientation to .|K|. (For a fuller discussion, see the section on orientations starting on p. 105 of [42].)  Corollary 10.1.19 .χ (|K|) = −rq. Proof The Euler characteristic for R is q times that of .RS . So there are 12rq vertices, .23rq edges, and .12rq triangles for R (see p. 300). Clearly, .|K| has .4rq fewer vertices and 2rq fewer edges. Thus .χ (|K|) = 8rq − 21rq + 12rq = −rq.  Corollary 10.1.20 The boundary of .|K| is the sub-complex consisting of the set of vertices .{p(i, j ), q(i, j ), r(i, j ), s(i, j )}, and all edges that contain at least 1 of them. Recall that every permutation is the composition of disjoint cycles that commute (see Propositions 1.1.1 and 1.1.2 in [59] or almost any abstract algebra book). This composition can include trivial cycles, that is, fixed points of the permutation. Let .ν(i) denote the number of disjoint cycles in the composition of the permutation −1 F (i) including trivial ones. This function will be used to count .G(i) = F (i − 1) the number of boundary components of .|K|. Proposition 10.1.21 The number of boundary components of .|K| equals ζ (G) =

r

.

ν(i).

(10.2)

i=1

Proof The only 2 types edges of the boundary sub-complex that have .c(i, j ) as a vertex are .c(i, j ), s(i − 1, j ) and .c(i, j ), q(i, j ) (see Figure 10.6). Hence, c(i, j ), q(i, j ) ∪ q(i, j ), p(i, j ) ∪ p(i, j ), d(i, F (i)j )

.

and c(i, j ), s(i − 1, j ) ∪ s(i − 1, j ), r(i − 1, j ) ∪ r(i − 1, j ), d(i, F (i − 1)j )

.

are arcs in the boundary of .|K| for all i and j . The proof of Theorem 10.1.18 shows that the entire boundary of .|K| consists of these arcs.

312

10 Existence of Transitive Flows

Replacing j by G(i)j = F (i − 1)−1 F (i)j

.

in the second arc and then reversing the order of the vertices in the second arc yield the boundary arc: d(i, F (i)j ), r(i − 1, G(i)j ) ∪

.

r(i − 1, G(i)j ), s(i − 1, G(i)j ) ∪ s(i − 1, G(i)j ), c(i, G(i)j ). Now the endpoint of the first arc is the same as the starting point of the new second arc. Thus .

c(i, j ), q(i, j ) ∪ q(i, j ), p(i, j ) ∪ p(i, j ), d(i, F (i)j ) ∪ d(i, F (i)j ), r(i − 1, G(i)j ) ∪ r(i − 1, G(i)j ), s(i − 1, G(i)j ) ∪ s(i − 1, G(i)j ), c(i, G(i)j )

is a boundary arc from the vertex .c(i, j ) to the vertex .c(i, G(i)j ). By iteration, there exist boundary paths from .c(i, j ) to .c(i, G(i)k j ) for .k ∈ Z+ . If .(j1 . . . jk ) is one of the .ν(i) disjoint cycles that make up .G(i), then the arc from .c(i, j ) to .c(i, G(i)k j ) is a simple closed curve and a boundary component. So for each i, there are .ν(i) boundary components, and the total number of boundary components of .|K| is .ζ (G).  Proposition 10.1.22 The bordered surface .|K| is a connected surface if the subgroup of .Sq generated by .{F (i) : i = 1, . . . , r} has only 1 orbit when it acts on .{1 ≤ j ≤ q}. Proof The sub-complex .|L| consists of q disjoint connected components .Lj each homeomorphic to .S1 by Proposition 10.1.17. As in the proof of that result, let .Bj =

r i=1 {(x, 0, i, j ) : x ∈ Ii }, and note that .(Bj ) = Lj . Since .(Rij ) is connected and intersects .Lj ,  Kj = 

r 

.

 Rij

i=1

is a connected sub-complex containing .Lj for .j = 1, . . . , q. Thus |K| =

q 

.

j =1

has at most q components.

Kj

10.2 Constructing Flows from Orbits

313

From the proof of Proposition 10.1.21, we know that c(i, j ), q(i, j ) ∪ q(i, j ), p(i, j ) ∪ p(i, j ), d(i, F (i)j )

.

is an edge path from starting at .c(i, j ) in .Lj and ending at .d(i, F (i)j ) in .LF (i)j . Thus .Lj and .LF (i)j are in the same component of .|K|. It follows that .Lj and .LF (in )···F (i1 )j are in the same component of .|K| for every finite sequence of integers in .{1, . . . , q}. If the subgroup of .Sq generated by .{F (i) : i = 1, . . . , r} has only 1 orbit when it acts on .{1 ≤ j ≤ q}, then for every j and .j  there exists .i1 , . . . , in such that .F (in ) · · · F (i1 )j = j  and .|L| is connected.  The result of the construction of .|K| which we are calling a rectangular surface is summarized in the final theorem. Items (a) and (b) are part of earlier propositions, and the last part is an immediate consequence of equation (2.4). Theorem 10.1.23 If the subgroup of .Sq generated by .{F (i) : i = 1, . . . , r} has only 1 orbit when it acts on .{1 ≤ j ≤ q}, then .|K| is a compact connected orientable bordered surface such that: (a) The number of boundary components of .|K| is given by .ζ (G). (b) The Euler characteristic is .χ (|K|) = −qr. (c) The genus is γ (|K|) =

.

2 + qr − ζ (G) . 2

(10.3)

10.2 Constructing Flows from Orbits The next task is to construct a flow on .X = |K|. Up to now the simplicial structure has dominated the construction of X. In this section, the rectangular structure will be dominant. The space X was constructed so that it contains a suspension flow that makes use of every .(Wij ) for its orbits. The strategy is to use the proof of Beck’s theorem (Theorem 1.3.3) to modify and extend the suspension flow to a flow on X. Since the components of a compact surface are always invariant sets of any flow on the surface, we will restrict our attention to rectangular flows on connected bordered surfaces constructed using Theorems 10.1.18 and 10.1.23. The construction of X started with the minimal cascade .h : [0, 1] → [0, 1] defined by .h(s) = s + a mod 1 with a irrational. Let  Y = [0, 1] \

r 

.

 O(ci ) .

i=1

Then Y is an h-invariant set and .(Y, h) is a cascade on a non-compact space.

314

10 Existence of Transitive Flows

There is a useful continuous locator function defined on the open set .[0, 1] \ {c1 , . . . , cr } by .(x) = i if and only if .x ∈ (ci , ci+1 ). Using the function ., we  : [0, 1] \ {c1 , . . . , cr } → Sq by setting .F (y) = can define a continuous function .F   F ((y)). Then the function .h(y, j ) = (h(y), F (y)j ) is a homeomorphism of .Y × {1, . . . , q} onto itself, so .(Y × {1, . . . , q},  h ) is another cascade.  can be continuously extended to .F  : Y × Z → Sq by the The function .F following formulas: (y, 0) = ι F

.

(y, 1) = F (y) F (hn−2 (y)) · · · F (h(y))F (y) for n > 1. (y, n) = F (hn−1 (y))F F (y, −n) = F (h F

−n

(y))

−1 

F (h

−n+1

(y))

−1

(h ···F

−1

(y))

−1

.

(h−n (y), n)−1 for n ≥ 1. =F

(10.4) (10.5) (10.6)

(y, n) is another example of a cocycle and satisfies the equation The function .F (y, n), (y, m + n) = F (hn (y), m)F F

.

(10.7)

which is a version of equation (1.2) written for homeomorphisms acting on the left  in this way, with a non-abelian range. By extending .F (y, n)j ).  hn (y, j ) = (hn (y), F

.

(10.8)

The cocycle equation has the following useful corollary: (x, −1) = ι = F (x, −1)F (h−1 (x), 1). (h−1 (x), 1)F F

.

(10.9)

The subset of R defined by  = {(y, s, (y), j ) : y ∈ Y, 0 ≤ s ≤ 1, and j = 1, . . . , q} Y

.

is dense in R and homeomorphic to .Y × {1, . . . , q} × [0, 1]. It follows from Theorem 3.1.15 and Proposition 10.1.14 that with .0 ≤ s < 1 (y, [s + t])j ) (y, s, (i), j )t = (h[s+t] (y), s + t − [s + t], (h[s+t] (y)), F (10.10) ) isomorphic to the suspension flow .(S(Y×{1, . . . , q},  defines a flow on .(Y h), R). ) as far as possible. We will use the formula (10.10) to extend the flow on .(Y Let .

V =

q  r 

.

j =1 i=1

{c(i, j ), d(i, j )},

(10.11)

10.2 Constructing Flows from Orbits

315

which is the set of vertices that lie on the embedded circles .Lj , .j = 1, . . . , q (Proposition 10.1.17). These vertices will eventually be the fixed points of the flow on X and are the important vertices in this section. Set ⎛ D1 = ⎝

q  r 

.

⎞  ({(ci , s, i, j ) : 0 < s < 1})⎠



j =1 i=1

⎛ ⎝

q  r 

⎞  ({(ci , s, i − 1, j ) : 0 < s < 1})⎠ .

(10.12)

j =1 i=1

D2 =

q  r ∞  

   {(h−k (ci ), s, (h−k (ci )), j ) : 0 ≤ s < 1} . (10.13)

k=1 j =1 i=1

D3 =

q  r ∞  

   {(hk (ci ), s, (hk (ci )), j ) : 0 < s ≤ 1} .

(10.14)

k=1 j =1 i=1

Although each of these sets is the union of many pieces, the pieces used to form Di are disjoint for .i = 1, 2, 3. It is helpful to observe that .D1 = ∂X \ V , and the two lines of its definition come from the two copies of each .ci used to construct R. It follows that

.

∂X = V ∪ D1 .

.

(10.15)

Furthermore, .D2 and .D3 are constructed from the negative orbits of .h−1 (ci ) and the positive orbits of .h(ci ), respectively, for .i = 1, . . . , r. Note, however, that .(h(ci ), 0, i, j ) is not in .D3 but equals .d(i, j ) in V . Now a straightforward examination of these sets proves the following: ), V , .D1 , .D2 , and .D3 form a partition of X. Lemma 10.2.1 The sets .(Y It will be convenient to denote points in X by .x and, as needed, use the fact that x = (x, s, i, j ) for some .(x, s, i, j ) in R. Recall from Proposition 10.1.16 and property (b) preceding Theorem 10.1.18 that for .x ∈ (P1 ) the R coordinates .(x, s, i, j ) are unique. For other points, it follows from the definition of .θ0 and Proposition 10.1.14 that there are 2 or 3 choices. In spite of these limitations, .x = (x, s, i, j ) are a useful coordinates system for X. Given .x = (x, s, i, j ) in .X \ V , using (10.12), (10.13), and (10.14), set .

316

10 Existence of Transitive Flows

⎧ ⎪ ⎪ ⎨1 − s + .g (x) = k−s ⎪ ⎪ ⎩∞

when x ∈ D1 when x ∈ D2

(10.16)

) ∪ D3 when x ∈ (Y

and

−

g (x) =

.

⎧ ⎪ ⎪ ⎨−s

when x ∈ D1

−k + 1 − s ⎪ ⎪ ⎩−∞

when x ∈ D3

(10.17)

) ∪ D2 . when x ∈ (Y

The value of .g + is always positive, and the value of .g − is always negative. An extended real-valued function h on a topological space is said to be upper semicontinuous provided that the set .{x : h(x) < α} is open for every .α ∈ R and lower semicontinuous provided that the set .{x : h(x) > α} is open for every .α ∈ R. (See [39] or [81] for a more substantive discussion of upper and lower semicontinuous functions.) Proposition 10.2.2 The functions .g + and .g − are lower semicontinuous and upper semicontinuous on .X \ V , respectively. Proof To prove that .g + is lower semicontinuous on .X \ V , it suffices to show that + .{x ∈ X : g (x) ≤ α} is a closed subset of .X \V when .α > 0. Since there are clearly ) ∪ D3 in .{x ∈ X : g + (x) ≤ α}, the proof is further reduced to no points of .(Y showing that .{x ∈ X : g + (x) ≤ α} ∩ D1 and .{x ∈ X : g + (x) ≤ α} ∩ D2 are closed subsets of .X \ V . If .α ≥ 1, then .D1 = {x ∈ X : g + (x) ≤ α} ∩ D1 . Although .D1 is not closed in X, the closure of .D1 is just .D1 ∪ V and .D1 is closed in .X \ V . The pattern of sets not being closed in X but being closed in .X \ V will occur repeatedly. For example, if .0 < α < 1, then {x ∈ X : g + (x) ≤ α} ∩ D1 =

.

q  r 

 ({(ci , s, i, j ) : 1 − α ≤ s < 1}) ∪

j =1 i=1 q  r 

 ({(ci , s, i − 1, j ) : 1 − α ≤ s < 1}) ,

j =1 i=1

and the only additional points in its closure are the vertices .d(i, j ). When .α ≤ 0, the set is empty. Therefore, .{x ∈ X : g + (x) ≤ α} ∩ D1 is a closed subset of .X \ V for all .α ∈ R. It follows from the definitions of .D2 and .g + that for .α > 0

10.2 Constructing Flows from Orbits

317

{x ∈ X : g + (x) ≤ α} ∩ D2 =

.

q  [α]  r 

{(h−k (ci ), s, (h−k (ci )), j ) : 0 ≤ s < 1} ∪

k=1 j =1 i=1 q  r 

{(h−[α]−1 (ci ), s, (h−[α]−1 ), j ) : 1 − (α − [α]) ≤ s < 1}.

j =1 i=1

Thus .{x ∈ X : g + (x) ≤ α} ∩ D2 consists of .([α] + 1)rq pieces homeomorphic to .[0, 1). Taking the closure of one of these pieces adds a point of the form −k (c ), 1, (h−k (c )), j ) and no other points. By Proposition 10.1.14 .(h i i (h−k (ci ), 1, (h−k (ci )), j ) = (h−k+1 (ci ), 0, (h−k+1 (ci )), F ((h−k+1 (ci )))j ),

.

and the point .(h−k (ci ), 1, (h−k (ci )), j ) is in .{x ∈ X : g + (x) ≤ α} ∩ D2 when .k > 1 and .k ≤ [d]. When .k = 1, (h−k (ci ), 1, (h−k (ci )), j ) = (h−1 (ci ), 1, (h−1 (ci )), j )

.

= (ci , 0, i, F (i)j ) = c(i, F (i)j ). Therefore, {x ∈ X : g + (x) ≤ α} ∩ D2 ∪ {c(i, j ) : i = 1, . . . , r and j = 1, . . . , q}

.

is the closure of .{x ∈ X : g + (x) ≤ α} ∩ D2 , which is a closed subset of .X \ V . This completes the proof of the lower semicontinuity of .g + . The proof that .g − is upper semicontinuous is very similar. The main difference is that .D2 and .D3 switch  roles altering the indexing and inequalities. Proposition 10.2.3 The set U=

.



 x, t : x ∈ X \ V and g − (x) < t < g + (x)

(10.18)

is an open subset of .(X \ V ) × R. Proof Consider .(x, τ ) ∈ U and choose .ε > 0 such that .τ + ε < g + (x) and .τ − ε > g − (x). By Lemma 10.2.2, the set .W + = {y ∈ X : g + (y) > τ + ε} is an open neighborhood of .x. Suppose .x is in .W + and .τ ≤ t  < τ + ε. Then t  < τ + ε < g + (x )

.

318

10 Existence of Transitive Flows

and .(x , t  ) is in U . Similarly, the set .W − = {y ∈ X : g − (y)) < τ − ε} is an open neighborhood of .x, and .(x , t  ) is in U when .x is in .W − and .τ − ε < t  ≤ τ . It now  follows that .W + ∩ W − × {t : |t − τ | < ε} is contained in U . We will show that U is a subset of X on which (10.19) defines a function .ϕ : U → X \ V , that is, for .x = (x, s, i, j ) ∈ U with .0 ≤ s < 1

(x, [s + t])j ) ϕ (x, s, i, j ), t = (h[s+t] (x), s + t − [s + t], (h[s+t] (x)), F (10.19) ) because is a well-defined function. This requires no verification for .x in .(Y ) × R is a flow isomorphic to the suspension flow .(S(Y × .ϕ restricted to .(Y {1, . . . , q},  h), R) by construction. − + If .x is in .D1 , then

.−s = g (x) < t < g (x) = 1 − s implies .0 < s + t < 1 and (x, 0) = ι ∈ .ϕ (x, s, i, j ), t = (x, s + t, i, j ) is another point in .D1 because .F  Sq is continuous on .[0, 1] mod 1. So .F (x, 0) can be used as needed when .[s +t] = 0 and .(x, s, i, j ) is in .X \ V . If .x is in .D2 , then .x = (h−k (ci ), s, i, j ) such that .k ≥ 1 and .0 ≤ s < 1. Then − + .−∞ = g (x) < t < g (x) = k − s implies .−∞ < s + t < k and .[s + t] ≤ k − 1. Thus .

(h[s+t]−k (ci ), [s + t])j ) (h[s+t]−k (ci ), s + t − [s + t], (h[s+t]−k (ci )), F

.

unambiguously defines a point in .D2 and is equation (10.19) for .x = h−k (ci ). If .x is in .D3 , then .x = (hk (ci ), s, i, j ) such that .k ≥ 1 and .0 < s ≤ 1. Then − + .−k + 1 − s = g (x)) < t < g (x) = ∞ implies .−k + 1 < s + t < ∞ and .[s + t] + k ≥ 1. Now (h[s+t]+k (ci ), [s + t])j ) (h[s+t]+k (ci ), s + t − [s + t], (h[s+t]+k (ci )), F

.

unambiguously defines a point in .D3 and is equation (10.19) for .x = hk (ci ). The next lemma follows from the verification that .ϕ is well defined on U . Lemma 10.2.4 The function .ϕ has the following properties: (a) If .(x, t) is in U and .x is in .Di , then .ϕ(x, t) is in .Di . (b) .g + (ϕ(x, t)) = g + (x) − t when .(x, t) is in U . (c) .g − (ϕ(x, t)) = g − (x) − t when .(x, t) is in U . The next proposition along with the observation that .ϕ(x, 0) = x shows that the function .ϕ given by equation (10.19) partially satisfies the axioms of a flow. Proposition 10.2.5 The function .ϕ has the following properties: (a) Given that .(x, t) is in U , the point .(ϕ(x, t), τ ) is in U if and only if the point .(x, t + τ ) is in U . (b) If .(x, t) and .(x, t + τ ) are in U , then

10.2 Constructing Flows from Orbits

319





ϕ ϕ x, t), τ = ϕ x, t + τ .

.

(c) The function .ϕ is continuous on U . Proof Part (a) is an immediate consequence of Lemma 10.2.4, and part (b) follows from the observation that .[s + t − [s + t] + τ ] = [s + t + τ ] − [s + t], and hence, s + t − [s + t] + τ − [s + t − [s + t] + τ ] = s + t + τ − [s + t + τ ].

.

Turning to part (c), consider an arbitrary point .x = (x, s, i, j ) in .X\V with .0 ≤ s < 1 as usual. Let .(xn , tn ) = ((xn , sn , in , jn ), tn ) be sequence in U converging to .(x, t) = ((x, s, i, j ), t) also in U . Because R is compact Hausdorff, it suffices to prove the following: If  .

lim

n→∞

 (xn , [sn + tn ])jn = (w, σ, κ, μ), h[sn +tn ] (xn ), sn + tn − [sn + tn ], in , F

then   (x, [s + t])j ) = ϕ(x, t). (w, σ, κ, μ) =  h[s+t] (x), s + t − [s + t], i, F

.

There are, however, cases, sub-cases, and sub-sub-cases to consider! First, if .s = 0, then .(xn , sn , in , jn ) must converge to .(x, s, i, j ) and .sn + tn converges to .s + t. If .s + t ∈ / Z, then .[sn + tn ] converges to .[s + t], that is, .[sn + tn ] = [s + t] for large n. So   (x, [s + t])j , (w, σ, κ, μ) = h[s+t] (x), s + t − [s + t], i, F

.

completing this sub-case. When .s = 0, the remaining sub-case is .s + t ∈ Z. Now .[sn + tn ] could converge to either .[s + t] = s + t or .[s + t] − 1. The first sub-sub-case is identical to the sub-case .s + t ∈ / Z. The key observation for the second is that .sn + tn − [sn + tn ] converges to 1 when .[sn + tn ] converges to .[s + t] − 1. Thus (x, [s + t] − 1)j ), (w, σ, κ, μ) = (h[s+t]−1 (x), 1, i  , F

.

which . identifies with (h[s+t]−1 (x))F (x, [s + t] − 1)j ). (h(h[s+t]−1 (x)), 0, i  , F

.

(y) = F (y, 1), equation (10.7) applies to the fourth Recalling that by definition .F coordinate and (x, [s + t] − 1)j ) = F (x, [s + t])j ). (h[s+t]−1 , 1)(F F

.

320

10 Existence of Transitive Flows

Therefore, (x, [s + t])j ). (w, σ, κ, μ) = (h[s+t] (x), s + t − [s + t], i, F

.

This completes the .s = 0 case. The second case is .s = 0. In this case, .(xn , sn , in , jn ) either converges to −1 (x), 1, i  , F (x, −1)j ), so there are also 2 sub-cases. .(x, 0, i, j ) or converges to .(h In the first sub-case, the two sub-sub-cases are .s + t = t ∈ / Z and .t ∈ Z, and they are handled as in the previous paragraph. In the second sub-case, that is, .(xn , sn , in , jn ) converges to (x, −1)j ), (h−1 (x), 1, i  , F

.

and the 2 sub-sub-cases are .s + t = t is not an integer and t is an integer. If .s + t = t is not an integer, then .[sn + tn ] converges to .[t] + 1 and .sn + tn − [sn + tn ] (x, [t])j ) converges to .t − [t]. It follows that .(w, σ, κ, μ) = (h[t] (x), t − [t], i  , F and .(w, σ, κ, μ) = ϕ(x, t). When t is an integer, .[sn + tn ] converges to t or .t − 1, and .(w, σ, κ, μ) is t (x, t)j ) or .(ht−1 (x), 1, i  , F (x, t − 1)j ). It follows that .(h (x), 0, i, F (x, t)j ) = (ht−1 (x), 1, i  , F (x, t − 1)j ) = ϕ(x, t) (ht (x), 0, i, F

.



to finish the proof. The .ϕ-orbit of .x in .X \ V is defined by Oϕ (x) = {ϕ(x, t) : g − (x) < t < g + (x)}.

.

(10.20)

Note that .D2 and .D3 each consist of exactly qr distinct .ϕ-orbits. Similarly, .Oϕ+ (x) = {ϕ(x, t) : 0 ≤ t < g + (x)} and .Oϕ− (x) = {ϕ(x, t) : g − (x) < t ≤ 0}. The last step in the construction is to modify the function .ϕ to obtain a flow on X with .V being the set of fixed points of the modified flow and without altering the orbits. This is accomplished by modifying the proof of Beck’s theorem (Theorem 1.3.3). The structure of .D2 and .D3 appearing in the proof of Proposition 10.2.2 provides a start for proving the initial key lemma. Lemma 10.2.6 (a) If .x is in .D1 ∪ D2 , then .lim t  g + (x) ϕ(x, t) exists and is in V . (b) If .x is in .D1 ∪ D3 , then .lim t  g − (x) ϕ(x, t) exists and is in V . The next two lemmas are analogs of Propositions 1.3.1 and 1.3.2 specifically for the function .ϕ and a metric d on X. Lemma 10.2.7 Let .y be an element of .X \ V . Given .ε > 0 and real numbers .α and .β such that .g − (y) < α ≤ 0 ≤ β < g + (y), there exists .δ > 0 such that .d(ϕ(x, t), ϕ(y, t)) < ε for all t such that .α ≤ t ≤ β, whenever .d(x, y) < δ.

10.2 Constructing Flows from Orbits

321

Proof Use the idea of the proof of Proposition 1.3.1 twice. First, to show that there exists .μ > 0 such that {x : d(x, y) < μ} × [α, β] ⊂ U,

.

because U is open (Proposition 10.2.3). Second, to show the existence of .δ given .ε using the continuity of .ϕ as in the original proof on page 21.  Lemma 10.2.8 Let .f : X \ V → R be a continuous function, let .y be an element of X \V , and let .α and .β be real numbers such that .g − (y) < α ≤ 0 ≤ β < g + (y) with .β − α > 0. Given .ε > 0, there exists .δ > 0 such that .{ϕ(x, t) : α ≤ t ≤ β} ⊂ X \ V and     β  β    . f (ϕ(x, s)) ds − f (ϕ(y, s)) ds  < ε,  α  α .

whenever .d(x, y) < δ and .α ≤ α  < β  ≤ β. Proof As in the proof of Proposition 10.2.7, there exists .μ > 0 such that .{x : β d(x, y) < μ} × [α, β] ⊂ U . Hence, . α f (ϕ(x, s)) ds is defined when .d(x, y) < μ. Since the function .f (ϕ(x, s)) is defined and continuous on .{x : d(x, y) < μ} × [α, β], the technique used to prove Proposition 1.3.1 can be used again to show that there exists .δ > 0 such that .|f (ϕ(x, s)) − f (ϕ(y, s))| < ε/2(β − α) for all s such that .α ≤ s ≤ β, whenever .d(x, y) < δ. Consequently,      β  β   β   . f (ϕ(x, s)) ds − f (ϕ(y, s)) ds  ≤ |f (ϕ(x, s)) − f (ϕ(y, s))| ds  α  α α  ≤

β

|f (ϕ(x, s)) − f (ϕ(y, s))| ds

α

 ≤

α

β

ε ds < ε, 2(β − α)

whenever .d(x, y) < δ.



We are now ready to show that the proof of Beck’s theorem can be applied with minor modifications to the “partial” flow .ϕ on X to obtain a flow on X. Theorem 10.2.9 There exists a flow .ψ on .X \ V with the following properties: (a) .Oψ (x) = Oϕ (x) for all .x in .X \ V . + − (b) .Oψ (x) = Oϕ+ (x) and .Oψ (x) = Oϕ− (x) for all .x in .X \ V . Proof To start, let

322

10 Existence of Transitive Flows

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨∞ .g(x) = min{−g − (x), g + (x)} ⎪ ⎪ ⎪ ⎪ g + (x) ⎪ ⎪ ⎪ ⎩−g − (x)

for x ∈ V ) for x ∈ (Y for x ∈ D1

(10.21)

for x ∈ D2 for x ∈ D3 .

Clearly, .g(x) ≥ 0 for all .x ∈ X. Note that .g(x) = 0 if and only if .x is in .X \ V because .g ± (x) = 0 for all .x ∈ X \ V . Then as in the original proof, set f (x) = inf{d(x, y) + g(y) : y ∈ X}

.

(10.22)

so .f (x) ≤ g(x) and .f (x) ≤ d(x, V ) = inf{d(x, y) : y ∈ V }. Obviously, .f (x) = 0 when .x is .V . Conversely, if .f (x) = 0, then there exists a sequence .yn such that both .d(x, yn ) and .g(yn ) converge to 0. Hence, .yn converges to .x. The definitions of .g ± and the structure of .D1 , .D2 , and .D3 used in the proof of Proposition 10.2.2 imply that .d(yn , V ) converges to 0, when .g(yn ) converges to 0. Therefore, .f (x) = 0 if and only if .x is in V . Next we will show that f is continuous. First observe that f (x) ≤ d(x, y) + g(y) ≤ d(x, x ) + d(x , y) + g(y)

.

for all .y ∈ X. Given .δ > 0, there exists .y such that .d(x , y) + g(y) ≤ f (x ) + δ. Hence, f (x) ≤ d(x, x ) + f (x ) + δ

.

for all .δ > 0, implying that .f (x) − f (x ) ≤ d(x, x ). Since the argument is symmetric in .x and .x , |f (x) − f (x )| ≤ d(x, x )

.

and f is continuous on X. As in the proof of Theorem 1.3.3, setting ρ(x) = max{1, 1/f (x)}

.

(10.23)

defines a continuous function on .X \ V bounded below by 1 such that .ρ(x) ≥ 1/d(x, V ) and .ρ(x) ≥ 1/g(x). Proposition 10.2.5 implies that the function  h(x, σ ) = 0

satisfies the cocycle equation (1.2):

σ

ρ(ϕ(x, t)) dt

.

(10.24)

10.2 Constructing Flows from Orbits

323

h(x, σ + τ ) = h(x, σ ) + h(ϕ(x, σ ), τ )

.

when .(x, σ ) and .(x, σ + τ ) are in U . (The notation here is not the best, but it is workable. The contexts of the homeomorphism .h(x) = x + a (mod 1) and the cocycle .h(x, σ ) are different, and both are consistent with earlier notation.) Since .ϕ(x, t) is defined on the open interval .(g − (x), g + (x)) for all .x ∈ X, in the present context, let .bx = g + (x) and .ax = g − (x). If .bx < ∞, then .limt→bx ϕ(x, t) is a point in .V (Lemma 10.2.6) and .bϕ(x,t) = bx − t (Proposition 10.2.5). Because .g(x) ≤ bx , it follows that .g(ϕ(x, t)) ≤ bx − t and that  h(x, τ ) =

.

τ



τ

ρ(ϕ(x, t)) dt ≥

0

0

1 dt = − ln(bx − τ ) + ln bx . bx − t

Thus .

lim h(x, τ ) = ∞,

τ →bx

and by similar reasoning, .

lim h(x, τ ) = −∞

τ →ax

when .ax > −∞. Since .ρ(x) ≥ 1 > 0 for all .x ∈ X \ V , .

lim h(x, τ ) = ∞ and

τ →bx

lim h(x, τ ) = −∞

τ →ax

(10.25)

when .bx = ∞ and .ax = −∞, respectively. Since .ρ(x) is a positive function on .X \ V , the function .t → h(x, t) is increasing, obviously continuous, and thus open, making it a homeomorphism of .(ax , bx ) onto .R. Let .H (x, t) denote its inverse, that is, .

H (x, h(x, t)) = t for all (x, t) ∈ U .

(10.26)

h(x, H (x, τ )) = τ for all (x, τ ) ∈ (X \ V ) × R.

(10.27)

So in this situation .H (x, τ ) is the unique real number .σ such that 

σ

τ=

ρ(ϕ(x, t)) dt.

.

0

Using the cocycle equation (1.2), .



h x, H (x, s) + H [ϕ(x, H (x, s)), t]



= h x, H (x, s) + h ϕ(x, H (x, s)), H [ϕ(x, H (x, s)), t]

(10.28)

324

10 Existence of Transitive Flows

=s+t = h(x, H (x, s + t)). Since there is exactly one real number .σ such that .h(x, σ ) = s + t, H (x, s + t) = H (x, s) + H (ϕ(x, H (x, s)), t).

(10.29)

.

To prove that H is continuous on .(X \ V ) × R, let .(xn , τn ) be a sequence converging to .(x, τ ) in .(X \ V ) × R. It must be shown that the sequence .H (xn , τn ) converges to .H (x, τ ) or equivalently .σn converges to .σ , when .σn and .σ are given by equation (10.28) for .τn and .τ , respectively. Lemma 10.2.8 will be used extensively in the rest of the argument. Let .ε > 0 satisfy .ax < σ − 3ε < σ < σ + 3ε < bx . Then the integrals τ

.

±



σ ±ε

=

ρ(ϕ(x, t)) dt 0

are defined and satisfy .τ − < τ < τ + . So there exists .N1 ∈ Z+ such that .τ − < τn < τ + for .n > N1 . Next set ξ

.

±



σ ±3ε

=

ρ(ϕ(x, t)) dt. 0

Clearly, .ξ − < τ − < τ < τ + < ξ − . There exists .N2 ∈ Z+ such that the integrals 



σ −2ε

σ +2ε

ρ(ϕ(xn , t)) dt and

.

0

ρ(ϕ(xn , t)) dt 0

are in the open intervals .(ξ − , τ − ) and .(τ + , ξ + ) when .n > N2 , respectively, by Proposition 10.2.8 because .ε < 2ε < 3ε. s Therefore, the increasing function . 0 ρ(ϕ(x, t)) dt for .σ − 2ε < s < σ + 2ε assumes every value in the open interval .(τ − , τ + ) when .n > N2 . If .n > N1 and − +  .n > N2 , then .τn is in .(τ , τ ), and there exists .s in .(σ − 3ε, σ + 3ε) such that 



σ −2ε

s

ρ(ϕ(xn , t)) dt
ξ for all n. By passing to a subsequence again, we can assume that either .σn > ξ for all n or .σn < −ξ for all n. So it suffices to obtain contradictions when .σn > 0 for all n and when .σn < 0 for all n. The same method can be used for both cases. What follows is the proof for .σn > 0 for all n. Since .τn is a convergent sequence, there exists .M > 0 such that .τn < M for all n. It follows from the definition of .ρ(x) that .ρ(x) ≥ 1, and hence, 

σn

0 < ξ < σn ≤

ρ(ϕ(xn , t)) dt = τn < M.

.

(10.31)

0

So .σn is a bounded sequence and can be assumed to converge to .σ ≥ ξ > 0. Writing .xn = (xn , sn , in , jn ) as usual with .(xn , sn , in , jn ) in .Rin jn , it can be assumed that .(xn , sn , in , jn ) converges to .(x, 0, i, j ) or .(x, 1, i, j ) in .−1 (x) by passing to a subsequence once again. So .xn converges to x, and either .sn or .1 − sn converges to 0 but not both. Again the same method can be used for both cases. What follows is the proof for the .sn converges to 0 case. Without loss of generality, we can assume that .sn < 1/2 for all n. Because R was constructed as a subset of .R4 , we can use the Euclidean metric on it, that is, the distance between .(x, s, i, j ) and .(y, t, i  , j  ) will be  (x − y)2 + (s − t)2 + (i − i  )2 + (j − j  )2 .

(x, s, i, j ) − (y, t, i  , j  ) =

.

If .(x, s, i, j )−(y, s, i, j ) < ε for particular points .(x, s, i, j ) and .(y, s, i, j ), then (x, t, i, j ) − (y, t, i, j ) < ε for .0 ≤ t ≤ 1. Let d be a metric for X. Because R and X are both compact metric spaces, . is uniformly continuous. There exists a real decreasing sequence .δk converging to 0 such that

.



δk

.

δk+1

1 dt > M t

and .0 < δk < min{σ, 1/2}. For each .δk+1 , there exists .νk > 0 such that

326

10 Existence of Transitive Flows

d((x, s, i, j ), (y, t, i, j )) < δk+1

.

when .(x, s, i, j ) − (y, t, i, j ) < νk by uniform continuity. Then for each k there exists .Nk ∈ Z+ such that .|xn − x| < νk when .n > Nk , so that d((xn , t, i, j ), (x, t, i, j )) < δk+1

.

for .0 ≤ t ≤ 1 when .n > Nk . Observe that .(x, t, i, j ) cannot be in Y for .0 < t ≤ 1. It follows from the definition of .D2 (equation (10.13)) that .(x, t, i, j ) is not in .D2 for .0 < t ≤ 1. It is, however, possible that .(x, t, i, j ) is in .D1 or .D3 for .0 < t ≤ 1. In either case, it follows from equations (10.17) and (10.21) that .g((x, t, i, j )) = t for .0 ≤ t ≤ 1/2. Using the last inequality in the previous paragraph, it follows that f ((xn , t, i, j )) ≤ d((xn , t, i, j ), (x, t, i, j )) + g((x, t, i, j )) ≤ δk+1 + t

.

and ρ((xn , t, i, j )) ≥

.

1 , δk+1 + t

when .0 ≤ t ≤ 1/2 and .n > Nk . For .0 ≤ t ≤ δk , we have .ϕ(xn , t) = (xn , sn + t, i, j ) because .0 < δk < 1/2 and .0 ≤ sn < 1/2 for all k and n. Choose k such that .δk < σ . Then there exists    .N such that .δk < σn and .sn + t < 1/2 when .n > N . When .n > max{Nk , N }, it follows that ρ(ϕ(xn , t)) = ρ((xn , sn + t, i, j )) ≥

.

1 δk+1 + t

for .0 ≤ t ≤ δk because .sn < 1/2 for all n and .δk < 1/2 for all k. Repeating the last calculation from the proof of Beck’s theorem yields 

σn

τn =

ρ((ϕ(xn , t)) dt

.



0 δk −δk+1

≥

ρ(ϕ(xn , t)) dt 

0 δk −δk+1

≥  =

0 δk δk+1

1 dt δk+1 + t

1 ds s

> M, and this contradiction completes the proof of the theorem.



10.2 Constructing Flows from Orbits

327

The proof of the last theorem in this section requires some additional comments about the structural relationship between the finite simplicial Euclidean complexes R and X. Recall that every .x in V is either a .c(·, ·) vertex or a .d(·, ·) vertex of X. If −1 (x) consists of 2 bottom corner vertices of .x is a .c(·, ·) type vertex of X, then . R in different components of R and one top vertex that is not a corner vertex. When −1 (x) consists of 2 top corner vertices of R in different .x is a .d(·, ·) type vertex, . components of R and one bottom vertex that is not a corner vertex. The proof will avoid treating these cases separately by using the ideas in the next paragraph. A bottom {top} corner vertex of .−1 (x) has in its component of R one nearest bottom {top} vertex of R. A bottom {top} non-corner vertex of .−1 (x) has in its component of R one nearest bottom {top} vertex of R on each side of it. These nearest vertices will be called adjacent vertices of .−1 (x). The original vertex in .−1 (x) and an adjacent vertex of it are always the endpoints of an edge called an adjacent edge of .−1 (x). Note that for each vertex of X in V there are 4 adjacent vertices and 4 adjacent edges of .−1 (x). These 4 adjacent edges of −1 (x) form 2 pairs of corresponding top and bottom edges as is evident from . Figures 10.6 and 10.5. Thus the setting of the next proof can simply be an arbitrary vertex .x of X in V and the adjacent edges of .−1 (x). Theorem 10.2.11 The flow .ψ can be extended to a flow on X with a finite set of fixed points by setting .ψ(x, t) = x for all .t ∈ R and .x ∈ V . Proof Obviously, .ψ(x, s + t) = ψ(ψ(x, s), t) and .ψ(x, 0) = x are now valid for all .x ∈ X and t and s in .R, if .ψ is extended to X by setting .ψ(x, t) = x for .x ∈ V and .t ∈ R. So the issue is the continuity of .ψ at .x in V . Given a sequence .yn in .X \ V converging to .x in V and a sequence .τn of real numbers converging to .τ ∈ R, it suffices to prove that .ψ(yn , τn ) converges to .ψ(x, τ ) = x. The key is that .σn = H (yn , τn ) converges to 0 (Theorem 10.2.10). Note that .−1 (yn ) consists of 1 or 2 points. A point of R in .−1 (yn ) can always be written in the form .(yn , sn , in , jn ). When .−1 (yn ) consists of 2 points, there is 1 point of the form .(yn , 0, in , jn ) and 1 of the form .(h−1 (yn ), 1, in , jn ). By agreeing to replace every .(yn , sn , in , jn ) of the second form by its equivalent point of the first form, there exists a sequence .(yn , sn , in , jn ) such that .(yn , sn , in , jn ) = xn for every sequence .xn converging to .x. Therefore, a convergent subsequence of −1 (x) by the continuity of .. .(yn , sn , in , jn ) can only converge to a point of . Because a subsequence is determined by a strictly increasing function of .Z+ into + .Z , every time we pass to a subsequence of .yn , we can and will automatically pass to subsequences of .τn , .σn , etcetera, without changing their relationships such as .σn = H (yn , τn ) or their limits. Since R is compact and the rq sets .Rij decompose R into compact components, it can be assumed, by passing to a subsequence, that .(yn , sn , in , jn ) = (yn , sn , i, j ) and converges to a point in .−1 (x) of either the form .(x, 0, i, j ) or .(x, 1, i, j ), which forces .yn to converge to x and .sn to converge to 0 or 1. Either .yn = 0 for infinitely many n or eventually .yn = 0 and .1 < sn < 0. Assume the former. By passing to subsequences one after another, it can be assumed that:

328

(a) (b) (c) (d)

10 Existence of Transitive Flows

yn = 0 for all n. Either .0 ≤ sn < 1/2 for all n or .0 < 1 − sn < 1/2 for all n. .|σn | < 1/2 for all n (Theorem 10.2.10). Either .(yn , 0, i, j ) is in an open adjacent edge .(x, 0, i, j ), (w, 0, i, j ) for all n or .(yn , 1, i, j ) is in an open adjacent edge .(x, 1, i, j ), (w, 1, i, j ) of .−1 (x) for all n. .

If .(yn , sn , i, j ), satisfying the conditions in the previous paragraph, converges to a point in .−1 (x) of the form .(x, 0, i, j ), then .−1/2 < sn + σn < 1 for all n by (b) and (c) above. The cases .0 ≤ sn + σn < 1 and .−1/2 < sn + σn < 0, however, must be considered separately. Because there are no vertices of R in the open adjacent edge .(y, 0, i, j ), (w, 0, i, j ) of .−1 (x), it follows that g − ((yn , sn , i, j )) ≤ −1 − sn < −1/2 < 1/2 < 1 − sn ≤ g + ((yn , sn , i, j )).

.

When .0 ≤ sn + σn , it follows that .ϕ((yn , sn , i, j ), σn ) = (yn , sn + σn , i, j ). When.−1/2 < sn + σn < 0, it follows that .[sn + σn ] = −1 and .



ϕ (yn , sn , i, j ), σ = (h−1 (yn ), −1)j ) = (h−1 (yn ), sn + σn − [sn + σn ], (h−1 (yn )), F (h−1 (yn ))−1 j ) (h−1 (yn ), sn + σn + 1, (h−1 (yn )), F

 are constant on open top by equations (10.19) and (10.9). The functions . and .F  −1 (h−1 (yn ))−1 j . and bottom edges, so for simplicity set .i = (h (yn )) and .j  = F Observe that (h−1 (x), 1, i  , j  ), (h−1 (w), 1, i  , j  )

.

is both an adjacent edge of .−1 (x) at .(h−1 (x), 1, i  , j  ) ∈ −1 (x) and the unique top edge corresponding with the bottom edge .(x, 0, i, j ), (w, 0, i, j ). Straightforward calculations show that ⎧ ⎪ ⎪ ⎨(yn , sn + σn , i, j ) − (x, 0, i, j ) .

when 0 ≤ sn + σn

lim

n→∞ ⎪ ⎪

⎩(h−1 (y ), s + σ + 1, i  , j  ) − (h−1 (x), 1, i  , j  ) n n n

= lim

⎧ 2 2 ⎪ ⎪ ⎨ (sn + σn ) + (yn − x)

n→∞ ⎪ ⎪

= 0.

⎩ (s + σ )2 + (h−1 (y ) − h−1 (x))2 n

when sn + σn < 0

when 0 ≤ sn + σn when sn + σn < 0

10.2 Constructing Flows from Orbits

329

The continuity of . now implies that .

lim ψ(yn , τn ) = lim ϕ(yn , σn ) n→∞ ⎧ ⎪ when 0 ≤ sn + σn ⎪ ⎨(yn , sn + σn , i, j ) = lim n→∞ ⎪ ⎪ ⎩(h−1 (y ), s + σ + 1, i, j ) when s + σ < 0 n n n n n ⎧ ⎪ when 0 ≤ sn + σn ⎪ ⎨(x, 0, i, j ) = ⎪ ⎪ ⎩(h−1 (x), 1, i  , j  ) when s + σ < 0

n→∞

n

n

= x. Doing the same analysis in the case that .(x, 1, i, j ) is in .−1 (x), the critical distance calculations are ⎧ ⎪ when 0 < sn + σn ≤ 1 ⎪ ⎨(yn , sn + σn , i, j ) − (x, 1, i, j ) . lim n→∞ ⎪ ⎪ ⎩ (h(yn ), sn + σn + 1, i  , j  ) − (h(x), 1, i  , j  ) when 1 < sn + σn < 3/2

= lim

⎧ 2 2 ⎪ ⎪ ⎨ (sn + σn − 1) + (yn − x)

n→∞ ⎪ ⎪

⎩

(sn + σn − 1)2 + (h(yn ) − h(x))2

when 0 < sn + σn ≤ 1 when 1 < sn + σn < 3/2

=0

and lead to the same conclusion. The proof for the case that .yn is eventually 0 is just a slightly different version of the case that .yn = 0, and the details are left to the reader. Finally, any sequence .ψ(yn , τn ) not convergent to .ψ(x, τ ) = x would lead to a  contradiction of one of the cases already established in the proof. Corollary 10.2.12 (a) If .x is in .(D1 ∪ D2 ), then .ω(x) is a point in V . (b) If .x is in .(D1 ∪ D3 ), then .α(x) is a point in V . (c) When .x is not in .∂X, the .ω-limit {.α-limit} set is a fixed point if and only if .x is in .D2 {.D3 }. (d) There are exactly qr orbits in .X\∂X whose .ω-limit {.α-limit} set is a fixed point. Proof It follows from the definition of .H (x, t) that .limt→∞ H (x, t) = g + (x). If .x is in .D1 ∪ D2 , then

330

10 Existence of Transitive Flows .

lim ψ(x, t) = lim ϕ(x, H (x, t)) =

t→∞

t→∞

lim

s→g + (x)

ϕ(x, s)

exists and is a point in V by Lemma 10.2.6. Part (c) follows from parts (a) and (b) because .(D1 ) ⊂ ∂X. Since .(D2 ) and .(D3 ) each consist of exactly qr distinct .ϕ-orbits in .X \ ∂X, they also each consist of exactly qr distinct .ψ-orbits in .X \ ∂X. So part (d) follows from part (c).  The flows constructed using Theorems 10.1.18 and 10.2.11 will be called rectangular T-flows when the surface is connected because these flows are constructed by gluing edges of rectangles together and using the vertical lines of the rectangles to construct the orbits of the flow, and the “T” is the subject of the next section.

10.3 Transitivity The key result in this section is that every rectangular T-flow is a transitive flow because it leads to a wealth of examples of transitive surface flows with the number of boundary components, Euler characteristic, and genus known (Theorem 10.1.23). Although the rectangular T -flows are all on bordered surfaces, they can be used to construct transitive flows on all possible compact connected orientable surfaces with .γ (X) > 1. In fact, this can be accomplished with only a finite number of fixed points. In the context of flows on compact connected bordered surfaces, stating that a flow .(X, R) is transitive is synonymous with X being a recurrent orbit closure of .(X, R) for some recurrent point x in X by the following general result: Proposition 10.3.1 A flow on a compact manifold with or without boundary of dimension at least 2 is transitive if and only if it is a recurrent orbit closure. Proof Let .(X, R) be a flow on a compact n-dimensional manifold with .n ≥ 2. Suppose v is a transitive point of .(X, R) forcing X to be connected by Proposition 1.1.3. Obviously, v is not a fixed point. The function .t → vt is either injective or v is a _ periodic point. Therefore, either .{vt : |t| ≤ k} is homeomorphic to .[0, 1] or .O(v) is homeomorphic to .S1 for large k (Theorem 1.2.1). In either case, if the interior of .{vt : |t| ≤ k} is not empty, then it must contain a set homeomorphic to both an open subset of .Rn , .n > 1, and open interval of .R, contradicting invariance of domain. Now Corollary 1.2.9 implies that X contains a residual set of points that are both transitive and recurrent. Since a recurrent orbit closure obviously contains a point with a dense orbit, Proposition 1.1.3 implies the flow is transitive.  It is easy to see that this result is false for compact 1-dimensional manifolds. Recall from Section 10.1 that the cascade .([0, 1], h) with .h(s) = s + a mod 1 is minimal. Because .ω(x) is always a closed invariant set, .ω(x) = [0, 1] mod 1 for all

10.3 Transitivity

331

x ∈ [0, 1] and every x in .[0, 1] mod 1 is positively recurrent. Likewise, every point of .[0, 1] mod 1 is negatively recurrent. These two facts are used to prove that the rectangular T-flows constructed in Section 10.2 are transitive.

.

Theorem 10.3.2 Let .(X, ψ) be a rectangular T -flow. If .x ∈ X \ V such that g + (x) = ∞ {.g − (x) = −∞}, then .x is a positively {negatively} recurrent point of .(X, ψ).

.

Proof By construction, X is a compact connected orientable bordered surface. Let ϕ be the partial flow defined by equation (10.19). By equation (10.16), .g + (x) = ∞ ) ∪ D3 , and .ϕ|((Y ) ∪ D3 ) is continuous by part (c) of if and only if .x is in .(Y Proposition 10.2.5. It will be sufficient to show that a point in .Oϕ (x) is positively recurrent because .(0, ∞) is in the domain of .ϕ(x, ·). It can be assumed that .x = (x, 0, i, j ) because .x is not in .D1 . There exists a sequence .mk of positive integers such that .hmk (x) converges to x because .([0, 1], h) is minimal and .hmk (x) ∈ / {O(ci ) : i = 1, . . . , r} for all k. By passing to a subsequence, it can be assumed that . hmk (x, j ) converges for .j = 1, . . . q to .(x, βj ) and .β is in .Sq because .(y, j ) − (y, j  ) ≥ 1 for all y when .j = j  . Thus .(x, βj ) is in .ω(x, j ) and .ω(x, βj ) ⊂ ω(x, j ) for all j . Now replacing j with .β n j shows that .ω(x, β n+1 j ) ⊂ ω(x, β n j ) for all j . Since there exists a smallest positive integer .κ such that .β κ = ι, it follows that

.

(x, j ) = (x, Gκ j ) ∈ ω(x, Gκ−1 j ) ⊂ · · · ⊂ ω(x, Gj ) ⊂ ω(x, j )

.

for .j = 1, . . . , q. Thus points .(x, j ), .j = 1, . . . , q of the cascade ([0, 1] \ {O(ci ) : i = 1, . . . , r} × {1, . . . , q},  h)

.

are positively recurrent. ) ∪ D3 , there is a sequence .(hnk (x), 0, ink , jnk ) Given .x = (x, 0, i, j ) in .(Y converging to .(x, 0, i, j ) such that . hnk (x, j ) converges to .(x, j ) that forces .ink = i n k for large k because .h (x) ∈ / {O(ci ) : i = 1, . . . , r} and .jnk = j for large k because (x, nk )j ) = (hnk , jnk ). Using equation (10.19), . hnk (x, j ) = (hnk (x), F (x, nk )j ) ϕ(x, nk ) = (hnk (x), 0, i  , F

.

converges to .(x, 0.i, j ) = x. Finally, ψ(x, h(x, nk )) = ϕ(x, H (x, h(x, nk ))) = ϕ(x, nk )

.

by equations (10.26) and (10.30), and .

lim h(x, nk ) = ∞

k→∞

332

10 Existence of Transitive Flows

by equation (10.25). Therefore, .x is a recurrent point of .(X, ψ).



Proposition 10.3.3 If .(X, ψ) is a rectangular T -flow, then the following hold: (a) (b) (c) (d)

Every point in .(D3 ) {.(D2 )} is positively {negatively} recurrent. ) are recurrent. Only the points in .(Y Only the points in .∂X \ V are neither positively nor negatively recurrent. There are no periodic points that are not fixed points.

Proof Part (a) is an immediate consequence of Theorem 10.3.2. Part (b) is a consequence of Lemma 10.2.6 and Theorem 10.3.2. Part (c) follows from equation (10.15) and Lemma 10.2.12. If .x is a periodic point of .(X, ψ) that is not a fixed + point of .ψ, then .Oψ (x) = Oϕ+ (x) is a compact subset of .X \ V . It is straightforward ), .D1 , .D2 , or .D3 to complete the to check that there are no compact .ϕ-orbits in .(Y proof of part (d).  Theorem 10.3.4 If .(X, ψ) is a rectangular T -flow, then .(X, ψ) is a transitive flow. Proof By Theorem 9.3.7, there exists at most .γ (X) distinct recurrent orbit closures that are not periodic orbits in X. Denote them by .Y1 , . . . , Yn . It follows from part (d) of Proposition 10.3.3 that every point of .X \ ∂X is at least positively or negatively recurrent if not both. Now Maier’s first theorem (Theorem 9.2.1) implies that .Yi \∂X and .Yj \ ∂X are disjoint subsets of .X \ ∂X when .i = j . Clearly, .Yi \ ∂X = Yi ∩ (X \ ∂X), and the sets .Yi \ ∂X are closed subsets of .X \ ∂X. Since the only periodic points are fixed points, .X \ ∂X is the disjoint union of the finite collection .Yi \ ∂X of closed sets, and .X \ ∂X is disconnected if .n > 1. By definition, the underlying space of a rectangular T -flow is connected, and .X \ ∂X is connected by Proposition 2.1.10. Thus .n = 1 and .X = Y1 , a recurrent orbit closure. A dense recurrent point is obviously a transitive point, and the flow is transitive by Proposition 1.1.3.  We now have a rough picture of the dynamics in a neighborhood of a component of the boundary as shown schematically in Figure 10.7. The interior of the hexagon is the hole in the surface. The dots are fixed points on the boundary, and the orbits connecting them are the orbits of the moving points on the boundary whose .ω− and .α− limit sets are distinct fixed points. The orbits approaching a fixed point either negatively or positively represent the positively or negatively recurrent points, respectively, that are not recurrent. Finally, the curved orbits represent the not necessarily distinct recurrent orbits that must swing close to the boundary orbits and the positively and negatively recurrent orbits while also coming close to every other orbit including themselves. Theorem 10.3.5 If .(X, ψ) is a rectangular T-flow with .ζ = ζ (G) boundary components, then there exists a transitive flow .(X∗ , ψζ ) on a compact connected orientable surface of the same genus as X and with a fixed point set F of cardinality θ : (X, ψ) → .ζ . Furthermore, there exists and a surjective homomorphism . (X∗ , ψζ ) such that .θˆ |X \ ∂X is an isomorphism onto .X∗ \ F .

10.3 Transitivity

333

Fig. 10.7 Schematic planar view of a rectangular T -flow near a boundary component

Proof Theorem 2.3.2 applies to .(X, ψ), and there exist a compact connected surface X∗ , an embedding h of X into .X∗ , and regular Euclidean balls .B1 , . . . , Bζ of .X∗ with the following properties:

.

(a) .Bj− ∩ Bk− = φ for .j = k. (b) For each component C of .∂X, there exists a unique j such that h(C) = ∂Bj− = Bj− \ Bj .

.

(c) X∗ \ h(X) =

n 

.

Bj .

j =1

In addition, there exists a flow .(X∗ , ψ ∗ ) such that .h(X) is a closed invariant set of ∗ ∗ .(X , ψ ) and h is an isomorphism of the flow .(X, ψ) onto the flow (X∗ \ (B1 ∪ · · · ∪ Bζ ), ψ ∗ |X∗ \ (B1 ∪ · · · ∪ Bζ )).

.

(These extensions to .Bi introduced a continuum of fixed points in each .Bi , but the second step will destroy all but one of them in each .Bi .) Theorem 2.1.14 applies to .(X∗ , ψ ∗ ) and .B1 with .D1 = B2 ∪ · · · ∪ Bζ to obtain a flow .(X∗ , ψ1 ) and a surjective homomorphism .θ1 : (X∗ , ψ ∗ ) → (X∗ , ψ1 ) such that .θ1 (B1− ) is a .ψ1 -fixed point .y1 ∈ B1 and .θ1 is an isomorphism of the flow .ψ ∗

334

10 Existence of Transitive Flows

restricted to .X∗ \ B1− onto the flow .ψ1 restricted .X∗ \ {y1 } such that .θ1 (x) = x for all .x ∈ D1 . Now Theorem 2.1.14 can be applied to .(X∗ , ψ1 ) and .B2 with .D2 = {y1 } ∪ B3 ∪ · · · ∪ Bζ . This produces a flow .(X∗ , ψ2 ) and a surjective homomorphism .θ2 : (X∗ , ψ1 ) → (X∗ , ψ2 ) such that .θ2 (B2− ) is a .ψ2 -fixed point .y2 ∈ B2 and .θ2 is an isomorphism of the flow .ψ1 restricted to .X∗ \ B2− onto the flow .ψ2 restricted to ∗ .X \ {y2 } such that .θ2 (x) = x for all .x ∈ D2 . Repeating this process .ζ times yields a flow .(X∗ , ψζ ) (see Figure 10.8) with exactly .ζ fixed points. Then  θ = θζ ◦ θζ −1 ◦ · · · ◦ θ1 ◦ h

.

is the required homomorphism, and .(X∗ , ψζ ) is easily seen to be transitive because .(X, ψ) is transitive.  Corollary 10.3.6 The lift of every positive {negative} semi-orbit of the flow (X∗ , ψζ ) to the universal covering flow .(X∗ , ψζ ) is either bounded or goes to infinity.

.

Proof Theorem 8.1.9 applies because .(X∗ , ψζ ) has a finite set of fixed points.



Corollary 10.3.7 For every positive integer .m ≥ 2, there exists a transitive flow (X∗ , ψ2 ) such that .γ (X∗ ) = m.

.

Proof Let .r = 2, .q = m, and a an irrational number in .[0, 1]. Let .(1 . . . q) denote a q cycle. Define F by .F (1) = (1 . . . q) and let .F (2) = ι. Then the flow .(X, ψ) given by Theorem 10.2.9 for these parameters is connected. It follows that .G(i) = F (i − 1)−1 F (i) mod 2 equals .F (1) = (1 . . . q) and .F (2) = (1 . . . q)−1 = (q q −1 . . . 2 1), and X has 2 boundary components, that is, .ζ (G) = 2. It now follows from equation (2.4) that γ (X) =

.

2 − rq − ζ (G) 2 + 2q − 2 2 − χ (X) − r = = = q = m. 2 2 2

(By coincidence, r is the number of boundary components in (2.4), and .ζ (G), the number of boundary components of X, equals r from the surface construction.)  An application of Theorem 2.1.14 completes the proof. Obviously, minimal flows are transitive, so Theorem 1.1.24 provides an example of a transitive flow on the torus with no fixed points. Using Theorem 1.3.3 to add a finite number of fixed points produces transitive flows on the torus with finite fixed point sets. Thus results in Chapter 1 take care of the .m = 1 case to prove: Theorem 10.3.8 There exists a transitive flow with a finite set of fixed points on every compact connected orientable surface of genus at least 1, making these surfaces recurrent orbit closures.

10.3 Transitivity

335

Fig. 10.8 Schematic planar view of the flow .(X∗ , ψζ ) near a fixed point

The construction of rectangular T-flows even works when .q = 1. In this case, Sq = {ι} is trivial, and its orbit of a point it is acting on is just that point. The Euler characteristic is .−r, and there are r boundary components because .G(i) = F (i − 1)−1 F (i) = ι. Hence, .γ (|K|) = (2 + r − r)/2 = 1 establishes:

.

Theorem 10.3.9 Given .r ≥ 2, there exists a transitive flow on a bordered surface of genus 1 with r boundary components and 2r fixed points. The above theorem leads to a little more information about the intersection of recurrent orbit closures. Let .(X, ψ) be a rectangular T-flow given by Theorem 10.3.9 with .r = 2, and collapse one of the boundaries to a fixed point to obtain a transitive flow .(Y, ϕ) with 1 boundary component containing exactly 2 fixed points and 2 orbits.



Consider

2 copies . (Y, i), ϕi of .(Y, ϕ) with .i = {1, 2} and .ϕi (y, i), t = ϕ(y, t), i . Then .σ (y, 1) = (y, 2) is an isomorphism of flows. One consequence of Theorem 2.1.20 is that one can form the connected sum .(Y, 1)#(Y, 2) because the Euclidean ball used to remove the boundary component becomes a regular Euclidean ball of the new surface. The result is a compact connected orientable surface (Proposition 2.1.18). Next one can form the connected sums of the flows by applying Proposition 2.1.16 because the gluing function .σ is an isomorphism of the 2 flows restricted to the 2 boundary curves. We now have an example of a flow on a surface of genus 2 containing 2 recurrent orbit closures whose intersection is an invariant simple closed curve with a finite nonempty set of fixed points. Whenever .G(i)j = F (i − 1)−1 F (i)j = j , which includes when .q = 1, the schematic planar views of a rectangular T -flow near a boundary component differ significantly from the usual even-sided figures with 4 or more sides as shown in Figures 10.7 and 10.9. Similarly, in .(X∗ , ψ ∗ ), the schematic picture of the orbits near a fixed point coming from a 2-sided boundary component satisfying .F (i − 1)−1 F (i)j = j is not as complex as Figure 10.8 as seen in Figure 10.10.

336

10 Existence of Transitive Flows

Fig. 10.9 View of a rectangular T -flow near a boundary component when .G(i)j = j

Fig. 10.10 View of the flow ∗ , ψ ) near a fixed point ζ where .G(i)j = j

.(X

If Figures 10.8 and 10.10 were phase plane diagrams for the solutions of autonomous differential equations, the indices of the fixed points would be .−2 and 0, which is the same as minus the number of positively recurrent orbits whose .αlimit set is the fixed point plus 1. This leads to a definition. Given a rectangular T-flow .(X, ψ), define the index of a fixed point w of ∗ .(X , ψζ ), using .| · | to denote cardinality, by   Ind(w) = −  O(x) : x = w, x ∈ ω(x), and α(x) = {w}  + 1.

.

(10.32)

Theorem 10.3.10 If .(X, ψ) is a rectangular T-flow with .ζ = ζ (G) boundary components and .(X∗ , ψζ ) is obtained from .(X, ψ) by Theorem 10.3.5 with fixed point set F , then .

Ind(w) = χ (X∗ ).

(10.33)

w∈F

Proof By Corollary 10.2.12 and Proposition 10.3.3, the orbits of the flow .(X, ψ) that are positively recurrent and not negatively recurrent coincide with the orbits whose .α-limit set is a single fixed point and there are exactly qr such orbits. The construction of .(X∗ , ψ ∗ ) only affects the orbits whose .α- and .ω-limit sets are single distinct fixed points. The flow .(X∗ , ψ ∗ ) has exactly .ζ fixed points because boundary components are transformed into fixed points in the construction. Therefore, . Ind(w) = −qr + ζ. w∈F

10.3 Transitivity

337

By Corollary 10.1.19, .χ (X) = −qr, and the Euler characteristic of .X∗ can be calculated as follows: χ (X∗ ) = 2 − 2γ (X∗ )

.

= 2 − 2γ (X)   2 − χ (X) − ζ = 2−2 2 = χ (X) + ζ = −qr + ζ because .ζ is also the number of boundary components of X.



The usual setting for this important classical theorem known as the Poincaré– Hopf theorem is a smooth vector field with a finite number of zeros on a compact smooth manifold. (Here Hopf refers to Heinz Hopf. There is also an Eberhard Hopf who made significant contributions to ergodic theory. They are both important twentieth century mathematicians.) Its significance is that there is a connection between the topology of smooth manifolds and their smooth vector fields. Theorem 10.3.10 obtains a similar connection between the topology of compact connected orientable surfaces and the fixed points of a rectangular T-flows on them. (For an introduction to the classical Poincaré–Hopf theorem, see Milnor’s book [64], which is dedicated to Heinz Hopf.) The equation  2  2 2 . R − x +y + z2 − r 2 = 0

(10.34)

when .R > r > 0 in .R3 is the equation of a torus in .R3 . The doughnut hole is perpendicular to the z-axis and centered at the origin. The intersection of the torus and the yz-plane consists of the 2 circles (y ± R)2 + (z − 0)2 = r 2

.

(10.35)

of radius r, and the circle .x 2 + y 2 = R 2 in the xy-plane is at the center of the torus. It follows from the implicit function theorem that !

" 2   2 2 2 2 .X = (x, y, z) : R − x + y +z −r =0 is a surface that is clearly compact and connected. Equation (10.34) is invariant under .(x, y, z) → (−x, y, z), .(x, y, z) → (x, −y, z), and.(x, y, z) → (x, y, −z). Thus the points satisfying this equation are symmetric with respect to the xy-plane, xz-plane, and zy-plane. Furthermore,

338

10 Existence of Transitive Flows

T (x, y, z) = (−x, −y, −z) is a homeomorphism of X onto itself such that .T 2 = ι. Since T has no fixed points in X, the group . = {T , ι} acts freely and properly on X, and .π : X → X/  is a covering map. The yz-plane divides X into 2 pieces .X+ and .X− depending on whether .x ≥ 0 or .x ≤ 0. Both .X+ and .X− are cylinders. The common boundary of .X+ and .X− is the 2 circles of radius r defined by equation (10.35). Observe that .T X+ = X− and .T X− = X+ and that T interchanges the two boundary circles in the yz-plane, reversing their circular orders. Thus .X+ is a fundamental region for the group . and .X/  is a compact connected surface. Moreover, .X/  is a Klein bottle because the orders of the ends of .X+ are reversed, and .X/  is not orientable. Next we extend this construction to compact surfaces of higher genus. Let c be a positive real number such that .R + r/2 < c < R + r. Now let .X1 = {(x, y, z) ∈ X : x ≤ c} and .X2 = {(x, y, z) ∈ X : x ≥ −c}. Then .

Y2 = X1 + (−R − r + c, 0, 0) ∪ X2 + (R + r − c, 0, 0)

.

is a compact connected orientable surface of genus 2. (Checking the details is tedious but not difficult.) In other words, in this setting, we can construct the connected sum .T2 #T2 in a concrete way. Setting .X3 = {(x, y, z) ∈ X : −c ≤ x ≤ c}, another relevant example is Y3 = X3 ∪ X1 + (−2(R + r) + 2c, 0, 0) ∪ X2 + (2(R + r) + 2c, 0, 0),

.

which is a copy of .T2 #T2 #T2 embedded in .R3 and a compact connected orientable surface of genus 3. Both examples are symmetric with respect to the xy-, xz-, and yz-planes. There is, however, an important difference between them, namely, the .x = 0 plane intersects .Y2 in a simple closed curve and .Y3 in 2 simple closed curves. This process can be continued to construct a sequence .Ym of compact connected orientable surfaces of genus m in .R3 that are invariant under .(x, y, z) → (−x, y, z), .(x, y, z) → (x, −y, z), and .(x, y, z) → (x, y, −z). Setting .Y1 = X, we have sequence in .R3 of all compact connected orientable surfaces, that is, of the form .T2 #T2 # . . . #T2 #T2 . The function .T (x, y, z) = (−x, −y, −z) is a homeomorphism of .Ym onto itself, and .G = {T , ι} acts freely and properly on .Ym . The intersection of .Ym with the yz-plane is a simple closed curve or 2 simple closed curves depending on whether m is even or odd. Let .Tm = T |Ym and .m = {Tm , ι}. Then the natural projection .πm : Ym → Ym / m is a 2-sheeted covering of a compact connected surface. It can be shown that .Ym / m is nonorientable for all m by identifying a band B in .Ym such that .πm (B) is a Möbius band in .Ym / m . Theorem 10.3.11 There exists a transitive flow on every compact connected nonorientable surface of genus at least 3, making these surfaces recurrent orbit closures. Proof If m is a positive integer, then .Ym+ = {(x, y, z) ∈ Ym : x ≥ 0} is a bordered surface of genus .[m/2] and the values of the genus of .Ym+ come in pairs for .m ≥

10.4 Locally Circular Cascades and Flows

339

2. Since .Ym double covers a compact connected nonorientable surface of genus m + 1 (Proposition 9.3.9), we can start with .m = 2. Corollary 10.3.7 implies the existence of transitive flows on compact connected orientable bordered surfaces of genus .m ≥ 1 with 2 or more boundary curves. Using the technique used in the proof of Theorem 10.3.5 to replace one or more of the boundary components with a fixed point, there exists a transitive flow on a compact connected orientable bordered surface of genus .[m/2] with 1 or 2 boundary component(s). It follows from the classification of bordered surfaces that there exists a transitive flow .(Ym+ , ϕ) for all .m ≥ 2. By Proposition 10.3.1, .(Ym+ , ϕ) is a recurrent orbit closure. Now apply Theorem 1.3.3 to the boundary of .Ym+ to obtain a flow .(Ym+ , ϕ  ) such that every point in the boundary of .Ym+ is a fixed point of .ϕ  . Since the recurrent points of .(Ym+ , ϕ) cannot be on the boundary, they remain recurrent after the time change. Therefore, +  .(Ym , ϕ ) is also a recurrent orbit closure. Next observe that ! ϕ  ((x, y, z), t) when (x, y, z) is in Ym+ .ψ((x, y, z), t) = T ϕ  (T (x, y, z), t) when (x, y, z) is in Ym− .

defines a flow on .Ym because T ◦ ϕ  ◦ (T × ι) : Ym− × R → Ym+

.

is a flow on .Ym− that agrees with the flow .ϕ  on .Ym+ ∩ Ym− . It is easy to check that T is an automorphism of .(Ym , ψ) that interchanges .Ym+ and .Ym− , making .(m , Ym , ψ) a bitransformation. Thus .(Ym , ψ) consists of 2 orbit recurrent closures whose intersection is .Ym+ ∩ Ym− . From general considerations, we know that .π : Ym → Ym / m is a double covering of compact connected nonorientable surface of genus .m + 1. Because  .(m , Ym , ψ) is a bitransformation, there is a flow .(Ym / m , ψ ) such that .π  is a homomorphism of .(Ym , ψ) onto .(Ym / m , ψ ) (Proposition 1.1.23). Now Theorem 2.2.6 implies that .(Ym / m , ψ  ) is recurrent orbit closure, and hence, a  transitive flow on a compact connected nonorientable surface of genus .m + 1. Since it has already been shown that there are no transitive flows on the projective plane and the Klein bottle, Theorem 10.3.11 is the best possible result for the existence of transitive flows on compact connected nonorientable surfaces.

10.4 Locally Circular Cascades and Flows Recall that a nontrivial minimal set is a closed nowhere dense minimal set, but not a periodic orbit, of a flow on a compact surface (Theorem 5.3.18 ). The minimal set of a Denjoy flow on the torus (Section 3.3) is an example of a nontrivial minimal set. In

340

10 Existence of Transitive Flows

this section, a class of more general nontrivial minimal sets will be constructed on compact connected orientable surfaces of arbitrary genus. The process begins with the construction of a special family of minimal cascades, as did the construction of the Denjoy flow on the torus. Let h be homeomorphism of .S1 onto itself such that .(S1 , h) is a Denjoy cascade and its Denjoy minimal set is a Cantor set C. For convenience, .(C, h) will denote the minimal cascade obtained from restricting h to C. As in the earlier sections of this chapter, .S1 will be written in its .[0, 1] mod 1 form. Given a positive integer r, by Theorem 3.3.5, we can assume that there exist at least r complementary intervals n .(ai , bi ) with .1 ≤ i ≤ r of C such that .h [(ai , bi )] = (aj , bj ) for all .n ∈ Z and .i = j . In other words, there are at least r distinct orbits of complementary intervals and at least r distinct pairs of doubly asymptotic orbits in C. By renumbering them, we can assume that they are arranged in counterclockwise order. Now consider a continuous function .F : C → Sq , the permutation group of q symbols, such that F is constant on the closed intervals [b1 , a2 ] ∩ C, [b2 , a3 ] ∩ C, . . . , [br , a1 ] ∩ C

.

(10.36)

and .F (ai ) = F (bi ) for .1 ≤ i ≤ r. Set .Cq = C × {1, . . . , q} and define  h(x, i) = (h(x), F (x)i),

.

(10.37)

where .F (x)i is the permutation .F (x) applied to the symbol i. If you calculate  hn (x, i) for small n, you see a cocycle appearing like equation (10.4). The inverse of . h is

.

 h−1 (x, i) = (h−1 (x), F (h−1 (x))−1 i).

.

(10.38)

Giving .{1, . . . , q} the discrete topology and .Cq the product topology, the sets .C ×{i} are open and closed subsets of .Cq for .1 ≤ i ≤ r and the functions . h and . h−1 are  continuous and .(Cq , h) is a cascade. Using .(x, y)-coordinates, there is a simple planar picture of .Cq that provides a convenient Euclidean metric. In the analysis of .(Cq ,  h), the cardinality of finite sets will be used. The cardinality of a finite set A will be denoted by .|A|. Let .π be the projection .π(x, i) = x. Clearly, .π is a q-to-1 closed homomorphism of .(Cq ,  h) onto .(C, h). h) has the properties that Since .(C, h) is minimal, every minimal set M of .(Cq ,  .π(M) = C and that .|{i : (x, i) ∈ M}| > 0 for each x in C. The rest of this section will be devoted to three projects. The first is to find h) to be minimal. The second is to necessary and sufficient conditions for .(Cq ,  embed the suspension .S(Cq ,  h) in a compact connected bordered surface X. The h), R) to X. third is to extend the flow .(S(Cq ,  The techniques used in Sections 10.1 and 10.2 will be reused in the second and third projects. We will not, however, present the entire proof, rather just the parts that differ substantially from Sections 10.1 and 10.2. The result will be the family

10.4 Locally Circular Cascades and Flows

341

of nontrivial minimal sets constructed in [50] and [55] that are different from the minimal sets of Denjoy flows. Proposition 10.4.1 Let x be in C, let .nk be a sequence of integers such that  h). Then hnk (x, i) converges for .1 ≤ i ≤ q, and let M be a minimal set of .(Cq ,  the following hold:

.

(a) There exists y in C and .β in .Sq such that .

lim  hnk (x, i) = (y, βi).

k→∞

(b) If .(x, i) is in M, then .(y, βi) is in M and |{j : (x, j ) ∈ M}| ≤ |{j : (y, j ) ∈ M}|.

.

(c) If .(x, i) is almost periodic, then .(y, βi) is almost periodic and |{j : (x, j ) is almost periodic}| ≤ |{j : (y, j ) is almost periodic}|.

.

Proof It follows from the construction of . h that for each k there exists a permutation βk such that . hnk (x, i) = (hnk (x), βk i). Since the sets .C × {j } are disjoint open and closed sets of .Cq , for each k the q points . hnk (x, i) are distributed precisely one to n each .C × {j } for .1 ≤ j ≤ q. Since .(h k (x), βk i) converges for .1 ≤ i ≤ q, the limit points .limk→∞  hnk (x, i) for .1 ≤ i ≤ q are distributed precisely one to each .C × {j } for .1 ≤ j ≤ q, that is, the function .i → j if and only if .limk→∞  hnk (x, i) is in .C × {j } is a bijection .β and in .Sq . Letting .y = limk→∞ x, we have .limk→∞  hnk (x, i) = (y, βj ) to complete part (a). For part (b), observe that .(y, βi) is in M whenever .(x, i) is in M. The inequality is now obvious because .β is a permutation. Recall that for a compact metric space a point is almost periodic is equivalent to its orbit closure being a minimal set by Theorem 1.1.11 from which the first part of (c) follows. Because minimal sets are either disjoint or identical, the rest of part (c) follows from part (b) by summing over the different minimal sets intersecting the set .{(x, i) : 1 ≤ i ≤ q}. 

.

Corollary 10.4.2 If M is a minimal set of .(Cq ,  h), then |{i : (x, i) ∈ M}| = |{i : (y, i) ∈ M}|

.

for all x and y in C. Furthermore, |{i : (x, i) is almost periodic}| = |{i : (y, i) is almost periodic}|

.

for all x and y in C.

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10 Existence of Transitive Flows

Proof For every x and y in C, there exist a sequence of integers .nk such that .hnk (x) converges to y because .(C, h) is a minimal cascade. By passing to a subsequence, . hnk (x, i) converges for .1 ≤ i ≤ q. Thus the proposition is symmetric in x and y.  Proposition 10.4.3 If .(a, b) is a complementary interval of C, then .(a, i) and .(b, i) are positively asymptotic, negatively asymptotic, or doubly asymptotic. Proof Recall that a and b are doubly asymptotic in .(C, h). There exists at most one m ∈ Z such that .hm [(a, b)] = (ai , bi ) for some i such that .1 ≤ i ≤ r. If .m < 0, then .(a, j ) and .(b, j ) are positively asymptotic for .1 ≤ j ≤ q because .F (hn (a)) = F (hn (b)) for all .n ≥ 0. (To see why .F (hn (a)) = F (hn (b)) for all .n ≥ 0 implies positive asymptotic, one needs to calculate the cocycle used evaluating the second term of . hn (x, i), which is similar to equation (10.4).) The same basic argument shows that .(a, i) and .(b, i) are negatively asymptotic when .m ≥ 0. If there does not exist m such that .hm [(a, b)] = (ai , bi ) for some i, then .(a, j ) and .(b, j ) are doubly asymptotic for .1 ≤ j ≤ q because .F (hn (a)) = F (hn (b)) for all .n ∈ Z. 

.

Proposition 10.4.4 Every point of .Cq is an almost periodic point of the cascade (Cq ,  h).

.

Proof Suppose .(x, j ) ∈ Cq is not almost periodic. Since .O(x, j )− contains a minimal set, there exists a sequence .nk of integers such that . hnk (x, j ) converges  to an almost periodic point .(y, j ). Without loss of generality, we can assume that . hnk (x, i) converges for all i. Part (a) of Proposition 10.4.1 implies that there exists .β in .Sq such that .limk→∞  hnk (x, i) = (y, βi). In particular, .βj = j  . If .(x, i  ) is  almost periodic, the .(y, βi ) must be almost periodic. Therefore, |{i : (x, i) is almost periodic}| + 1 ≤ |{i : (y, i) is almost periodic}|,

.

contradicting Corollary 10.4.2.



So each .(Cq ,  h) is a pointwise almost periodic cascade. Pointwise almost periodic continuous group actions on compact metric spaces are always the disjoint union of minimal sets. Because the projection .π(x, i) = x maps a minimal set of .(Cq ,  h) onto C, there are at most q minimal sets of .(Cq ,  h). So the minimal sets h) are closed and open disjoint sets of .Cq . When the cascade .(Cq ,  h) is of .(Cq ,  minimal, it will be called a locally circular minimal cascade. Theorem 10.4.5 The cascade .(Cq ,  h) is minimal if and only if the subgroup of .Sq generated by the range of F acting on .{1, . . . , q} has exactly 1 orbit. Proof Let .(a, b) be a complementary interval of C. Then .(a, i) and .(b, i) are asymptotic in at least one direction (Proposition 10.4.3), and there exists a sequence . hnk such that .limk→∞  hnk (a, i) = (x, j ) = limk→∞  hnk (b, i). Thus .(a, i) and .(b, i) are in the same minimal set because .(Cq ,  h) is pointwise almost periodic by Proposition 10.4.4.

10.4 Locally Circular Cascades and Flows

343

Number the minimal sets of .(Cq ,  h) from 1 to p. Define .fi : C → {1, . . . , p} by .fi (c) = k if and only if .(c, i) is in the .k th minimal set of .(Cq ,  h). Since the minimal sets that intersect .C × {i} partition .C × {i} for fixed i into disjoint closed sets, these sets are also open sets of .Cq . Hence, .fi−1 (k) is an open set of C and .fi is a continuous function. Let .κ : S1 → S1 be a Cantor function for C (Corollary 3.3.3). Recall that .κ(a  ) = κ(b ) if and only if .(a  , b ) is a complementary interval of C. Since .fi (a  ) = fi (b ) when .(a  , b ) is a complementary interval of C (Proposition 10.4.3), .fi passes to the quotient and there exists a continuous function .fi : S1 → {1, . . . , p} such that .fi = fi ◦ κ. Because .S1 is connected, .f  and hence f are constant functions. h) for .i = 1, . . . , q, which Therefore, .C × {i} is contained in a minimal set of .(Cq ,  will lead to the existence of a bijective correspondence between the minimal sets of .(Cq ,  h) and the orbits of the range of F acting on .{1, . . . , q}. From the definition of the function .F : C → Sq , it is clear that the group generated by the range of F is the group .[F (a1 ), . . . , F (ar )]. Then each of the following sets: C × {i}, C × {F (aj1 )i}, C × {F (aj2 )F (aj1 )i}, . . . , C × {F (ajn ) . . . , F (aj2 )F (aj1 )i}

.

is contained in a minimal set of .(Cq ,  h). The subgroup .[F (a1 ), . . . , F (ar )] of .Sq acts on .{1, . . . , q}. For each i in .{1, . . . , q}, there is an orbit .[F (a1 ), . . . , F (ar )]i of i, and the orbits are disjoint. It follows from the above that given i the sets .C × j and .C × i are in the same minimal set when j is in the orbit of i. Conversely, if .C ×{j } and .C ×{i} are in the same minimal set, then, given .x ∈ C, there exists .n > 0 such that . hn (x, i) is in the open set .C × {j } because .(x, j ) and .(x, i) are in the same minimal set and .C × {j } is an open set. Then  hn (x, j ) = (hn (x), F (hn−1 (x))F (hn−2 (x)) · · · F (h(x))F (x)i),

.

that is, j = F (hn−1 (x))F (hn−2 (x)) · · · F (h(x))F (x)i

.

and j is in the i orbit of the group .[F (a1 ), . . . , F (ar )]. Therefore, there is a bijective correspondence between the minimal sets of .(Cq ,  h) and the orbits of .[F (a1 ), . . . , F (ar )] acting on .{1, . . . , q} such that the minimality of .(Cq ,  h) and the group action of .[F (a1 ), . . . , F (ar )] on .{1, . . . , q} having exactly 1 orbit.  By choice, the complementary intervals .(ai , bi ) of C are precisely those such that .F (ai ) = F (bi ). Note, however, that .F (ai ) = F (bi ) does not guarantee that .F (ai )j = F (bi )j for all j . If .F (ai )j = F (bi )j , then .(ai , j ) and .(bi , j ) of .Cq are also positively asymptotic because .F (hn (ai )) = F (hn (bi )) for all .n ≥ 1. So .F (ai )j = F (bi )j is the interesting case.

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10 Existence of Transitive Flows

We are also assuming that .hn [(ai , bi )] = (ai  , bi  ) for all n when .i = i  . Starting with a complementary interval .(ai , bi ) of C and a specific j such that .1 ≤ j ≤ q, the points .(ai , j ) and .(bi , j ) of .Cq are negatively asymptotic by Proposition 10.4.3. Since .F (x) is in .Sq , for each .x ∈ C, the function .f is a bijection of .{(x, i) : 1 ≤ i ≤ q} onto .{(h(x), j ) : 1 ≤ i ≤ q}. Thus when .F (ai )j = F (bi )j , there exists k such that .F (ai )k = F (bi )j . Consequently, .(ai , k) and .(bi , j ) are positively asymptotic, and the permutation .G(i) = F (ai )−1 F (bi ) links them by .G(i)j = k. Note that .G(i)j = j when .F (ai )j = F (bi )j and .G(i) = ι when .F (ai ) = F (bi ). This sets the stage for the next step. Starting at .(ai , j ), we will construct an asymptotic cycle. The first link is the negatively asymptotic pair .(ai , j ) and .(bi , j ). Applying . h to .(bi , j ), the second link is the positively asymptotic pair .(h(bi ), F (bi )j ) and .(h(ai ), F (bi )j ). Applying . h−1 produces the negatively asymptotic pair .(ai , F (ai )−1 F (bi )j ) and −1 .(bi , F (ai ) F (bi )j ) or more simply .(ai , G(i)j ) and .(bi , G(i))j ). Unless .G(i)j = F (ai )−1 F (bi )j = j , the process continues. If the construction does not stop, then the next 2 links are the negatively asymptotic pair .(ai , G(i)j ) and .(bi , G(i)j ) and the positive asymptotic pair .(h(bi ), F (bi )G(i)j ) and .(h(ai ), F (bi )G(i)j ). The next step is the negatively asymptotic pair .(ai , G2 (i)j ) and .(bi , G2 (i)j ) provided .G2 (i)j = j . There exists a smallest positive integer k such that .Gk (i)j = j and the G orbit of j contains k points starting with j . So the first link is the negatively asymptotic pair .(ai , j ) and .(bi , j ), and the last link is the positively asymptotic pair .(h(bi ), F (bi )Gk−1 (i)j ) and .(h(ai ), F (bi )Gk−1 (i)j ). Each asymptotic cycle contains k pairs of negatively asymptotic pairs of points and k pairs of positively asymptotic pairs of points. Furthermore, the asymptotic cycles associated with the complementary interval .(ai , bi ) are completely determined by the orbits of .G(i) acting on .1 ≤ j ≤ q. The orbits of .G(i) also determine the decomposition of .G(i) into disjoint commuting cycles, and the number of orbits is the #number cycles in the decomposition. The latter was denoted by .ν(i) and .ζ (G) = ri=1 ν(i). The overarching goal for the rest of the section is to use the locally circular minimal sets and ideas from Sections 10.1 and 10.2 to construct a large class of flows with interesting nontrivial minimal sets isomorphic to the suspension of a locally circular minimal set. To start, let r be a positive integer greater than 1, and then by Theorem 3.3.5, there exists a Denjoy cascade .(S1 , h) having a Cantor minimal set C with r, or more if desired, complementary intervals .(ai , bi ), .1 ≤ i ≤ r of C such that .hn [(ai , bi )] = (aj , bj ) for all .n ∈ Z and .i = j . In other words, there are at least r distinct orbits of complementary intervals and at least r distinct pairs of doubly asymptotic orbits in C that will be our focus here. By renumbering them, we can assume that they are in consecutive order mod 1. Let .ci be the center of the complementary interval .(ai , bi ). By conjugating with a rotation, it can be assumed that .c1 = 0. Using the notation of Section 10.2, observe that the set

10.4 Locally Circular Cascades and Flows

345

 Y = [0, 1] \

r 

.

 O(ci )

(10.39)

i=1

is an h-invariant set that includes the Denjoy minimal set C and that .(Y, h) is a cascade on a non-compact space. The starting point is again the two sets E = {c1 , . . . , cr } ∪ {h−1 (c1 ), . . . , h−1 (cr )} ⊂ [0, 1)

.

and E  = {c1 , . . . , cr } ∪ {h(c1 ), . . . , h(cr )} ⊂ [0, 1)

.

and the special relationship between the two types of points in each one as shown in Figure 10.1. In this context, the following version of Proposition 10.1.1 holds: Proposition 10.4.6 The function .h(s) = s+a mod 1 has the following properties: (a) The function h maps E bijectively to .E  . (b) If b and d are consecutive points mod 1 of E, then .h(b) and .h(d) are consecutive points mod 1 of .E  . (c) If b and d are consecutive points mod 1 of E, then h maps the closed interval .[b, d] mod 1 homeomorphically to the closed interval .[h(b), h(d)] mod 1. (Can be in 2 pieces.) (d) The function h provides a bijective correspondence between the .2 r closed intervals whose endpoints are consecutive points mod 1 of E with the .2 r closed intervals whose endpoints are consecutive points mod 1 of .E  . Note that the lengths of the complementary intervals are no longer equal in item (c) because it is no longer true. Except for Proposition 10.1.14, however, most of the remaining construction in Section 10.1 can still be used. Moreover, Proposition 10.1.14 shows that the translation that allows the use of the simplicial map for irrational rotations also maps .(x, 1, i, j ) to .(h(x), 0, i  , F (i)j ), which points in the right direction. As in Section 10.1, set .I1 = [c1 , c2 ] = [0, c2 ], I2 = [c2 , c3 ], . . . , Ir = [cr , 1], and set .Ri = Ii × [0, 1] × {i} so that the rectangles .Ri are compact and have disjoint vertical edges .{ci } × [0, 1]. Then set .Ri,j = Ri × {j } for .1 ≤ j ≤ q so R=

q  r 

.

Rij ,

j =1 i=1

and points in R can again be written as .(x, t, i, j ). Except for using a Denjoy cascade instead of an irrational rotation, the basic structure here is the same as in as Section 10.1. The rest of this section extends the ideas and results of Section 10.1 to Denjoy cascades.

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10 Existence of Transitive Flows

Next consider a continuous function .F : C → Sq , the permutation group of q symbols, such that F is constant on the consecutive open intervals .(c1 , c2 ), (c2 , c3 ), . . . , (cr , c1 ) and .F (ai ) = F (bi ) for .1 ≤ i ≤ r. Such functions certainly satisfy the conditions on p. 340. Letting .Cq = C × {1, . . . , q}, equation (10.37) defines a locally circular cascade on .Cq . By Theorem 10.4.5, q and F can be selected so that .(Cq ,  h(x, i)) is a minimal locally circular cascade. Now the function . h(x, i) = (h(x), F (x)i) can be extended to a homeomorphism of .Y × {1, . . . , q} onto itself, so .(Y × {1, . . . , q},  h ) is another cascade. In addition, part (d) of Proposition 10.4.6 still implies: Corollary 10.4.7 Mapping the top edge .(β1 , 1, i, j ), (β2 , 1, i, j ) to the bottom edge .(h(β1 ), 0, i  , F (i)j ), (h(β2 ), 0, i  , F (i)j ) is a bijective correspondence between the top edges of R and the bottom edges of R. The foundation is now laid for triangulating R, defining the abstract simplicial complex .K, and proving that the finite Euclidean simplicial complex .|K| is a surface exactly as done in Section 10.1. These facts will be useful, but not the whole story. The surface, however, does not connect the flow lines of the suspension of a locally circular minimal set. Without these flow lines connecting properly, the proofs in Section 10.2 are worthless. Modifying the construction of the surface so the orbits connect properly is the crucial part of this section. For the present discussion, think of the points in E as simply markers for the vertices of .[0, 1] for the top edge so that 2 consecutive markers in the same component of R are an edge. Denote the points of .[0, 1] by s running from 0 to 1, that is, the identity parameterization of .[0, 1]. Now this parameterization of .[0, 1] can be broken up into distinct parameterizations of each top edge. Starting with the simplicial complex R from the previous section, consider a specific top edge .(β1 , 1, i, j ), (β2 , 1, i, j ) in .Rij . It is parameterized by .ξ(s) = (s, 1, i, j ) on the closed interval .[β1 , β2 ], and then its corresponding bottom edge    .(h(β1 ), 0, i , F (i)j ), (h(β2 ), 0, i , F (i)j ) is naturally parameterized as .ξ (s) =  (h(s), 0, i , F (i)j ) for s in .[β1 , β2 ]. Given 2 consecutive points .β1 and .β2 in E, they uniquely determine (β1 , β2 ) = (β1 , 1, i, j ), ((ci + ci+1 )/2, 2/3, i, j ), (β2 , 1, i, j ) ,

.

which is the only triangle with .(β1 , 1, i, j ), (β2 , 1, i, j ) as an edge, when .(β1 , 1, i, j ), (β2 , 1, i, j ) is a top edge of .Rij . For simplicity, let .((ci + ci+1 )/2, 2/3, i, j ) = u(i, j ). The straight line segments in R beginning at .u(i, j ) and ending at .ξ(s) for .s ∈ [β1 , β2 ] are a fan of lines emanating from .u(i, j ) and covering .(β1 , β2 ). Let .u(i, j ) and .(s, 1, i, j ), with s fixed in .[β1 β2 ] be the endpoints of an element of this fan of line segments. Although not a simplex of .Rij , it is a 1-simplex that could be added to the simplicial complex R with the addition of a vertex and an edge. Also

.t → t (s, 1, i, j ) − u(i, j ) + u(i, j ) is an affine function that parameterizes this line segment .0 ≤ t ≤ 1. Similar comments apply to the line segment with endpoints   .(h(s), 0, i , F (i)j ) and .u(i , F (i)j ) in .|K|.

10.4 Locally Circular Cascades and Flows

347

Recall that .|K| is a finite Euclidean simplicial complex, but in a higher dimension. Thus every triangle is in a Euclidean plane, and it makes sense to consider an affine function mapping a line in such a plane onto another such line. In this context, the affine function .t





(s, 1, i, j ) − u(i, j ) + u(i, j ) → t (h(s), 0, i  , F (i)j ) − u(i  , F (i)j + u(i  , F (i)j )

is a simplicial map of the 1-simplex .u(i, j ), (s, 1, i, j ) onto the 1-simplex u(i  , F (i)j ), (h(s), 0, i  , F (i)j ) because it maps the vertices of the first simplex onto the vertices of the second with .t = 0 .t = 1. Now define . : (β1 , β2 ) → (β1 , 1, i, j ), u(i, j ), (β2 , 1, i, j ) by

.



 t (s, 1, i, j ) − u(i, j )) + u(i, j ) .

= t (h(s), 0, i  , F (i)j ) − u(i  , F (i)j + u(i  , F (i)j )

.

(10.40) (10.41)

for .(s, t) ∈ [β1 , β2 ] × [0, 1]. (Here the coordinates .(s, t) are like polar coordinates (r, θ ) with t meaningless when .s = 0, which can be resolved by passing to a quotient.) The following are left as exercises:

.

(a) . is a homeomorphism of a top triangle .(β1 , β2 ) of R onto the triangle .(β1 , 1, i, j ), u(i, j ), (β2 , 1, i, j ). (b) The functions . and . are equal on the edges .(β1 , 1, i, j ), u(i, j ), and .(β2 , 1, i, j ), u(i, j ). (c) . maps each point .(s, 1, i, j ) in the edge .{(s, 1, i, j ) : s ∈ [β1 , β2 ]} to the point   .(h(s), 0, i , F (i)j ) in the edge .{(h(s), 0, i , F (i)j ) : s ∈ [β1 , β2 ]}. (d) If A = {(β1 , β2 ) : β1 and β2 are consecutive elements of E}

.

and .R is any triangle in the triangulation of R, then the function ! 

 =

.



when R ∈ A



R ∈ /A

is continuous. These properties set the stage for the following result: Theorem 10.4.8 If .(C, f) is a locally circular minimal cascade, then . (R) is a compact connected bordered orientable surface such that the number of boundary components equals .ζ (G), the number of boundary components of .|K|, and γ ( (R)) =

.

2 + qr − ζ (G) = γ (|K|). 2

(10.42)

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10 Existence of Transitive Flows

Therefore, . (R) is homeomorphic to .|K|. Proof There is no change in the vertices, edges, triangles, and boundary curves of  (R). All it does is change the identification of the points in the top open edges of R so that .(s) is identified with .(h(s)) in the corresponding bottom edge of R. In the process, . moves the interior points of . ∈ A around differently than  . did. In fact, . is not a simplicial map of R onto .|K|, but the invariants of the surface are unchanged because the triangular pieces fit together in the same way. (Figures 10.4, 10.5, and 10.6 again show how the pieces fit together to form a surface.)  .

In the rest of this section, X will be the compact connected orientable surface  (R). The function F coming from the construction of the locally circular cascade  mod 1 on Y (equation .(Cq ,  h) can obviously be extended to a continuous function .F  can be (10.39)). Since .Y ∩ O(ci ) = φ for .i = 1, . . . , r, the extended function .F    used to extend .h to .Y × {1, . . . , q} by setting .h(y, j ) = (h(y), F (y)j ). Clearly, . h is a homeomorphism of .Y × {1, . . . , q} onto itself, so .(Y × {1, . . . , q},  h ) is another cascade.  can also be continuously extended to .F  : Y × Z → Sq Moreover, the function .F using the cocycle formulas on p. 314. Then equations (10.7) and (10.8) hold in the present context. Obviously, .Cq is a compact invariant set of .(Y × {1, . . . , q},  h ) by the choice of .c1 , . . . , cr , so the locally circular minimal .(Cq ,  h) is a sub-cascade of .(Y × {1, . . . , q},  h ). As in Section 10.2 with the locator function .(y) = i if and only if .y ∈ Ii , .

 = {(y, s, (y), j ) : y ∈ Y, 0 ≤ s ≤ 1, and j = 1, . . . , q} Y

.

is a dense subset of R and homeomorphic to .Y × {1, . . . , q} × [0, 1]. It follows from Theorem 3.1.15 that equation (10.10), that is, (y, [s + t])j )  (y, s, (i), j )t =  (h[s+t] (y), s + t − [s + t], (h[s+t] (y)), F

.

) isomorphic to the suspension flow .(S(Y×{1, . . . , q},  defines a flow on . (Y h), R), h), R). This which in turn contains a compact minimal flow isomorphic to .(S(Cq ,  sets the stage for following Section 10.2 verbatim for . from equation (10.11) on p. 314 up to but not including Theorem 10.2.9 on p. 321. The version of Theorem 10.2.9 for locally circular minimal sets contains additional information. Theorem 10.4.9 There exists a flow .ψ on .X \ V with the following properties: (a) .Oψ (x) = Oϕ (x) for all x in .X \ V . + − (b) .Oψ (x) = Oϕ+ (x) and .Oψ (x) = Oϕ− (x) for all .x in .X \ V . (c) There exists a minimal set of .X \ V isomorphic to .(S(Cq ,  h), R).

10.4 Locally Circular Cascades and Flows

349

Proof Use the proof of Theorem 10.2.9 up to equation (10.23). Here we have to ) that modify .ρ so that it is constant on .(M, R), the minimal flow contained in .(Y   is isomorphic to .(S(Cq , h), R). So M is compact because .(S(Cq , h), R) is compact (Proposition 3.1.1). Let f be the continuous function given by equation (10.22). Since .M ∩ V = φ, there exists .λ > 0 such that .λ ≤ f (x) for all .x ∈ M because .f (x) = 0 if and only if .x is in V . Setting ρ(x) = max{1/λ, 1/f (x)}

.

(10.43)

defines a continuous function on .X \ V such that .ρ(x) ≥ 1/d(x, V ) and .ρ(x) ≥ σ 1/g(x). The function .h(x, σ ) = 0 ρ(ϕ(x, t)) dt still satisfies the cocycle equation, and we have gained .ρ(x) = 1/λ for all .x ∈ M. Thus .h(x, σ ) = σ/λ for all .x ∈ M and .σ ∈ R. The rest of the proof of Theorem 10.2.9 can be used to prove the existence of a flow .ψ  on .X \ V satisfying parts (a) and (b) noting only that equation (10.25) is just as an immediate consequence of .ρ(x) > 1/λ > 0 as it is of .ρ(x) > 1 > 0. Then .h(x, σ ) = σ/λ for all .x ∈ M and .σ ∈ R implies .H (x, s) = λs for all .x ∈ M and   .s ∈ R. It follows that .ψ (x, s) = ϕ(x, λs). Therefore, .ψ(x, s) = ψ (x, s/λ) is the desired flow.  If we can extend .ψ  to X by setting .ψ  (x, s) = x for .x in V and .s ∈ R, then the same holds for .ψ. So we will continue to work with .ψ  . Following Section 10.2, the next step is to prove: Theorem 10.4.10 If the function .ρ is defined by equation (10.43), then the function H can be extended to a continuous function on .X × R by setting .H (x, t) = 0 for all .x ∈ V and .t ∈ R. Proof With a small change and an observation, the proof of Theorem 10.2.10 works here. Specifically, replace equation (10.31) with  0 < ξ < σn ≤ λ

.

σn

ρ(ϕ(xn , t)) dt = λτn < λM,

0

which also implies that .τn is a positive sequence converging to .τ > 0 and .σn is a bounded sequence. The proof of Theorem 10.2.10 also makes critical use of .ρ(x) > 1/f (x), which  clearly holds when .ρ is defined by equation (10.43). Once the map . or . is constructed so that a point in a top edge is always identified with its h-image on a bottom edge, the underlying simplicial complex structure is no longer important. Only in the . case did this require extra work. Now having the structure of a bounded compact surface and the required identifications, the focus shifts to the rectangular structure, in particular, the vertical line segments of each .Rij because they can be used to build orbits and a flow as was done in Section 10.2 and can be repeated here.

350

10 Existence of Transitive Flows

Theorem 10.4.11 Given a minimal locally circular cascade .(Cq ,  h), there exists a flow .ψ on a bounded surface X with the following properties: (a) (b) (c) (d)

The set of fixed points of .ψ is V . There exists a minimal subset M of .(X, ψ) isomorphic to .(S(Cq , fˆ), R). .Oψ (x) = Oϕ (x) for all .x in .X \ V . + − + − .O (x) = Oϕ (x) and .O (x) = Oϕ (x) for all x in .X \ V . ψ ψ

Proof It follows from Theorem 10.4.9 that parts (b), (c), and (d) hold for .ψ. It suffices to show that the proof of Theorem 10.2.11 applies to .ψ  , and then set  .ψ(x, s) = ψ (x, s/λ). The proof of Theorem 10.2.11 makes no direct use of the function .ρ, but early in the proof there is one critical application of Theorem 10.2.10, which Theorem 10.4.10 replaces. The rest of the proof of Theorem 10.2.11 uses only the rectangular structure of .X = (R), namely, the .(x, t, i, j ) coordinates, and can be used to obtain the same  result for .X =  (R). Theorem 10.4.12 Given a minimal locally circular cascade .(Cq ,  h), there exists a flow .(W, R) with a finite number of fixed points on a compact connected surface containing an isomorphic copy of the suspension flow .S(Cq ,  h), which is a nontrivial set of .(W, R). Proof Apply Theorem 10.4.11 and then apply Theorems 2.3.2 and 2.1.14 as in the  proof of Theorem 10.3.5. The unified construction of flows on surfaces that are either transitive or contain a nontrivial minimal sets may be new. An unpublished report [50] contains a construction using rectangles of flows on compact surfaces containing nontrivial minimal sets that cannot occur in a flow on the torus. The related idea of the genus of a nontrivial is explored in [57].

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Index of Special Symbols

Symbol e xG .X/G _ .Y o .Y .Z .C 1 .S + .Z .R .Px 2 .D .A \ B .f |W .ι .ϕ .O (x) + .O (x) − .O (x) + .R .α(x) .ω(x) .χ(X) n .R .u n .Er (u)

Description identity element of a group written multiplicatively orbit of x for a continuous group action orbit space of a continuous G action on X closure of .Y ⊂ X, a topological space interior of .Y ⊂ X, a topological space integers complex numbers the unit circle .{z ∈ C : |z| = 1} the positive integers the real numbers .{g ∈ G : xg = x} for a continuous right action the unit disc .{z ∈ C : |z| ≤ 1} denotes the points in set A and not in set B the function .f : X → Y restricted to .W ⊂ X identity function of a set onto itself standard symbol for a flow orbit of x for a flow or cascade positive orbit of x for a flow or cascade negative orbit of x for a flow or cascade positive real numbers alpha limit set of x omega limit set of x Euler characteristic of X n-dimensional Euclidean space Euclidean norm open Euclidean ball of radius r centered at .u ∈ Rn

Page 3 3 3 4 4 4 4 4 5 6 7 9 10 11 11 15 16 16 16 16 17 17 20 30 30 30

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356

Index of Special Symbols Symbol .B .S

n

n

.Z2 .P

n

.T

n

.0 .H

n

.∂X .D

n

.A

B

.X#Y .X



.1 (X, q) .1 (X) .g∗  .ι .HX .

.[G, G] .Ab(G) .γ (X) .P#P ∗ .X .(S (X, f ), R) .[ · ] .[S, . . . ] .deg(f ) .λ .B = λ[−α, α] i .λ

.λ .C∞ .JI .JE .X∞ .[x, y]λ .(x, y)λ .(xσ, xτ )ϕ .[xσ, xτ ]ϕ .Z[i] .LT

Description (open) unit ball of dimension n unit sphere of dimension n integers mod 2 projective space of dimension n torus of dimension n the identity .(0, . . . , 0) of .Rn upper half-space of dimension n boundary of the manifold X unit disk or closed unit ball disjoint union of two sets A and B connected sum of manifolds X and Y removing a spherical boundary component fundamental group at a base point fundamental group of a path connected space induced fundamental group homomorphism  for a covering .π : X →X identity map of .X a group of all homeomorphisms of X onto itself a group of covering transformations hereafter commutator subgroup abelianization of G which equals .G/[G, G] genus of a compact connected surface X the Klein bottle surface with boundary components removed suspension flow of the cascade .(X, f ) the greatest integer function a group generated by .S, . . . degree of f mapping .S1 to itself local section of a flow flow box interior points of the local section .λ closure of .λi one-point compactification of .C or .S2 interior of embedded circle J in .C exterior of embedded circle J in .C one-point compactification of a space X closed segment between x and y on an arc .λ open segment between x and y on an arc .λ open orbit segment between .xσ and .xτ closed orbit segment between .xσ and .xτ Gaussian integers axis of T in the covering group for .T2

Page 30 31 31 31 31 31 32 33 34 38 38 40 43 43 43 45 45 45 52 52 53 54 55 60 60 65 66 83 84 84 84 91 91 91 94 94 94 94 94 112 112

Index of Special Symbols

357

Symbol

Description line determined by z and w .s{z, w} line segment determined by z and w .[T ] cyclic subgroup generated by T u .β universal lift of the loop .β I .2 × 2 identity matrix .GL(2, C) .2 × 2 complex invertible matrices .TA Möbius transformation given by .A ∈ GL(2, C) .SL(2, C) .2 × 2 complex special linear group .K .2 × 2 matrix group consisting of .{I, −I } 2 .U upper half plane .G Möbius transformations mapping .B2 onto itself  .G subgroup of .SL(2, C) given by equation (6.4) .dh (ζ1 , ζ2 ) hyperbolic metric on .B2 .Sβ model parabolic Möbius transformation .LR real h-line .{x : −1 < x < 1} : R → G  .F isomorphism to hyperbolic functions fixing .±1 (ρ) model hyperbolic Möbius transformation .Tρ .F .H a subgroup of hyperbolic functions fixing .±1 .LI imaginary h-line .{iy ∈ C : −1 < y < 1}   Z2 

 × Z2 = G .G equals .G   Z2 .(A, μ) ∗ (B, ν) multiplication for semi-direct product .G .T(A,μ) z equals .TA z when .μ = 0 and .TA z when .μ = 1

.G the smallest subgroup containing .G and .z → z .T(A−1 A,1) typical reflection in h-line .T(F model paddle motion (ρ),1) .T(A−1 F typical paddle motion (ρ)A,1) .( ) limit set of . , a group of h-rigid motions .Dζ ( ) Dirichlet region centered at .ζ for the group .

. c conformal subgroup of a subgroup . of .G

P .J positive side of a control curve .J N .J negative side of a control curve .J + .J alternate notation for positive side of .Ji i − .J alternate notation for negative side of .Ji .{z, w}

i

. v0 , v1 . v0 , v1 , v2 .Sq .|K| .ζ (G) .|A|

edge joining the vertices .v0 and .v1 triangle with vertices .v0 , .v1 , and .v2 permutation group of q symbols geometric realization of .K a number of boundary components of .|K| cardinality of a finite set A

Page 113 113 113 124 150 150 150 150 150 150 151 152 156 162 164 164 164 166 167 170 170 171 171 172 172 172 173 186 193 216 216 237 237 298 298 301 304 312 340

Index

A A fixed point, 7 Abelianization, 52 Accessible points, 74 Adjacent edge of −1 (x), 327 Adjacent vertices of −1 (x), 327 Almost periodic point, 7 Alpha limit set, 16 An accumulation point, 8 Annular region, 34 Anosov dichotomy, 218 Anosov’s theorem, 223 Arc, 93 Arcwise connected, 93 Asymptotic covering map, 122, 239 Asymptotic cycle, 344 Automorphism, 11 Axis, 112, 164

B Bebutoff’s theorem, 86 Beck’s theorem, 22 Bendixson sack, 110 Bitransformation group, 11 Bordered surface, 55 Boundary of a manifold, 32 Bounded ω-limit point, 245 Buffer vertices, 298, 301

C Cantor function, 75 Cantor ω-limit set at infinity, 259 Cantor set, 8, 9

Cascades, 5 Chart, 30 Circle representative, 123 Closed curve, 43, 123 Closed nowhere dense set, 8 Closed unit ball, 34 Closure, 4 Cocycle equation, 23 Coherent topology, 38 Collar neighborhood of embedded circle, 286 Collar neighborhood of ∂X, 40 Commutator subgroup, 52 Complementary intervals, 74 Components, 43 Condensation point, 8 Conformal subgroup, 193 Connected sum, 38 Consecutive crossings, 101 Consecutive mod 1, 296 Continuous group action, 3 Continuous left group action, 3 Continuous right group action, 3 Contractible subspace, 227 Control band, 130 Control curve, 127, 212 Coordinate chart, 30 Covering group, 45 Covering homomorphism, 50 Covering isomorphism, 50 Covering map, 44 Covering space, 44 Covering space automorphisms, 45 Covering transformation, 45 Cross ratio, 155

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. G. Markley, M. Vanderschoot, Flows on Compact Surfaces, Birkhäuser Advanced Texts Basler Lehrbücher, https://doi.org/10.1007/978-3-031-32955-5

359

360 Curve, 43 Cyclic subgroup, 113, 174

D Deck transformations, 45 Degree of a circle map, 66 Degree of a weak control curve, 216 Denjoy cascade, 79 Denjoy flow, 80 Denjoy’s theorem, 78 Dirichlet region, 186 Discontinuous group action at z, 182 Discrete group, 9 Discrete metric space, 9 Discrete subgroup, 182 Discrete subset, 9 Double cover of projective space, 44 Doubly asymptotic, 74

E Edge, 298 Elliptic Möbius transformation, 162 Embedded, 40 Embedded copy, 40 Embedding, 40 Equation of torus, 337 Equidistant curve, 168 Euclidean ball of a manifold, 30 Euclidean metric, 30 Euclidean norm, 30 Euclidean space, 30 Euler characteristic, 20 Evenly covered, 43 Existence of universal covering, 50 Exterior of embedded circle, 91

F Fiber, 44 Fibers of Normal Coverings, 47 Finite-sheeted, 44 First return time, 90 Flow, 12 Flow box, 84 Free action, 7 Fuchsian covering group, 184 Fuchsian group, 183 Fundamental Dirichlet region, 188 Fundamental group, 43 Fundamental region, 185

Index G Gaussian integers, 112 Genus, 52 Global section, 88 H h-line, 146 Homomorphism, 11 Homotopic functions, 43 Homotopic loops, 235 Homotopy Lifting Property, 44 Horocycle, 162 h-perpendicular bisector, 185 Hyperbolic line, 146 Hyperbolic metric, 160 Hyperbolic Möbius transformation, 162 I Inaccessible points, 74 Interior, 4 Interior of embedded circle, 91 Invariance of domain, 32 Invariant metric, 219 Invariant set, 6 Irrational limit of semi orbit, 121, 239 Isolated ω-limit set at infinity, 254 Isomorphism, 11 Isomorphism of metric groups, 4 J Jordan separation theorem, 91 K Klein bottle, 54 L Left component of a local section, 262 Left syndetic, 7 Lift, 44 Lifted flow, 46 Lifting criterion, 45 Limit set of , 173 Linear fractional transformation, 149 Local cross sections, 84 Local section, 83 Local section at a point, 85 Locally circular minimal cascade, 342 Locally connected, 43 Locally path-connected, 43 Locally simply connected, 43

Index Locator function, 314 Loop, 43, 123 Lower semicontinuous, 316 Loxodromic, 162

M Maier’s first theorem, 276 Maier’s second theorem, 282 Maier’s third theorem, 290 Manifold of dimension n, 30 Manifold with boundary, 32 Maximal cyclic subgroup, 189 Metric group homomorphism, 4 Minimal set, 6 Minimal transformation group, 6 Mobile, 175 Möbius band, 51 Möbius transformation, 149 Monotonic sequence, 101 Moving point, 85

N Natural projection, 3 Negative bounding line, 130 Negative side, 105, 128, 216 Negatively asymptotic, 74 Negatively recurrent, 18 Nontrivial minimal set, 80 Normal covering map, 47 Null-homotopic, 124 Null-homotopic loop, 43

O Omega limit set, 16 One-point compactification, 94 Open Euclidean ball, 30 Open ω-limit set at infinity, 259 Open recurrent ω-limit set at infinity, 259 Open simplex, 306 Open star of a vertex, 309 Orbit, 3 Orbit closure, 6 Orbit equivalence relation, 3 Orbit space, 3

P Paddle motion, 172 Pairs of consecutive crossing times, 236 Parabolic Möbius transformation, 162 Path, 43

361 Path component, 43 Path Lifting Property, 44 Path-connected, 43 Path-homotopic, 43 Peano continuum, 93 Perfect set, 8 Period, 15 Periodic flow, 26 Periodic point, 7 Periodic transformation group, 13 Permutation group, 301 Poincaré return function, 90 Poincaré theorem, 89 Poincaré–Bendixson theorem, 110 Points at infinity, 146 Polyhedron, 20 Positive bounding line, 130 Positive orbit goes to infinity, 218 Positive side, 105, 128, 216 Positively asymptotic, 74 Positively recurrent, 18 Primitive element, 125, 189 Projective space, 31 Proper action, 9

Q Quotient metric, 219 Quotient space, 4

R Rational Boundary Point Lemma, 238 Rational h-line, 206 Rational limit of semi orbit, 121, 239 Rational line, 112 Rational point of S1 , 121, 206 Rectangular flows, 330 Recurrent, 18 Recursive behavior, 16 Reflection in an h-line, 172 Regular covering map, 47 Regular Euclidean ball, 35 Remote ω-limit point, 245 Residual set, 5 Right component of a local section, 262 Right syndetic, 7 Rotation number, 80

S Same component test, 92 Saturated set, 3 Schöenflies Theorem, 92

362 Semi-direct product, 170 Simple closed curve, 123 Simply connected, 43 Standing Assumption 1, 97 Standing Assumption 2, 127 Standing Assumption 3, 127 Standing Assumption 4, 150 Straight line flow, 12 Strictly almost periodic, 260 Strictly monotonic sequence, 101 Surface, 32, 51 Suspension flow, 60 T Tessellation, 185 Topological manifold, 30 Torus, 31 Trace of a matrix, 161 Trace of a Möbius transformation, 162 Transformation group, 3 Transitive point, 5 Triangle, 298 Triangulation, 20, 52 Trivial continuous group action, 7

Index Trivial subgroups of R, 14 Type I/II Fuchsian group, 183 Type of a line, 122 Type of an h-line, 207

U Unbounded positive orbit, 218 Unique Lifting Property, 44 Unit ball, 30 Unit circle, 4 Unit disk, 34 Unit sphere, 31 Universal covering, 50 Universal covering group, 50 Universal lift, 124, 206 Upper half space, 32 Upper semicontinuous, 316

W Weak control curve, 215 Weil’s theorem, 114, 178, 210 Whitney’s theorem, 95