Equivariant Degree Theory [Reprint 2012 ed.] 9783110200027, 9783110175509

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de Gruyter Series in Nonlinear Analysis and Applications 8

Editors A. Bensoussan (Paris) R. Conti (Florence) A. Friedman (Minneapolis) K.-H. Hoffmann (Munich) L. Nirenberg (New York) A. Vignoli (Rome) Managing Editor J. Appell (Würzburg)

Jorge Ize Alfonso Vignoli

Equivariant Degree Theory

w DE

G_ Walter de Gruyter · Berlin · New York 2003

Authors Jorge Ize Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas Universidad Nacional Autonoma de Mexico 01000 MEXICO D. F. MEXICO Mathematics

Alfonso Vignoli Department of Mathematics University of Rome "Tor Vergata' Via della Ricerca Scientifica 00133 R O M A ITALY

Subject Classification 2000: 58-02; 34C25, 37G40, 47H11, 47J15, 54F45, 55Q91, 55E09

Keywords: equivariant degree, homotopy groups, symmetries, period doubling, symmetry breaking, twisted orbits, gradients, orthogonal maps, Hopf bifurcation, Hamiltonian systems, bifurcation

© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication

Data

Ize, Jorge, 1946— Equivariant degree theory / Jorge Ize, Alfonso Vignoli. p. cm. - (De Gruyter series in nonlinear analysis and applications, ISSN 0941-813X ; 8) Includes bibliographical references and index. ISBN 3-11-017550-9 (cloth : alk. paper) 1. Topological degree. 2. Homotopy groups. I. Vignoli, Alfonso, 1940- II. Title. III. Series. QA612.I94 2003 514'.2-dc21 2003043999

ISBN 3-11-017550-9 Bibliographic information published by Die Deutsche

Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

© Copyright 2003 by Walter de Gruyter G m b H & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg Typeset using the authors' T g X files: I. Zimmermann, Freiburg Printing and binding: Hubert & Co. G m b H & Co. Kg, Göttingen

J. I. expresses his love to his wife, Teresa, and his sons, Pablo, Felipe and Andres.

Α. V. wishes to dedicate this book to his beloved wife Lucilla, to his son Gabriel and to Angela, who kept cheering him up in times of dismay and frustration becoming more frequent at sunset.

Preface

The present book grew out as an attempt to make more accessible to non-specialists a subject - Equivariant Analysis - that may be easily obscured by technicalities and (often) scarcely known facts from Equivariant Topology. Quite frequently, the authors of research papers on Equivariant Analysis tend to assume that the reader is well acquainted with a hoard of subtle and refined results from Group Representation Theory, Group Actions, Equivariant Homotopy and Homology Theory (and co-counter parts, i.e., Cohomotopy and Cohomology) and the like. As an outcome, beautiful theories and elegant results are poorly understood by those researchers that would need them mostly: applied mathematicians. This is also a self-criticism. We felt that an overturn was badly needed. This is what we try to do here. If you keep in mind these few strokes you most probably will understand our strenuous efforts in keeping the mathematical background to a minimum. Surprisingly enough, this is at the same time an easy and very difficult task. Once we took the decision of expressing a given mathematical fact in as elementary as possible terms, then the easy part of the game consists in letting ourselves to go down to ever simpler terms. This way one swiftly enters the realm of stop and go procedures, the difficult part being when and where to stop. In our case, we felt relatively at ease only when we arrived at the safe harbor of matrices. Of course, you have to buy a ticket to enter. The fair price is to become a jingler with them. After all, nothing is given for free. We have enjoyed (and suffered) with the fact that so many beautiful results can be obtained with so little mathematics. Our hope is that you will enjoy (and not suffer) reading this book. Acknowledgments. We would like to thank our families for their patience and support during the, longer than expected, process of writing the book. Very special thanks to Alma Rosa Rodriguez for her competent translation of ugly hieroglyphics to beautiful MgX. Thanks to our colleagues, Clara Garza, for reading the manuscript, to Arturo Olvera for devising and running some of the numerical schemes which have given evidence to some of our results and to Ana Cecilia Pérez for her computational support. We are grateful to L. Vespucci, Director of the Library at La Sapienza, for her help in our bibliographical search. Last but not least, let us mention the contributions of our friend and collaborator Ivar Massabó with whom we started, in 1985, the long journey through equivariant degree. During the last two years, the authors had the partial support of the CNR, of the University of Rome, Tor Vergata, given through the scientific agreement between IIMAS-UNAM and Tor Vergata, and of several agencies on the Italian side, including

viii

Preface

CANE, and from CONACyT (grant G25427-E, Matemáticas Nolineales de la Física y la Ingeniería, and the agreement KBN-CONACyT) on the Mexican side. México City and Rome, February 2003

Jorge Ize Alfonso Vignoli

Contents

Preface

vii

Introduction

xi

1

Preliminaries 1.1 Group actions 1.2 The fundamental cell lemma 1.3 Equivariant maps 1.4 Averaging 1.5 Irreducible representations 1.6 Extensions of Γ-maps 1.7 Orthogonal maps 1.8 Equivariant homotopy groups of spheres 1.9 Symmetries and differential equations 1.10 Bibliographical remarks

1 1 5 8 12 17 25 29 35 42 57

2

Equivariant Degree 2.1 Equivariant degree in finite dimension 2.2 Properties of the equivariant degree 2.3 Approximation of the Γ-degree 2.4 Orthogonal maps 2.5 Applications 2.6 Operations 2.6.1 Symmetry breaking 2.6.2 Products 2.6.3 Composition 2.7 Bibliographical remarks

59 59 61 67 69 72 77 78 78 79 85

3

Equivariant Homotopy Groups of Spheres 3.1 The extension problem 3.2 Homotopy groups of Γ-maps 3.3 Computation of Γ-classes 3.4 Borsuk-Ulam results 3.5 The one parameter case 3.6 Orthogonal maps 3.7 Operations 3.7.1 Suspension

86 86 102 108 119 136 156 165 165

χ

Contents

3.8 4

3.7.2 Symmetry breaking 3.7.3 Products 3.7.4 Composition Bibliographical remarks

Equivariant Degree and Applications 4.1 Range of the equivariant degree 4.2 Γ-degree of an isolated orbit Example 2.6. Autonomous differential equations Example 2.7. Differential equations with fixed period Example 2.8. Differential equations with first integrals Example 2.9. Time dependent equations Example 2.10. Symmetry breaking for differential equations Example 2.11. Twisted orbits 4.3 Γ-Index for an orthogonal map Example 3.4. Bifurcation Example 3.5. Periodic solutions of Hamiltonian systems Example 3.6. Spring-pendulum systems 4.4 Γ-Index of a loop of stationary points Example 4.1. The classical Hopf bifurcation Example 4.3. Hopf bifurcation for autonomous differential equations. Example 4.4. Hopf bifurcation for autonomous systems with symmetries Example 4.5. Hopf bifurcation for time-dependent differential equations Example 4.6. Hopf bifurcation for autonomous systems with first integrals Example 4.7. Hopf bifurcation for equations with delays 4.5 Bibliographical remarks

171 180 188 195 197 197 211 222 232 234 237 238 240 245 255 256 266 288 289 301 303 304 308 324 325

Appendix A

Equivariant Matrices

327

Appendix Β

Periodic Solutions of Linear Systems

332

Bibliography

337

Index

359

Introduction

Nonlinearity is everywhere. But few nonlinear problems can be solved analytically. Nevertheless much qualitative information can be obtained using adequate tools. Degree theory is one of the main tools in the study of nonlinear problems. It has been extensively used to prove existence of solutions to a wide range of equations. What started as a topological (or combinatorial) curiosity has evolved into a variety of flavors and represents, nowadays, one of the pillars, together with variational methods, of the qualitative treatment of nonlinear equations. In the simplest situation, the "classical" degree of a continuous map f ( x ) from R" into itself with respect to a bounded open set Ω such that f ( x ) is non-zero on 3Ω is an integer, deg(/; Ω), with the following properties: (a) Existence. If deg(/; Ω) φ 0, then f (x) = 0 has a solution in Ω. (b) Homotopy invariance. If one deforms continuously f(x), the boundary, then the degree remains constant.

without zeros on

(c) Additivity. If Ω is the union of two disjoint open sets, then deg(/; Ω) is the sum of the degrees of f ( x ) with respect to each of the pieces. If one has in mind studying a set of equations, those properties have a striking conceptual importance: a single integer gives existence results by loosening the rigidity of the equations and allowing deformations (and not only small ones). In other words, one does not need to solve explicitly the equations in order to get this information and one may obtain it by deforming the equations to a simpler set for which one may easily compute this integer. Furthermore, one has a certain localization of the solutions or one may obtain multiplicity results for these solutions. Thus, in dimension one, the degree is another way to view die Intermediate Value Theorem of Calculus and, in dimension two, it is nothing else than the winding number of a vector field, familiar from Complex Analysis. If, furthermore, one requires the property (d) Normalization. The degree of the identity with respect to a ball containing the origin is 1, then, one may show that the degree is unique. Now there are many ways to construct the degree. As a consequence of the uniqueness, they are all equivalent and depend more on the possible application or on the particular taste of the user. For instance, one may take a combinatorial approach, or

xii

Introduction

analytical (through perturbations or integrals), or topological (homotopical, cohomological) or an approach from fixed point theory. Classical degree theory, or Brouwer degree, would have remained a simple curiosity if it were not for the extension to infinite dimensional problems, in particular to non-linear differential equations. This extension has required some compactness, starting from the Leray-Schauder degree with compact (or completely continuous) perturbations of the identity, continuing with k-set contractions, A-proper maps, 0-epi maps (these terms will be defined in Chapter 1) and so on. In most of these extensions the compactness is used to construct a good approximation by finite dimensional maps. One of the by-products of the construction presented here is to pinpoint a new way to see where the compactness is used. Now the subject of this book is also that of symmetry. This is a basic concept in mathematics and words like symmetry breaking, period doubling or orbits are familiar even outside our discipline. In fact, many problems have symmetries: in the domains and in the equations. Very often these symmetries are used in order to reduce the set of functions to a special subclass: for instance look for odd (or even) solutions, or radial, or independent of certain variables. They are also used to avoid certain terms in series expansions or, in connection with degree theory, in order to get some information on this integer, the so-called Borsuk-Ulam results. However, since any continuous (i.e., not necessarily respecting the symmetry) perturbation is allowed, the ordinary degree will not give a complete topological information. This very important point will be clearer once the equivariant degree is introduced and computed in many examples. In this book we shall integrate both concepts, that of a degree and that of symmetry, by defining a topological invariant for maps which commute with the action of a group of symmetries and for open sets which are invariant under these symmetries, i.e., for equivariant maps and invariant sets. More precisely, a map f{x), from M" to R m for instance or between two Banach spaces, is said to be equivariant under the action of Γ (a compact Lie group, for technical reasons) if Πγχ)

= y

fix)

for all y in Γ, where y and γ represent the action of the element y in R " and R m respectively. Think of odd maps (y = y = — Id) or even maps (y = — Id, y = Id), or any matrix y expressed in two bases. The set Ω will be called invariant if, whenever χ is in Ω, then the whole orbit Γ χ is also in Ω. By looking only at maps with these properties, including deformations of such maps, one gets an invariant, d e g r ( / ; Ω), which is not an integer anymore (unless m = η, Γ = {e}, in which case one recovers the Brouwer degree) but with properties (a)-(c) valid and (d) replaced by a universality property. Since the construction of this equivariant degree is quite simple, we shall not resist the temptation to present it now. Let f(x) be an equivariant map, with respect to the actions of a group Γ, defined in an open bounded invariant set Ω and non-zero on 3Ω. Since Ω is bounded, one may choose a very large ball Β containing it. Then one constructs an equivariant extension / of / to Β. The new map f(x) may have

Introduction

xiii

new zeros outside Ω. One takes an invariant partition of unity φ ( χ ) with value 0 in Ω and 1 outside a small neighborhood Ν of Ω, so small that on N\Q the map fix) is non-zero (it is non-zero on 3 Ω ) . Take now a new variable t in / = [0, 1] and define

f ( f , x ) = (2f + 2φ{χ) — 1,

/(*)).

It is then easy to see that /(?, x) = 0 only if Λ; is in Ω with / ( x) = fix) = 0 and, since φ(χ) = 0, one has t = 1/2. In particular, the map f i t , x) is non-zero on 3(7 χ Β) and defines an element of the abelian group (this group will be studied in Chapter 1) n£„(Sm) of all Γ-equivariant deformation (or homotopy) classes of maps from 3 ( I χ B ) into K m + 1 \{0}. We define the Γ-equivariant degree of fix) with respect to Ω as the class of f{t, χ) in n £ „ ( S m ) : degr(/; Ω) =

[/V

This degree turns out to have properties (a)-(c), where having non-zero degree here means that the class [/]p is not the trivial element of construction, this degree has the Hopf property,

(Sm).

Furthermore, by

which is that if Ω is a ball and [/]p

is trivial, then f\;>n has a non-zero Γ-equivariant extension to Ω . In other words, d e g r ( / ; Ω ) gives a complete classification of Γ-homotopy types of maps on spheres. This property implies also that d e g r ( / ; Ω ) is universal

in the sense that, if one has

another theory which satisfies ( a } - ( c ) such that, for a map / and a set Ω, one has a non-trivial element, then d e g r ( / ; Ω ) will be non-zero. The simplest example is that of a non-equivariant map from M" into itself. Then we shall see that [ / ] r is the Brouwer degree of f with respect to / χ Β. Since f is not zero on / χ (Β\Ω), this degree is that of / with respect to 7 χ Ω , where / is a product map. A simple application of the product theorem implies that [ / ] r = deg(/; Ω), a result which is, of course, not surprising but which indicates that our approach has the advantage of a very quick definition, with an immediate extension to the case of different dimensions, including infinite ones. A second simple example is that of a ^ - a c t i o n on M" = R k χ W ~ k = M m , where x = (y, z) and f i y , z) = ifoiy,

z), f\(j,

z)) with /o even in ζ and f\ odd in z. It

turns out that in this case Π f Z ( S " ) = Ζ χ Ζ, and that deg% ( f ; Ω ) is given by two integers: deg(/o(;y, 0); Ω Π M*) and deg(/; Ω ) . As a consequence of the oddness of /i, with respect to z, one has f\ (jc, 0) = 0 and it is clear that these two integers are well defined. The set { λ , 0} is the fixed point subspace of the action of Z2 and it is not surprising that these two integers are important. What is less intuitive is that if Ω is a ball then these two integers characterize completely all Z2-maps defined on Ω. A third example is that of an S1 -action on M.k χ Cm R * fixed and acts as &χρ(ίη]φ),

1

χ · · · x C'"p, where S1 leaves

for j = 1 , . . . , p, on each complex coordinate of CmJ.

This is an important example because if one writes down the autonomous equation

dX dt

fiX)

= 0,

k

X in R ,

Introduction

xiv

for X(t) = Σ xne,n\ that is for 2π-periodic functions, then the fact that f(X) does not depend on t implies that its component fn(X) on the rc-th mode has the property that fn(X(t

+ 2, or x¡ e Κ if W(rej) = {e} or The action of Γ on the elements of the basis is given by γ e¡ = e x p i l ( ( N j , Φ) + 2 π {Κ,

Lj/M))ej,

as in (1.4) and Remark 1.1, with NJ = (n{,...,nJnf

and

V ¡M = ( / / / m i , . . . ,

l¡/ms)T.

Then yX = Σ xjYej and γΧ = X gives yej = e¡ if χ¡ Φ 0. Hence, Γχ = Ρ) Tej, where the intersection is over those j's for which Xj Φ 0. Thus, W(Ye¡) < ΐ ν ( Γ χ ) . Lemma 2.1. VT" = {X e V : Ψ ( Γ χ ) < oc}. Proof. If

is finite, then W ( r e p is a finite group and Γ^ contains T". In this

case, Γχ contains also T", that is, X belongs to VT". Conversely, if X is fixed by T", then νν(Γχ) is a factor of Z m , χ · · · χ Zm¡ and hence is finite. • Denote by Hj =

and define Hj-i

— H\ Π ••• Π //,_), H0 = Γ. Then H}-\

acts on the space Vj generated by e¡ (Vj = MorC), with isotropy Hj-i Π Hj = Hj, if

6

1 Preliminaries

Xj φ 0, and Hj-\/Hj acts freely on V/\{0}. Then, from Lemma 1.1, this Weyl group is isomorphic either to Sl, to {e}, or to Z p , ρ > 2. Let kj be the cardinality of this group: kj = \Hj-\/Hj\. If the group is S 1 , then kj = oo, while kj = 1 means that Hj^i = Hj. If kj = 2 and Vj is complex, then Vj splits into two real representations of Hj-\/Hj = Z2, while if Vj is real, then kj = 1 or 2. Consider G = {X 6 V : \x¡\ = 1 for any j}, a torus in V. Let Η = H\ Π ÍÍ2 Π • · · Π Hm+r be the isotropy type of G, where there are m of the Vj ' s which are complex and r which are real (hence dim V = 2m + r). Let k be the number of j's with kj = 00. Let Δ = {X e G : 0 < Argx¡ < 2n/kj for all ; = 1 , . . . , m + r}. That is, if kj = 1 there is no restriction on Xj (in C or R), while, if kj = 00, then Xj e and, if Xj € IR and kj = 2, then x¡ is positive. Let Δ ν = (X e V : 0 < A r g X j
1, then one has to cut G into k\ equal pieces in order to generate G. If the result is true for η - 1, let G = Gn~ 1 χ {\xn\ = 1}, Δ = Δ„_ι χ {0 < Arg*„ < 2n/kn) and write Γ/Η = (Τ/Hn-\)(Hn-\/Η), recalling that these groups are abelian. By the induction hypothesis, the images of Δ η _ι under Τ/Ηη-\ cover properly Gn-\. Furthermore, from the case η = 1, the set [xn : |jcrt[ = 1} is covered properly by the images of {jc„ : 0 < Argx n < 2 π / k n ] under Hn-\/H, a group which fixes all points of Cn-\. Hence, if (X„_i, jc„) is in G, there are y„_i in T/Hn-\ and Yn in Hn-i/H such that Χ„_ι = γη-\Χ°η_ν with in C„_ 1, γ~\χη = Yn with 0 < Argx^ < 2ix¡kn and γηΧη-\ = Χη-\· Then (Χ„_ι,λ„) = (γη-ιΧ°_ν Yn-ìY'^Xn) = yn-iVn(X%-i,xj}), i-e-, G is covered by the images of Δ under Γ / Η . If(Xn-i,xn) = γ\(Χ\χι) = γ2(Χ2,χ2), with ( X j , x j ) in Δ and y¡ in Γ / Η , 1 2 2 1 then (Χ',λ; ) = y f ' ^ Í X , χ ) . Thus, Χ = γΧ2, χ1 = γχ2. By the induction hypothesis, Χ 1 = X2 and γ belongs to Hn-1, but then χ1 = χ2 and γ belongs to H.

• This fundamental cell lemma will be the key tool in computing the homotopy groups of Chapter 3.

1.2 The fundamental cell lemma

7

Example 2.1. Let S 1 act on e} via βη'ψ, with nj > 0. Then, Hj — {φ = Ink/tij, k — 0 , . . . , « / — 1} = Z„.. Let ñj — (η \ : • · • : rij) be the largest common divisor (l.c.d.) of n\,..., nj, then Hj = [φ = 2πk/ñj, k = 0 , . . . , ñj — 1} = Ζf¡ . Thus, k\ = oo, kj = ñj-i/ñj. Note that, since Γ/Η = (T/Hi)x(Hi/H2)x---x(.Hm+r-i/H) if dim Γ/Η = k, then there are exactly k coordinates (which have to be complex) with kj = oo. In fact, since Hj is the isotropy subgroup for the action of Hj-\ on χy, each factor, by Lemma 1.1, is at most one-dimensional. Lemma 2.3. Under the above circumstances, one may reorder the coordinates in such a way that kj = oo for j = 1 , . . . , k and kj < oo for j > k. Proof. Assuming k > 0, there is at least one coordinate with dim Γ/Hj = 1: if not, Hj > T" for all j's and hence Η > Tn with \Γ/Η\ < oo. Denote by z\ this coordinate, then Γ/Η = (Γ/Ηι)(Ηι/Η), with dim H\/H = k - 1. If Hy/H is a finite group, i.e., k = 1, then one has a decomposition into finite groups with kj < oo for j > 1. On the other hand, if k > 1, then, by repeating the above argument, one has a coordinate Z2 with H\ /Hi of dimension 1. • The following result will be used very often in the book. Lemma 2.4. Let T" act on V = C m via expi (Nj, Φ), j = 1 , . . . , m. Let A be the m χ η matrix with NJ as its j-th row. Then: (a) dim Γ/Η = k if and only if A has rank k. (b) Assuming kj — oo for j = 1 , . . . , k and that the k χ k matrix Β with B¡j = nlj, 1 < i, j < k, is invertible, then one may write ΛΦ = τ

Γ

Γ

, with Φ = Φ + ΛΦ,

τ

where Φ — (Φ , Φ ) and Φ — {ψ\,..., • [0, 1], with support in a ball centered at yj = fiyjxj), respectively yj~l fiyjXj), of radius 1 /2N and such thatE2, one has that the left hand side is μ(γΐ, >'2), hence it is of the form c(x\, X2){y\, yi), where c is independent of x\, X2, y i , yi. Taking = χι, y\ = yi, the left hand side is / Γ 2(yx\, yi)2dy and c is positive. Take now, e j , e i , e ν an arbitrary collection of orthonormal vectors in E. Then, f r o m Parseval's inequality, one has

Ν Y^yx,ejf