Convergence in Ergodic Theory and Probability 9783110889383, 9783110142198

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Ohio State University Mathematical Research Institute Publications 5 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Convergence in Ergodlc Theory and Probability Editors

V. Bergelson P. March J. Rosenblatt

w DE

Walter de Gruyter · Berlin · New York 1996

Editors VITALY BERGELSON

PETER M A R C H

JOSEPH R O S E N B L A T T

Mathematics Department The Ohio State University Columbus, OH 43210, USA

Mathematics Department The Ohio State University Columbus, OH 43210, USA

Mathematics Department University of Illinois Urbana-Champaign Urbana, IL 61801, USA

Series Editors: Gregory R. Baker, Karl Rubin Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174, USA Walter D. Neumann Department of Mathematics, The University of Melbourne, Parkville, VIC 3052, Australia 1991 Mathematics Subject Classification: Primary: 28-06, 60-06 Secondary: 28D05, 28D10, 28D20, 60B15, 60F05, 60F10, 60G10, 60J10, 60J15 Keywords: Ergodic theory, probability theory, measure-preserving transformations, Fourier transforms, random walks, Markov chains, large deviations, martingales, and distributions ® 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 Convergence in ergodic theory and probability / editors, V. Bergelson, P. March, J. Rosenblatt p. cm. - (Ohio State University Mathematical Research Institute publications, ISSN 0942-0363 ; 5) Papers from the Conference on Convergence in Ergodic Theory and Probability, held at Ohio State University, in June 1993. ISBN 3-11-014219-8 (acid-free paper) 1. Convergence - Congresses. 2. Inequalities (Mathematics) - Congresses. 3. Probabilities - Congresses. 4. Ergodic theory - Congresses. I. Bergelson, V. (Vitaly), 1950. II. March, P. (Peter), 1955. III. Rosenblatt, J. (Joseph), 1946IV. Conference on Convergence in Ergodic Theory and Probability (1993 : Ohio State University) V. Series. QA295.C67 1996 515'.42-dc20 96-21518 CIP

\

Die Deutsche Bibliothek - Cataloging-in-Publication Data Convergence in ergodic theory and probability / ed. V. Bergelson ... - Berlin ; New York : de Gruyter, 1996 (Ohio State University, Mathematical Research Institute publications ; 5) ISBN 3-11-014219-8 NE: Bergelson, Vitaly [Hrsg.]; International Mathematical Research Institute : Ohio State University ...

© Copyright 1996 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storaj*gliaadj§trieval system, without permission in writing from the publisher. Printfed-pjC^i^&B^Printing: Arthur Collignon GmbH, Berlin. Binding: Lüd§qt?& B a u e ^ ^ H , Berlin. Cover design: Thomas Bonnie, Hamburg.

Dedication

This volume of articles is dedicated to Professor Louis Sucheston. His commitment to research and teaching in mathematics and his persistent efforts to develop the ergodic theory and probability groups at The Ohio State University have had a great impact on the mathematical community as a whole and the personal lives of the editors of this volume in particular. We are all indebted to him and grateful for his support of our work.

Preface

This volume is a sample of the mathematics that was presented and discussed at the Conference on Convergence in Ergodic Theory and Probability, held at The Ohio State University, Fawcett Center for Tomorrow, in June, 1993. The conference was the third in a recent series of such meetings, the first being held at The Ohio State University in the Spring of 1988 (organized by Louis Sucheston and Gerald Edgar) and the second being held at Northwestern University in the Fall of 1989 (organized by Alexandra Bellow and Roger Jones). The organizers of the meeting which gave rise to this volume would like to express their appreciation for the financial support given this conference by The Ohio State University Mathematical Research Institute and the Mathematics Department at The Ohio State University, the College of Mathematical and Physical Sciences at The Ohio State University, the National Science Foundation, and the Department of Defense (Army). Without this generous help, this meeting would not have been possible. The organizers would also sincerely like to thank the de Gruyter Publishing Company and their editor, Dr. Manfred Karbe, who have made the publication of this volume of articles a reality. Can there ever be creation without communication? Vitaly Bergelson Peter March Joseph Rosenblatt

Contents

J. Aaronson and B. Weiss A Ζ d Ergodic Theorem with Large Normalising Constants

1

M. A. Akcoglu, J. R. Baxter, and S. E. Ferrando A Repeated Filling Scheme for Upcrossings

15

E. Akin, J. Auslander, and K. Berg When is a Transitive Map Chaotic?

25

M. Atlagh et Μ. Weber Une Nouvelle Loi Forte des Grands Nombres

41

J. R. Baxter and N. C. Jain An Approximation Condition for Large Deviations and Some Applications

63

V. Bergelson and I. J. Häland Sets of Recurrence and Generalized Polynomials

91

Ν. H. Bingham The Strong Arc-Sine Law in Higher Dimensions

Ill

M. Boshernitzan, R. L. Jones, and M. Wierdl Integer and Fractional Parts of Good Averaging Sequences in Ergodic Theory

117

R. E. Bradley The Uniqueness of Induced Operators

133

N. Etemadi On Convergence of Partial Sums of Independent Random Variables

137

R. A. FefFerman Harmonic Analysis of Operators Associated with a Multiparameter Group of Dilations

145

S. R. Foguel Markov Matrices

157

χ

Contents

Ν. Ε. Frangos and P. Imkeller Existence and Continuity of the Quadratic Variation of Strong Martingales

179

C. D. Fuh and T. L. Lai Convergence Rate in the Strong Law of Large Numbers for Markov Chains

185

H. Furstenberg and B. Weiss A Mean Ergodic Theorem for ^

f(Tnx)g(Tnlx)

193

R. L. Jones and M. Wierdl Convergence of Ergodic Averages

229

M. Lacey Bourgain's Entropy Criteria

249

S. P. Lalley Self-Intersections of Closed Geodesies on a Negatively Curved Surface: Statistical Regularities

263

D. Li and Μ. B. Rao A Note on the Law of the Iterated Logarithm for Weighted Sums of Independent Identically Distributed Random Variables

273

M. Lin and J. Olsen Besicovitch Functions and Weighted Ergodic Theorems for LCA Group Actions

277

V. Losert A Remark on the Strong Sweeping Out Property

291

D. Maharam Ergodic Invertible Liftings

295

G. Peskir and M. Weber The Uniform Ergodic Theorem for Dynamical Systems

305

M. A. Pinsky Pointwise Fourier Inversion and Related Jacobi Polynomial Expansions

333

P. Revesz The Distribution of the Particles of a Branching Random Walk

345

B.-Z. Rubshtein On a Random Ergodic Theorem for Bistochastic Operators

365

Contents

xi

B.-Z. Rubshtein Α Central Limit Theorem for Conditional Distributions Y. Sagher and Ν. Xiang Complex Methods in the Calculation of Some Distribution Functions

373

381

Y. Sagher and K. Zhou Exponential Integrability of Rademacher Series

389

D. Schneider and M. Weber Weighted Averages of Contractions Along Subsequences

397

Y. Sun On Uniform Distribution of Sequences

405

R. Wittmann Ergodic Theorems for Exit Laws Q. Zhang On Skew Products of Irrational Rotations with Tori

421

435

A 7Ld Ergodic Theorem with Large Normalising Constants Jonathan Aaronson and Benjamin Weiss

Abstract We show, inter alia that for a non-integrable, non-negative stationary, ergodic Zd random field, there are no normalising constants for the pointwise ergodic theorem.

0. Introduction Let (X, S(X), m) be a Lebesgue probability space, let [Tg : g e Zd} be a free, probability preserving action of TLd on X, and suppose that / : X Κ is integrable. The pointwise ergodic theorem ([Kr]) states that -U„/ na

—• E{f\3(T))

a.e.

where Snf ·•=

Y] j—'

foTg

(||(ni, ...,nd)\\

:= max \nj\) 1 0 such that T~Snf — > 1 a.e. (1) bn This is the statement in the abstract, as a stationary l/ random field is a stochastic process of {Yg: g £ Zd) of form Yg = f ο Tg where / and Τ are as above. Our main theorem is Theorem A (Zd-actions alternative). Let (Χ, Έ, m, Τ) be a free, ergodic, probability preserving Zd-action on a Lebesgue probability space, and suppose that f:X~>M. is measurable. Let a: [0, oo) —> [0, oo) be continuous, strictly increasing, and satisfy + )0 < a(x) + a(y), and a(x) = o(x) as χ -»• oo. Then either, lim s u p , , ^ ^ilMll _ ^ ο a.e. as η ^ oo. or The authors would like to thank IMPA, Rio de Janeiro for inspiration and hospitality while this work was conceived.

2

J. Aaronson and B. Weiss

As a corollary, we obtain a version of the ergodic theorem with large normalising constants (Feller's theorem): Theorem F. Let (X, S , m , T) be a free, ergodic, probability preserving -action on a Lebesgue probability space, and suppose that f :X —• Κ is measurable. Let b: Ν [0, oo) satisfy ^ f °o as η t oo. Then either, lim s u p ^ ^ j j ^ j = oo o.e., or

0 a.e. as η ->• oo.

As far as we know, these are the first multi-dimensional versions of these results. There are previous results for d = 1. Theorem F, in case { / ο Tg : g e Z} are independent, identically distributed random variables, is due to W. Feller [Fe]. The general stationary one dimensional results were obtained in [Aa2]. The nonexistence of a sequence of constants satisfying (1) was obtained by Chow and Robbins [C-R] in the independent case, and in general in [Aal]. In § 1, we consider the multi-dimensional independent case, also proving a lemma which is needed for the proof of Theorem A. This proof is carried out in §2, the main tool being a "random Kac inequality" (Theorem 2.1). The Kac formula (obtained by M. Kac in [Kac]) says that if Τ is an ergodic measure preserving transformation of (X, !B,m), A e S + (here, and throughout, 3+ denotes the collection of sets in Έ with positive measure: 3 + = {A € 3 : m(A) > 0}), and φ = φa'. A Ν is t h e i r s / return time function: 1 : Tnx e A}, then

No such formula is available for actions. We attempt to provide a substitute with Theorem 2.1. To complete this introduction, we deduce Theorem F from Theorem A, and, afterwards in Proposition 0, we show a little more than the nonexistence of a sequence of constants satisfying (1) from Theorem F (as in [C-R]). Proof of Theorem F. Suppose that lim s u p ^ ^ j ^ j j < oo on some set of positive measure, and in particular suppose that A € B+, Μ > 1 are such that S„f < Mb{nd) on Α V« > 1. Setting a = b~l we have ^ on A, for η > 1, we have

| as χ f , and therefore a(x + y) «11*111) < 0 0 V e > 0

whence hm sup g ^ j

· Ν such that

N(x,r).

Suppose η > N(x,r), ||c|| < n. By the random Kac inequality, 3 d e 7Ld such that Tc+dX ^ A, and < (p(Td+cx, oc+

2

< •

be as in the claim, then V Ν >

= 4 # (

_

| J

B { T

2

z , B{Tgx,Ggr)nT{_N

g

x , a

g

r ) \

Ν]1{χ)φΰ

< 4#Γ[_3Λ, 3n]2{jc} =

2

3 6 N

.

•

3. Counterexamples In this section, we show that generalization of Theorem 1.1 to an arbitrary non-negative, stationary, ergodic h d random field is impossible. Theorem 3.1. probability There

Let

space

Τ

be

X .

a f r e e , ergodic,

Suppose

is a measurable

a ( x

function

E { a { f ) ) =

+

probability y )

f : X — >

oo,

but


0, and choose M e Ν such that a(M)e > choose Ν e N such that < 8 for every η > N. By [K-W], 3 A € B(X) such that {TaA : ||a|| < 2 N ) are disjoint, and ( J

χ

that

Proof.

m i

standard

T

a

A ,

and

V n

>

TV,

χ

€

X .

Next,

A TLd Ergodic Theorem with Large Normalising Constants

To see this, note that for χ $ U||a|| Ν,

d

< (2n) M, whence a(2dndM)

a(Snf(x)) nd

nd

~

N.

To prove the theorem, fix €k = f a(fk)dm V*> 1 Jx Jx > k ^ oo ask —>oo Let k(x) = min {k > 1 : χ φ ΒΓ V r > k) m(Bk) < oo. It follows that for ί > κ{χ), η > Nt,

which is finite a.e. because

00

a(SnF)
(S„/*)+ £ «($„/*) *=i k=e+1 oo k=i+1

z

i + l j

4. Concluding remarks Suppose that Τ is an ergodic probability preserving TLd action on (X, !B,m), and suppose that / : X ->· Μ is measurable. Let a: [0, oo) - * [0, oo) be continuous, strictly increasing, and satisfy a(x + y) < a(x) + a(y), and a(x) = o(x) as χ —• oo. In this section, searching for a suitable generalization of Theorem 1.1, we'll try to characterise the situation lim sup —-ra(\Snf\] = oo a.e. «-•oo η \ J

12

J. Aaronson and B. Weiss

Let us define a Kac function for a set A e !B+ to be a measurable function φ: A satisfying

Ν

00

U

[J

Tg[

φ

-

α Τ Ϊ Θ ,

T

j

( f

-

g

-

p )

=

TJ β for all r with vr < 00. It is easy to see that in this case ur, vr will also be a sequence of upcrossing times at χ for the functions / , g, ρ, in the sense of Definition 1. Accordingly, a bound for the number of upcrossings according to Definition 1 will certainly give a bound for the number of upcrossings of an interval in the context of the Ratio Ergodic Theorem. Definition 1 is a special case of the definition of upcrossings in the more general setting used by Bishop. For simplicity we will not deal with Bishop's most general case. However, we will widen the definition of an upcrossing sequence, as follows. r

r

U r

Definition 2. Let f , g , ρ be nonnegative functions in L 1 . Let versions be chosen for the L 1 -functions T H f ~ g), g ρ ) , T ^ i f - p ) , j = 0,1 Let u v , r = 1 , 2 , . . . be elements in { - 1 , 0 , 1 , . . . , 00}, such that r t

00,

(6)

v r 5 u r +i for all r.

(7)

u

r


0 be a real number. Let / = φ, g = φ, ρ = c9. For any χ, let (wr)r=i,2.... be a sequence taking values in

18

Μ. A. Akcoglu, J. R. Baxter, and S. E. Ferrando

{0, 1 , 2 , . . . , oo}, such that w r < w r +\ if w r < oo, and such that the sequence (w r ) consists of exactly those integers η > 0 with η Τηφ{χ)>οΥ^Τ^Θ{χ). j=0

(11)

Let ur = wr — 1, vr = wr for all r = 1,2, Let ζ = (u r , iv)r=i,2,...· Then ζ is a sequence of generalized upcrossing times at χ for / , g, ρ so the function ω is an upper bound for the number of times that (11) holds. Theorem 1 implies in particular that the integral of ωρ is finite, so that ωρ is finite almost everywhere for every choice of c > 0. Thus yn Ψ lim = 0 a.e. on {Θ > 0}.

Σ;=ο ™

This is a well-known fact which is usually established in the course of proving the Ratio Ergodic Theorem (cf. Lemma 2.3 in Section 3.2 of [6]). Remark. As another illustration of the use of generalized upcrossing times, we note that Theorem 1 implies the main combinatorial result (Lemma 1) in [1],

Consequences of filling Although the upcrossing inequality for the Ratio Ergodic Theorem appears similar to that for submartingales, it might be noted that there is extra information in the filling scheme. Let's consider again the upcrossings of an interval (of, β) as described following Definition 1. Here the filling scheme statement of the upcrossing inequality is that the function (β — ά)ωθ can be completely filled by the mass of the function φ (see (31) below). A standard argument then shows that the ratio limit using φ and θ is greater than or equal to the ratio limit using (β — α)ωθ and θ. In particular if there are at least k upcrossings everywhere on the space then the ratio limit using φ and θ is at least k(ß — a). Similarly if there are at least k upcrossings on most of the space then the ratio limit using φ and θ is nearly k(ß — a) on most of the space. Statements of this sort are of course not generally true for submartingales.

2. Afillingscheme We continue to use the same space L 1 and operator Τ as in Section 1. Let / , g, ρ be given nonnegative functions in L 1 . We will define an alternating filling scheme. Intuitively, the function ρ will represent a hole, which we alternately fill, at each point, with mass from / and from g. As usual, the operator Τ is the shovel that we use to move the mass. We will define our alternating filling scheme soon, by inductively defining sequences of functions φη, θη, πη, κη, p„, ση, η = 0 , 1 , — The functions φη, θη, Ήηι Κη

A Repeated Filling Scheme for Upcrossings

19

are nonnegative L1 -functions. The functions pn, σ„ are bounded measurable functions taking nonnegative integer values. Our construction of these functions takes place in steps, labelled by the time n. Each step η has three parts, which we refer to as turns. During the first turn of step n, we attempt to fill a copy of the hole using the mass of / , if this is appropriate. During the second tum of step n, we attempt to fill a copy of the hole using the mass of g, if this is appropriate. During the third turn of step n, we move the unused mass of both / and g, using the operator T. The function φη represents the unused mass of / just before step n. The function πη represents the unfilled portion, just before step n, of the copy of the hole that we will try to fill with /-mass during the first turn of step n. The value pn (x) is the number of times that a copy of the hole has been completely filled at χ with /-mass prior to step n. The functions θη, κη, and ση have similar interpretations using g instead of / . For convenience we will also define some other functions which are expressible in terms of the functions already mentioned. We now begin the inductive definition of all these functions. It should be noted that we construct φ η +\, π>ι+ΐ» Όι+ΐ» Pn+i> 1 during step π, since these functions refer to conditions just before step η+ \. We also construct functions en, hn, /„, g„ during step n. These functions describe results of operations during step n.

Initial definitions Let r in all cases, proving (b) for η = m and proving the lemma. • As immediate corollaries, we see first that for any ζ e IX, lim pn(x) > length(f),

n-*oo

almost everywhere on Η (ζ), and hence that lim

pn(x)

η—too

(31)

> ω

almost everywhere. Combining this fact with Lemma 2 gives / ωράμ,


f is defined by ωf = {(x, y) | y € ω/(χ)}. Although each cof(x) is closed in X, the relation ωf is not generally closed in Χ χ X. We define: Ω/ = p I ( J / " c X x I

(3)

N>0n>N

Thus, y 6 Ω / ( χ ) iff for every neighborhood U of y, every neighborhood V of χ and every Ν e Ν there is an η > Ν and χ e V suchthat fn(x) e U. Again, for metrizable X, y € Ω/(χ) iff there is a sequence {n*} in Ν and a sequence {xk} in X such that njt ->· oo, Xk ->• jc and f"k (xk) y. Clearly Ω / is a closed relation containing wf and thus containing ω/. The containment may be proper. A subset A of X is called positively invariant if / ( A ) c A, negatively invariant if -1 / ( A ) c A, and invariant if / ( A ) = A. A set is positively invariant iff its complement is negatively invariant. An invariant set is positively invariant but it need not be negatively invariant unless / is injective. A set that is both positively and negatively invariant, such as X, need not be invariant unless / is surjective. In the important case when f:X X is a homeomorphism, a set is invariant iff it is both positively and negatively invariant. If A c X is a closed positively invariant set then (A, f\a) is called a subsystem of (X, / ) . It is clear that each a>f(x) and each Ω / ( χ ) is positively invariant and, since X is compact, they are in fact invariant. In this paper we will be concerned with the related notions of recurrence and transitivity. We say that χ is a recurrent point for / if jc G wf{x). If χ satisfies χ e Ω / ( χ ) then χ is said to be non-wandering. The system (X, / ) , or the map / , is called topologically transitive if ω/(χ) = X for some χ e X, and such a point χ is called a transitive point. Clearly a topologically transitive map on a compact space is surjective. (Note that topo-

When is a Transitive Map Chaotic?

27

logical transitivity requires more than the existence of a dense orbit, which is sometimes taken as the definition; in particular a transitive point is recurrent.) In case (X, / ) is topologically transitive, we write Trans/ for the set of transitive points of (X, / ) . The system ( X , f ) is said to be minimal if every point is a transitive point (Trans/ = X). A closed positively invariant subset Y c X is called minimal if the subsystem (F, / | y ) is minimal. The following result collects several elementary facts about compact (metrizable) systems. We will outline some key points of the proof. See the book of Akin [1] (Theorem 4.12) for further details. Theorem 1.1. Let (X, / ) be a compact system P. (1) The following conditions are equivalent: a. (X, / ) is topologically transitive. b. Ω/(*) = Χ for all χ e X. c. Trans/ is a countable intersection of open dense sets. d. Every open, non-empty negatively invariant subset of X is dense. e. Every closed, proper positively invariant subset of X has empty interior. (2) a. b. c. d. e.

The following conditions are equivalent: (X, / ) is minimal. X has no proper nonempty open negatively invariant subsets. X has no proper nonempty closed positively invariant subsets. X has no proper closed nonempty invariant subsets. Of(x) = X for every χ € X.

Most of these results are known and not difficult to prove. See the book of Akin [1] (Theorem 4.12) for details. We sketch the implication b —»· c in (1), which is not quite standard. If y e Χ, Ν € Ν, and ε > 0, let V = V(y, Ν, ε) consist of all χ e X such that d(fk(x), y) < ε for some k > N. Clearly V is open and it follows easily from b that it is dense. Moreover Trans/ is the intersection of the sets V(y, Ν, ε), for positive integers Ν, positive rationale ε and y in a countable dense subset of X. The implication c —• b follows from the Baire Category Theorem. The rest of the proof of (1) (a b, d ·Φ=» e, a -»• e, and d -»• b) is straightforward, as are the proofs of the equivalences in (2). The usual definition of a minimal system is the condition c of (2) in the above theorem. A straightforward Zorn's Lemma argument shows that any nonempty closed positively invariant set contains a minimal set. A point χ is called a minimal point if it is contained in a minimal set Y or, equivalently, if (Ö f(x), f ) is a minimal system. Obviously a minimal point is a recurrent point. It's well known that the converse is false, and we'll see some examples illustrating this later. It is useful to define the following sets of integers. If χ e X and A c X let R(x,A) = {η € Ν I fn(x) 6 A}. Clearly N \ Ä ( x , A ) = R(x,X\A). If U is a neighborhood of χ then R(x,U) is called the set of return times of the point χ to

28

Ε. Akin, J. Auslander, and Κ. Berg

the neighborhood U. Clearly a point χ is recurrent iff R(x,U) neighborhood U of x.

is infinite for every

Definition. A subset Τ c Ν is called relatively dense if there is a positive number L such that [n,n + L] Π Γ # 0 for all neN. Lemma 1.2. Suppose that U is an open subset of X such that R(x,U) is infinite but not relatively dense. Then there is a point y € U Π ω/(χ) such that U Π cüf(y) = 0. Proof. For η = 1,2, . . . there is an increasing sequence {£„} of positive integers such that fk"(x) e U and fkn+i(x) £ U for y" = 1,2 n. By compactness we may kn assume that f {x) y 6 U. Clearly y e cof(x). Now let j > 0. Then for all kn+J η > j, f (x) g U so f j ( y ) 0 it follows that a>f(y) Π U = 0. • Lemma 1.3. (1) Let Κ φ 0 be a closed positively invariant subset of ωf (JC). Let Ε be a closed set disjoint from K. Then R{x, E) is not relatively dense. (2) Suppose that Y is a minimal subset of aif{x) such that R(x, U) is relatively dense for every open set U containing Y. Then Y is the unique minimal set contained in cüf(χ). Proof Let U = X \ E. For any positive integer ν there is, since Κ is positively invariant, an open set V containing Κ such that /• / (V) c U for 0 < j < v. Since fn(x) G V for some n, the result follows. Let AT be a minimal subset of cof(x). If Κ φ Y then Κ ΠΥ = 0. There is thus an open set U containing Y such that U Π Κ = 0. By (1), R(x, U), and thus R(x, U), is not relatively dense contrary to our assumptions. • The above lemma may be used to establish the following classical theorem, whose proof we omit. It will also be used later, in the proof of Theorem 4.1. Theorem 1.4. Let (X, / ) be a compact dynamical system and let χ € X. The point χ is minimal iff R(x, U) is relatively dense for every neighborhood U of χ. For a transitive system (X, / ) we have already seen that the transitive points are residual (Theorem 1.1). This next theorem shows that non-minimal systems have a dense set of non-transitive points. Theorem 1.5. If the compact dynamical system (X, f ) is transitive but not minimal then the set of non-transitive points is dense. Proof Let V be a non-void open set. Since the set of transitive points is dense, we can choose χ £ V Π Trans/. The point χ is not minimal so there is a neighborhood U of χ suchthat R(x,U) is not relatively dense. Clearly we may choose U so that U c V. By Lemma 1.2 there is a point y e U such that wf(y) Π U = 0. In particular, y ψ Trans/.

•

When is a Transitive Map Chaotic?

29

II. Equicontinuity and almost equicontinuity We call a dynamical system ( X , f ) equicontinuous if the family of mappings { / " I η e N} is uniformly equicontinuous. In terms of a metric d for X, this means that for every ε > 0 there is a δ > 0 such that whenever x, y e X with d(x, y) < > ) in X with the product topology (the topology of pointwise convergence). This same set of points, with the topology of uniform convergence, will be denoted by C(N, X). A metric D for C(N, X) can be defined in terms of a metric d on X by D(x,x) = supJ(jc, , jc,·). Later we will let D denote the uniform metric on other function spaces. For example, C(X, X) will denote the space of continuous mappings from X into itself and D(g,h) : = sup d(g(x),h(x)). xeX

2

Define Of(x) = (x, f ( x ) , f (x), ...). We can regard 0 / as a map from X to X N or to C(N, X). With the product topology on the range, the orbit map 0 / is clearly continuous; this is not in general the case for the uniform topology. Given a metric d it is useful to introduce the metric df on X: df(x,y)=supd(fn(x),

fn(y)).

/JGN

Clearly convergence with respect to df implies convergence with respect to d but the converse is not generally true. Theorem 2.1. The following are equivalent: a. The system (X, f ) is equicontinuous. b. The map G/: X C(N, X) is continuous. c. The metrics d and df are equivalent (and therefore uniformly equivalent). d. The topology on X defined by the metric df is compact. e. There is a metric d* equivalent to d such that d*(f(x), f(y)) < d*(x, y) for all x, y € X. Proof. Conditions b and c are clearly reformulations of a. That c e (choose d* = d f ) , and e ->• a (choose 8 = ε in the definition of equicontinuity) are obvious. Clearly, c — d. Finally, d e since the identity map ( X , d f ) ->- (X, d) is continuous and thus, if (X, d f ) is compact, a homeomorphism. • The notion of equicontinuity can be localized in the obvious way. That is, χ e X is called an equicontinuity point (or (X, f ) is equicontinuous at χ) if for every ε > 0 there is a δ > 0 such that whenever y e X satisfies d(x, y) < δ then d(fn(x), fn(y)) < ε for all η e N. If every JC € X is an equicontinuity point then the compactness of X implies that the system (X, / ) is equicontinuous.

30

Ε. Akin, J. Auslander, and Κ. Berg

The proof of the following proposition is immediate. Proposition 2.2. The following are equivalent: a. χ is an equicontinuity point. b. The orbit map Of. X —• C(N, X) is continuous at x. c.

For every ε > 0 there is a 8 > 0 such that d(x,y)

< 8 implies df(x, y) < ε.

The next result is less obvious. Theorem 2.3. (1) Suppose χ is an equicontinuity point for f . Then a. · oo and Xk x such that fnk(xjt) y. Let ε > 0 and let 8 > 0 correspond to I in the definition of equicontinuity at x. Choose k so large that d(x,xk) < 8 and d(fnk(xk), y) < 8. By the choice of 8 we have d(fnk(x), fnk(xic)) < §· Then d(fnk(x), y) < ε and it follows that y e o)f(x). b. This is an immediate consequence of a. c. Let U be a neighborhood of x. We show that the return time set R(x, U) is relatively dense. Let ε > 0 be such that U contains B-$e{x), the open 3ε ball around x, and choose 8 > 0 corresponding to ε in the definition of equicontinuity at x. (In particular δ < ε.) Let ζ be a minimal point with d(x,z) < δ. By Theorem 1.4 R(z, Be(z)) is relatively dense. If k e R(z, Βε(ζ)), then d(x, fk(x)) < d(x, z) + d(z, fk{z)) + d(fk(z), /*(*)) < 3ε (since χ is an equicontinuity point). Therefore R(z, Βε(ζ)) C R(x, U) and so the latter set is relatively dense. Since U is arbitrary, χ is a minimal point, again by Theorem 1.4. (2) a b and c -» d are immediate, b -»• c follows from (l)a, and d -»· a is an easy exercise. • Recall that for any map between metric spaces, the set of continuity points of the map is a G$ (a countable intersection of open sets). If we apply this result to the map 0 / we obtain from Theorem 2.3 that the set of equicontinuity points of (X, / ) is a G$. We get sharper results by explicitly describing this Gg. To this end we introduce for k = 1, 2, the sets G* = {χ € X \ there exists a neighborhood U of χ such that X\,X2 e U d(fn(xι), fn{xi)) < \ for all η > 0}. Note that χ € Git iff χ has a neighborhood

When is a Transitive Map Chaotic?

31

whose df diameter is < ρ Also, it is easy to see that each Gjt is an open negatively invariant set and that HG* is the set of equicontinuity points of / . As an application of this representation of the set of equicontinuity points, we prove (in a somewhat different way) a result by Ausländer and Yorke [2, Corollary 2] which establishes, for minimal systems, a dichotomy between sensitivity and equicontinuity. We first define two terms that are central to this paper. Definition. A topologically transitive system is called almost equicontinuous if there is at least one equicontinuity point. A topologically transitive system is called sensitive if there is an ε > 0 such that whenever U is a non-void open set there exist x,y e U such that d(fn(x), fn(y)) > ε for some η e N. Equivalently, every nonempty open set has df diameter greater than ε The term chaotic, used in the literature with various meanings, was used in [2] to denote what we have here called sensitive. Theorem 2.4. Let (X, / ) be topologically transitive. If (X, / ) is almost equicontinuous then the set of equicontinuity points coincides with the set of transitive points (and so the set of equicontinuity points is a dense Gs )• In particular, a minimal almost equicontinuous dynamical system is equicontinuous. If (X, f ) has no equicontinuity points then it is sensitive. In particular a minimal system is either equicontinuous or sensitive. Proof. Let χ 6 Transf. Suppose that every G* is nonempty. Then (since G* is open) there is an η* > 0 such that /"*(*) e G*. Since G* is negatively invariant, χ e G*. Thus χ G HG*, which is the set of equicontinuity points of / . Furthermore, since / is transitive Ω / ( χ ) = X for all χ € X. If χ is an equicontinuity point cof(x) = Ω/(Λ;) by the previous theorem so o)f(x) = X. Each Gk is open and negatively invariant and their intersection is the set of equiconitinuity points. If there are no equicontinuity points then some Gk is empty by the argument of the first paragraph. It then follows that every nonempty open set has df diameter greater than | and so (X, / ) is sensitive. • The construction and analysis of (necessarily non-minimal) almost equicontinuous topologically transitive systems that are not equicontinuous will be carried out in the next two sections. The next result is the generalization of the theorem of Banks, et al, which we mentioned in the introduction. It appears as Theorem 1.3 in [5]. Theorem 2.5. Let (X, / ) be a topologically transitive system which is not minimal. Suppose that the set Μ of minimal points is dense. Then (X, f ) is sensitive. Proof. If (X, / ) is not sensitive then there is an equicontinuity point x, and χ is the limit of a sequence of minimal points. By Theorem 2.3 (lc), χ is a minimal point. But the equicontinuity point χ is a transitive point, so (X, / ) is minimal. •

32

Ε. Akin, J. Auslander, and Κ. Berg

Our next theorem relates the metric df to the set of transitive points. It will be used in the next section. There is no assumption of almost equicontinuity. Theorem 2.6. The metric df is complete, and Trans/ is closed with respect to d f . Proof. Let {**} be a df Cauchy sequence in X. Since d < d f , is a d Cauchy sequence so {x*} converges in the metric d to some χ e X. We will show that, in fact, [xk] converges to χ with respect to d f . Let ε > 0. Since {jcjt} is df Cauchy, there is an m e Ν such that if k,k' > m, then df(xk,Xk') < ε. Thus for each fixed η € Ν, d(fn(xk), /"(**')) < ε for lc,k' > m. Now let k' oo. n Then d(f (xk), /"(*)) < ε for all k > m. Since this holds for all η 6 Ν we have df(xk, χ) < ε for k > m. Therefore [xk] df converges to x. Thus df is complete. Now let {xk} be a sequence in Trans/ which df converges to χ € X. We will show that χ e Trans/. Let y e X, let Μ e Ν and let ε > 0. Choose k e Ν such that df(x,xk) < §, and, using transitivity, choose m > Μ such that d(fm(xk), y) < |. Then d(fm(x), y) < d(fm(x), fm(xk))+ d(fm(xk), y) < df(x, xk) +d(fm(xk), y) < ε. •

ΠΙ. Uniform rigidity Now for something completely different. In this section we will focus on a strong form of recurrence. We say that the compact system (X, / ) is uniformly rigid if there is a sequence {n*} suchthat nk ->· oo and /"* converges uniformly to Ιχ, the identity map of X. The notion of uniform rigidity was introduced by Glasner and Maon [4] who constructed minimal nonequicontinuous examples and also investigated some related notions, which are mentioned below. Somewhat earlier, Weiss and Katznelson [6] (without explicitly referring to the property) constructed a nonminimal example. We will analyze this example in Section 4. In this section, we develop some consequences of uniform rigidity (Theorem 3.4). This is obviously an extremely strong property, and there is room for a number of weaker properties. Recall that the point χ is recurrent if χ e cof(x). If X is topologically transitive, clearly any χ e Trans/ is recurrent. The system (X, / ) is called pointwise recurrent if every point is recurrent. Obviously a minimal system is pointwise recurrent; the construction of transitive nonminimal examples takes some work. Other conditions which have been studied [4] include weak rigidity - every finite product / χ · · · χ / is pointwise recurrent, and rigidity - the identity map Ιχ is the pointwise limit of some sequence / " ' . Lemma 3.1. Suppose (X, f ) is pointwise recurrent. Then the map f is surjective and every positively invariant set is both invariant and negatively invariant. Proof. Let A be any closed positively invariant set. For each a € A, a e ω/(a) c A, so A is the union of omega limit sets. As we pointed out in Section 1, every omega

When is a Transitive Map Chaotic?

33

limit set is invariant and so A is invariant. Applying this result to A = X we see that / is surjective. To show that A is negatively invariant, suppose that f ( x ) e A. Then x € ω/(χ) = eof(f(x)) C A . • Let C = C(X, X) denote the set of all continuous maps of X to itself, with the topology of uniform convergence. A metric D for C(X, X) is given by: D(f, g) = su P;t&xd(f(x),g(x)). Lemma 3.2. (1) (X, f ) is uniformly rigid iff for every ε > 0 there is an η > 1 such that D(fn, \χ) < ε. (2) If f is a homeomorphism and η ε Ν then D(fn, Ιχ) = D(f~n, Ιχ). Proof (1) is an easy consequence of the definition. As for (2), note that if f\, f2, g: X X with g surjective then D(f\ og,f2og) = D(f\, f2). If we put f\ = fn, f2 = Ιχ n n and g = f~ we obtain D(\x, f~ ) = D(fn, Ιχ). • Lemma 3.3. Let (X, / ) be a compact dynamical system such that f χ / is pointwise recurrent. Then (1) / is a homeomorphism. (2) If χ € X, (Df-Hx)f(x) is positively invariant and therefore, by Lemma 3.1, negatively invariant. Since χ € (of(x) it follows that cof~l(x) c a)f(x). If x,y € X, (/"' (*), fn'(y)) ->· (x,y) for some sequence {«, } with oo. It follows that d / ( f ( x ) , f(y)) > d{x, y). But clearly df(x, y) = sup{d(x, y), d f ( f ( x ) , f(y))} so df(x,y)

= df(f(x),f(y)).

•

Theorem 3.4. Suppose (X, f ) is uniformly rigid. Then: (1) The restriction of f to any subsystem is uniformly rigid. Any finite Cartesian product f χ · · · χ / on Xk is uniformly rigid. Any power fn on X is uniformly rigid. (2) / is a homeomorphism and (X, f ~ l ) is uniformly rigid In fact {/""*} converges uniformly to l χ whenever {/"*} does. (3) If χ € X then ω / Ο ) = a)f~x(x), so Trans/ = Transy-i. (4) The metrics df and dj-1 coincide on X, and f is an isometry for this metric. If {/"*} converges uniformly to Ιχ with respect to the given metric d then it also does so with respect to the metric d f . Proof The first two statements are immediate. To prove the third, note that /*' lχ nk uniformly implies that f < -> \χ = Ιχ uniformly P. (2) follows from (1) of Lemma 3.3 and (1) of Lemma 3.2.

34

Ε. Akin, J. Auslander, and Κ. Berg

By (2) of Lemma 3.3, ωf !(JC) C ω/(χ). But / 1 is also uniformly rigid, so cof(x)£a>f-l(x). By (3) of Lemma 3.3, / is a df isometry. It follows that / - 1 is a df isometry. Then for all k e N, df{x, y) = df(f~k(x), f~k(y)) and therefore df[x, y) = supketidf(f-k(x), r k ( y ) ) > s u p k ^ d ( f - k ( x ) , f ~ k ( y ) ) = df-x(x,y). Therefore - I df > df-1. Since / is also uniformly rigid we have df-1 > df as well. The proof of the final statement in (4) follows easily from the definitions. • We now bring together the two parts of our story. The next lemma will be crucial in accomplishing this. Lemma 3.5 (Hinge Lemma). Let (X, / ) be a topologically transitive system and let χ e Trans/. Let ε, 8 > 0. Then the following are equivalent: (HI) If y € X and d(x, y) < 8 then d(fk(x), fk(y)) < ε for all k e N. (H2) If η eN and d(x, fn(x)) < 8 then d(fk(x), fn+k(x)) < e for all k € N. n n (H3) If η e Ν and d(x, f (x)) < 8 then d(y, f (y)) < ε for all y e X. Proof We prove HI H2 H3 H2 HI. n HI H2: Apply HI to λ = χ and y = f (x). H2 H3: We see from H2 that d{y, fn(y)) < ε for all y = fk{x). By continuity the inequality holds for all y e ω/(χ) = X. H3 H2: Apply H3 to χ = λ and y = fk{x). H2 HI: Fix U N . We see from H2 that d(fk(x), fk(y)) < ε for each y = fn (x) that satisfies d{x, y) < 8. Any y satisfying d(x, >>) < 8 is the limit of some sequence fn'(x) with d(x, /"' (JC)) < 8, and continuity implies k k d(f (x),f dy)) 1χ. (3) For some transitive point x, the convergence / " ' (JC) —• χ implies the uniform convergence /"' —> Ιχ.

When is a Transitive Map Chaotic?

35

(4) There is a transitive point χ such that for every ε > 0 there exists 8 > 0 such that d(fn(x),x) < 8 implies d(fn+k(x), fk(x)) < ε for all Ice Κ (5) The topologies induced by the metrics d and df on Trans/ agree. (6) The restriction of the orbit map 0/ to Trans/ defines a continuous map from Trans/ to C(N, X). Moreover, these conditions imply that (X,f) is uniformly rigid. Proof. (1) -»· (2) follows from the implication (Hl) ->· (H3) in the Hinge Lemma. (If ε > 0 choose 8 > 0 to correspond to ε in the definition of equicontinuity at the transitive point x.) (2) (3) since (X, f ) is topologically transitive. (3) —> (4) follows from (H3) (H2) in the Hinge Lemma. (4) —• (1): We use the Hinge Lemma again, this time the implication (H2) (HI). Let χ be the transitive point whose existence is assumed in (4). It is sufficient to show that χ is an equicontinuity point. Let ε > 0, and let 8 > 0 correspond to ε as in (4). Then (H2) of the Hinge Lemma is satisfied for this ε and 8. So for every ε > 0 there is a 8 > 0 so that (HI) holds, which is exactly equicontinuity at x. We have shown that the first four statements are equivalent. We will now show that (1) -> (6) (5) (4). (1) (6): Recall that almost equicontinuity is equivalent to the property that the transitive points are the points of continuity of the orbit map 0 / . Thus 0 / is certainly continuous when restricted to Trans/. (6) ->• (5): To prove (5) it is sufficient to show: if [xj} is a sequence in Trans/, x e Trans/, and d(x, χy) -»• 0 then d/(x, Xj) 0. But this follows immediately from the assumption of (6), namely the continuity of the orbit map 0 / on Trans/. (5) —• (4): First recall that if χ € Trans/ then so is fn(x) for all η € Ν. Now fix χ g Trans/, and let ε > 0. Since the metrics d and df are equivalent on Trans/ there is a 8 > 0 such that whenever y € Trans/ with d(x,y) < 8 then d/(x,y) < ε, so d(fk(x), fk(y)) < ε for all k e N. In particular, if y = / " ( * ) where d(fn(x),x) < 8, then d(fk(x), fn+k(x)) < ε for all k e Ν which proves (4). Finally, since a transitive point is recurrent, uniform rigidity follows from (2). • The following corollary summarizes properties of almost equicontinuous systems. It is a direct consequence of the theorems in this section and Theorem 2.6. Corollary 3.7. If the system (X, / ) is almost equicontinuous, then f is a homeomorphismand (X, / ) is uniformly rigid. The system (X, f ~ l ) is also almost equicontinuous, Trans/ = Trans/-1, and a point is a transitive point iff it is an equicontinuity point (for either f or / - 1 ) . The set of transitive points is residual, and on it the metric df = df-1 is complete and induces the same topology as d. The restriction of f to Trans/ is an isometry with respect to d f .

36

Ε. Akin, J. Auslander, and Κ. Berg

After this extensive and delicate analysis of almost equicontinuous nonminimal systems, it is time to show that this class is nonempty, and this is what we do in the final section. For an alternative construction see [5, Prop 1.5].

IV. The examples of Weiss and Katznelson We will now construct a class of examples of uniformly rigid systems, some of which are almost equicontinuous (and not equicontinuous). The construction we present below was outlined by Weiss and Katznelson [6], who were, in turn, inspired by a much earlier construction of Nemitsky [7]. The examples are all subsystems of the so called Bebutov system, which is defined by the shift map S on / N . ( / = [0,1], the closed unit interval, / N is provided with the product topology - the topology of pointwise convergence - and the shift map is defined by S(a)(n) = a(n + 1) for or e / N and η G N.) A metric which generates the product topology is given by d (α, β) = sup π=0

.

(1)

1

It will also be convenient to consider the uniform topology on / N , with the metric D given by D(a, β) = sup |α(η) — β{η)\. ne Ν

(2)

It follows that D(ct, β) = max {J (α, β), D(S(a), S(ß))}.

(3)

Now for the construction. Let L >2 and let ao be any function from [—1,1] to I satisfying αο(-1) = a o ( l ) = 1 and |αο(*ι) - aote)l
0 define ap by ap(s) = a\Then (for s, s\, S2 e M): 0 < ap(s) < 1, ap(-p)

= ap(p) = \,ap(s+2p)

\ap(si) -ap(s2)\ In particular, setting s\ = ρ and

= ap(s),

L < — |ii - J2I. Ρ

(5)

= ρ + s, we see that for |s|
aoo > a Pi from (6) we immediately obtain: \-ei Pj Applying the same elementary result to (5), since ^ < L, we also obtain a Lipschitz estimate for a00: |aoo(j + r) - aoo(s)\ < L\r\ for all j, r € R. Now we are ready to define our uniformly rigid system. For a fixed choice of the sequence {£,}, we define the sequence α € IN by a(n) = {floo(w)}. We call this the WK sequence associated with the function üq. If we begin with the constant function ao = 1, then aoo = 1 and the associated WK sequence, which we denote by Θ, is the constant sequence a^in) = 1 for all n. Of course θ is a fixed point of S: S(9) = θ. Theorem 4.1. Let a be the sequence defined in the previous paragraph and let X be the orbit closure of a in the Bebutov system (/N, S). Then the subsystem (X, S) is a uniformly rigid system which has the fixed point {0} as its unique minimal set. If α 0 (0) φ 1 then (X, S) is not minimal. Proof. We first show that for all β e X and all i we have D(S2P'(ß),ß) 0 there is a 8 > 0 such that a\(t) < ai(0) + 8 implies | a i ( j + / ) — αι(ί)| < ε for all s e Μ Proof of the Lemma. Let e > 0. ao(t) < a 0 (0) + 8 then |r| < Write t = u + t\ where u is an αΐ(0) + δ > αι(ί) = αο(ίι) so |ίι| \ai(s + ti)-ai(s)\ a\(t). even integer and t\ € [—1,1]. Then ao(0) + 0 there is a 8 > 0 such that 1 a^t) > aPi(t) = α , ( ^ ) for all pi so \aPi, (s +1) - aPi (j)| = |ai(^· + - α ΐ(^:)Ι < ε by Lemma 4.3. Talcing the supremum over all i then establishes the required property for a^ (by (10)). Now we show that the system (X, S) is almost equicontinuous. For, if ε > 0, let 8 > 0 be chosen to correspond to ε as above. Then, if d(Sn (a), a) i(AUB)) f(A))+f(B)).

A la lumiere des resultats pr£cedents, on peut se demander si (1.1.1) peut s'&ioncer ä l'aide de la density naturelle sur les entiers; et si 1'on peut caracteriser la propri&d suivante: (1.1.3)

p.s.

' 1) une suite de variables aleatoires definies sur (Ω, 21, P) et a valeurs dans N. S'il existe une suite (α*, k > 1) de reels strictement positifs avec lim ak = +oo, telle que: k-y oo a)

Ve > 0,

lim Ρίω e Ω : k-t-oo 1

ak

_ i| < ε ) - ι

44

Μ. Atlagh et Μ. Weber

et

»

Σ ^ 1 0 assez petit, posons (2.1.8)

Αε = Α + εΙ = {x: d(x,A)

0. Soit ε > 0. Introduisons pour tout i e Ν les quantites: (2.1.9)

*, Π Ωι avec i > sup(j'o() e A;

done, pour tout 0 element de V/, et pour tout r element de Ιη,,μί+ιω on a: W(t0) VF

(ω) = Yte(co)VÖ - Υα(ω)

+ Yu(a>) G A + ε/ = A£.

Soit η un entier dans Ιη,μί+Ιω (il en existe car i est supörieur ä ι μ ) , done η est un Element de Ιμ,ϊ et, par (2.1.1), Κ(ΣΙ=ι η){ω) 7= = y/n

w m — +Jn

(

avec θ g Vh et done Vn d'oü (2.1.10). Etape 2. Posons UmÜ) = W(M') stationnaire U = {W(M')M~2, t > sürement, bß =

t > 0. L'ergodicit6 du processus gaussien 0}, carfortementm^langeant, impliquequepresque

1 3 lim - Υ] J

f lßi = / Jaι

1 ; lim - V

/* \Di = /

et o). Alors, par (2.1.10)', il existe un ensemble Ω2 de probability 1 et contenu dans Ω ι tel que sur Ω2: c a r d j l < i < j : 3n g IM,i Π Ν : lim i n f 00

IΥ» - Υίθ4θI

< IY u -Yml

g A