Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions

270 Pages·1997·7.73 MB·English

by  Ethan Akin (auth.)

Recurrence in
Topological Dynamics
Furstenberg Families
and Ellis Actions
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Recurrence in
Topological Dynamics
Furstenberg Families
and Ellis Actions
Ethan Akin
The City College
New York, New York
Springer Science+Business Media, LLC
Llbrary of Congress Cataloglng-ln-Publlcatlon Data
Akln. Ethan. 1946-
Recurrence In topologlcal dynamlcs : Furstenberg famllles and
Ellls actlons I Ethan Akln.
p. c •. -- (The unlvers1ty ser1es 1n .athemat1cs)
Includes b1bl1ograph1cal references and 1ndex.
ISBN 978-1-4419-3272-3 ISBN 978-1-4757-2668-8 (eBook)
DOI 10.1007/978-1-4757-2668-8
1. Topolog1cal dyna.1cs. 2. Po1nt mapp1ngs (Mathe.at1cs)
1. T1tle. II. Ser1es: Un1vers1ty ser1es 1n mathe.at1cs (Plenu.
Press)
QA611.5.A38 1997
514' .322--dc21 97-24363
CIP
ISBN 978-1-4419-3272-3
© 1997 Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1997
Softcover reprint of the hardcover Ist edition 1997
http://www.plenum.com
AU rights reserved
10987654321
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise,
without written permission from the Publisher
For Dr. Karl Gruber, Ph.D.
In memory of a warm friend.
Preface
In the long run of a dynamical system, after transient phenomena have passed away,
what remains is recurrence. An orbit is recurrent when it returns repeatedly to
each neighborhood of its initial position. We can sharpen the concept by insisting
that the returns occur with at least some prescribed frequency. For example, an
orbit lies in some minimal subset if and only if it returns almost periodically to
each neighborhood of the initial point. That is, each return time set is a so-called
syndetic subset ofT= the positive reals (continuous time system) or T = the
positive integers (discrete time system). This is a prototype for many of the results
in this book. In particular, frequency is measured by membership in a family of
subsets of the space modeling time, in this case the family of syndetic subsets of
T. In applying dynamics to combinatorial number theory, Furstenberg introduced
a large number of such families.
Our first task is to describe explicitly the calculus of families implicit in
Furstenberg's original work and in the results which have proliferated since. There
are general constructions on families, e.g., the dual of a family and the product
of families. Other natural constructions arise from a topology or group action
on the underlying set. The foundations are laid, in perhaps tedious detail, in
Chapter 2. The family machinery is then applied in Chapters 3 and 4 to describe
family versions of recurrence, topological transitivity, distality and rigidity. Many
different notions are unified by this family viewpoint. In particular, various ideas
of mixing appear as family versions of transitivity. In Chapter 5 we use Gelfand
theory to describe how such properties are carried over to compactifications of the
original system.
For compact systems Robert Ellis developed something completely different.
He defined the enveloping semigroup of a system by taking the pointwise closure
of the set of maps of the flow. The result is an important example of a class of
compact semigroups which we call Ellis semigroups. The idempotents play an
important role in the resulting theory. For example, a point is recurrent exactly
vii
viii Preface
when it is fixed by certain idempotents of the enveloping semigroup. We review
the theory of Ellis semigroups and their idempotents in Chapter 6.
The two viewpoints are brought together in Chapter 7. As a specific example
consider the discrete case with T the set of positive integers. Addition on T
extends to define a nonabelian composition on the Stone-Cech compactification
f3T, giving this space the structure of an Ellis semigroup. A compact, discrete
dynamical system is an action ofT on a compact space and such an action extends
to an action of the semigroup. On the other hand, the closed subsets of f3T
correspond to filters on T with the points associated to ultrafilters, i.e., maximal
filters. Filters and their duals are families and the family constructions for them
have natural semigroup interpretations. Furthermore, the semigroup structure
yields special results for such families. On the other hand, there are important
families which are neither filters nor filterduals. Such families can be used to
describe aspects of f3T.
Finally, in Chapter 8 we apply all the earlier work to study equicontinuous and
related systems.
For help with this work I would like to thank my fellow travelers in topological
dynamics, particularly Joseph Auslander, Ken Berg and Eli Glasner with whom I
have shared many conversations both real and virtual. Especially fruitful were my
meetings with the two artists who inspired all this: Hillel Furstenberg and Robert
Ellis. At the production end, Kate March once again translated my pen scratchings
into a manuscript, and at Plenum it was a joy to meet and work with Sy Marchand
(it was a pleasure even to lose an editorial argument with him). To my wife Jean:
Thank you for your continuing support of my mathematical enthusiasms, not to
mention your patience when I used math as an excuse to avoid other work.
Contents
Introduction
1. Monoid Actions 11
2. Furstenberg Families 23
3. Recurrence 53
4. Transitive and Central Systems 75
5. Compactifications 101
6. Ellis Semigroups and Ellis Actions 133
7. Semigroups and Families 155
8. Equicontinuity 193
Appendix. Semicontinuous Relations and Almost Open Maps 229
References 261
Index 263
ix
Introduction
Iff : X ----> X is a continuous map and x E X, we say that y is a limit point for
the associated dynamical system with initial value x, or y is an co limit point of x,
wheny is a limit point of the orbit sequence {F(x): nET} where Tis the set of
nonnegative integers. This means that the sequence enters every neighborhood of
y infinitely often. That is, for any open set U containing y, the entrance time set
N(x,U) ={nET: F(x) E U} is infinite.
It is often useful to keep track of just how frequently these entrance times
occur. In Recurrence in Ergodic Theory and Combinatorial Number Theory,
Hillel Furstenberg used families of subsets of T to keep track of the frequencies.
A family :.F for T is a collection of subsets, i.e., a subset of the power set P of
T, which is hereditary upward. That is, if F1 c F2 and F1 E :.F then F2 E :.F. A
family is proper if it is a proper subset of P, i.e., :.F #0, P. In view of heredity,
this says that :.F is proper when T E :F and 0 rj_ :F. For a proper family :.F, we
say that y is an :.F co limit point of x if N(x, U) E :F for all neighborhoods U of y.
In Chapter 2 we present the elementary theory implicit in Furstenberg's work for
such families; in Chapters 3 and 4, we apply it to dynamical systems in general
and to topologically transitive systems in particular.
This family approach goes back at least to Gottschalk and Hedlund (1955),
who introduced admissible subset collections to unify several notions of recurrence
-exactly our purpose.
When the state space X is compact, the semig roup theory of Robert Ellis can be
applied. For our purposes, this is best described here as an extension oft he action of
Ton X to an actionof,BT, the Stone-CechcompactificationofT, on X. Forx EX,
we can define the orbit map % : T ---->X by ({Jx ( n) = _r (x). Since T is discrete, this
is a continuous map which therefore extends to a map <l>x : ,BT----> X. For p E ,BT,
we write p(x) = <l>x(p), thus regarding elements of ,BT as functions on X. Ellis
observed that ,BT has a natural semigroup structure satisfying (pq)(x) = p(q(x)).
However this hybrid between the algebra and the topology on ,BT appears at first
E. Akin, Recurrence in Topological Dynamics
© Springer Science+Business Media New York 1997
2 Introduction
to be rather mulish. Right translation is continuous, but left translation is not; i.e.,
p pq is continuous, q pq is not. Similarly p p(x) is continuous, but p
f-+ f-+ f-+
itself, x p(x), is usually not. Perhaps unsurprisingly the composition in {3T,
f-+
while it extends addition on the dense subset T, is not commutative. Despite these
infelicities, the semigroup structure on f3T has proved very fruitful in studying
dynamical systems.
The reader should be aware that Ellis uses right rather than left actions, so
his semigroup structure on {3T is the reverse of ours. His left translations are
continuous, and his compatibility equation reads (xq)p = x(qp). Thus the view in
Chapters 6 and 7 is the mirror image of the Ellis way.
We can regard the points of {3T as ultrafilters on T. A filter is a proper family
that is closed under the operation of intersection. An ultrafilter is a maximal filter.
Thus the two approaches meet. The Furstenberg theory applies to ultrafilters as
it does to all families; some of the Ellis constructions are nicely expressed in a
family way. Furthermore the family approach is more general. Various family
constructions do not preserve the filter property. On the other hand, the semigroup
structure with its associative law and collections of idempotents reveal properties
about certain special families that would not be otherwise apparent.
An outline of our book follows.
1. Monoid Actions
Once you abandon compactness assumptions, you discover that various dynamic
notions like equicontinuity and chain recurrence are really uniform space notions.
The very definitions require a uniform structure. The associated topological spaces,
i.e., completely regular spaces, appear in full generality when you take arbitrary
products and subsets. Each dynamical system in this book is a uniform action of an
abelian uniform monoid on a uniform space, written ({) : T x X ---. X. This chapter
presents set-up work that enables the reader to make sense of these phrases.
An abelian topological group has a unique translation invariant uniform struc
ture obtained from neighborhoods of the identity. An abelian uniform monoid is a
submonoid T of an abelian topological group; i.e., T is closed under addition and
zero is in T, but inverses may not be. Also T satisfies a mild technical condition,
the Interior Condition, on the set of tails. FortE T, the associated tail I; is the im
age ofT under translation by t, 1; = {s + t : s E T}. Any discrete abelian monoid,
e.g., the nonnegative integers, Z+, and any abelian topological group; e.g., R, is
uniform monoid. The nonnegative reals R+ under addition is also uniform.
Note: After Chapter 1 all monoids are assumed to be abelian unless otherwise
mentioned; all topologies are assumed to be Hausdorff.
An action is a function ({) : T x X ---. X such that time t maps defined by
f(x) = ({)(t,x) satisfy the composition property fi o f2 = fi +t2. For({) to be a