Minimal Flows and Their Extensions (North-Holland Mathematics Studies, 153)

MINIMAL FLOWS AND THElR EXTENSlONS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (122)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

153

MINIMAL FLOWS AND THEIR EXTENSIONS

Joseph AUSIANDER Department of Mathematics University of Maryland CollegePark MD 20742 U.S.A.

1988

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

0

OXFORD

TOKYO

Elsevier Science Publishers B.V., 1988

All rights reserved. No part of this publications may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 044470453 1

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Library of Congress Cataloging4n-PublicationData Auslander, Joseph, 1930Minimal flows and their extensions. (North-Holland mathematics studies ; 153) (Notas de matedtica ; 122) Bibliography: p. 1. Topological dynamics. I. Title. 11. Series. 111. Series: Notas de matemztica (Rio de Janeiro, Brazil) ; no. 122. QAl.N86 no. 122 iQA611.51 510 s 15141 88-1 51 14 ISBN 0-444-70453-1 ( U . S . )

PRINTED IN THE NETHERLANDS

V

Introduction In this monograph, I present certain developments in the abstract theory of topological dynamics, with emphasis on the study of minimal flows and their extensions. The main theme is the internal structure of minimal flows and the relations among different minimal flows, rather than such questions as their location and stability as subsets of a larger flow (which arise in those areas of topological dynamics which are inspired by the qualitative theory of differential equations). By a

flow we

mean the jointly continuous action of a topological

group on a compact Hausdorff space. A minimal subset of a flow is one which is "dynamically indecomposable" - a closed invariant set which contains no proper closed invariant non-empty subsets.

If the space itself

is minimal under the action of the group, we call it a minimal flow. Almost by definition, the aim of the subject is the classification and construction of all minimal flows. This has been achieved only in the case of equicontinuous minimal flows, and we study these in detail. A fruitful way of looking at minimal flows in general is to consider how

they are related to equicontinuous flows, and we investigate some classes of flows which are "close" to the equicontinuous ones.

Most notable

among these are the distal minimal flows, the study of which culminates in the beautiful structure theorem of H. Furstenberg. This leads to the study of distality, and its opposite, proximality, in arbitrary minimal flows, and also has inspired more general structure theorems. If the "objects" of topological dynamics are the flows, the "morphisms" are the continuous equivariant maps - the homomorphisms (or

Introduction

vi

extensions).

Most of the notions of topological dynamics can be

"relativized" - one speaks of distal and proximal extensions for instance. Indeed, the relative notions are essential in formulating and proving the structure theorems in the subject. Another major theme in this book is what might be called the "universal" approach. That is, we study entire classes of minimal flows, rather than flows in isolation. An important concept in this direction is disjointness, which is a kind of independence condition. For example, we show that the weakly mixing minimal flows are precisely those which are disjoint from every equicontinuous minimal flow. This approach is also exemplified by the use of the universal minimal flow, of which every minimal flow is a factor, and which acts on every minimal flow.

As was mentioned earlier, the theory is developed in the context of the action of an arbitrary group on a compact Hausdorff space. However, in some cases (the Furstenberg structure theorem, the equicontinuous structure relation, and disjointness) it is assumed in addition that the phase space is metrizable and/or that the flow admits an invariant measure. Chapter 13 pursues a topic which is somewhat apart from the main development of the book.

It is devoted to a remarkable theorem of

Kakutani which shows that a large class of (real) flows can be represented as flows on a function space. Some generalizations to other acting groups are also presented. Examples of minimal flows, which, as in all branches of mathematics, indicate the boundaries of the theory and suggest general theorems, are woven into the theoretical development. In most cases in the examples the acting group is the integers (powers of a single homeomorphism) o r the real numbers (a one parameter flow).

Introduction

vii

The book should be accessible to a student who has had courses in real analysis (including the elements of functional analysis) and general topology. I have included appendices on uniform spaces and convergence in topology (by means of nets) since these topics are unfortunately no longer included in elementary graduate education in the United States. Of course, I have been influenced by and am indebted to many books on topological dynamics. A bibliography follows this introduction. The monograph of Nemytskii and the book of Nemytskii and Stepanov are important early contributions and contain much material which is related t o stability theory of differential equations. The A . M . S . Colloquium volume of Gottschalk and Hedlund on which the author was "raised" as a graduate student, develops the notation which is used in the present work and played a significant role in the development of the subject. The monograph of Ellis, which appeared in 1970, was an extremely important contribution, and treats many of the same topics as we do, but from a somewhat uncompromising algebraic viewpoint. The book of Bronstein also covers much of the same ground as the present work.

The more specialized

monographs of Glasner and Furstenberg should also be mentioned. The latter, in particular, is a tour de force in which topological dynamics and ergodic theory are applied to obtain deep results in combinatorial number theory. Finally, a number of books in ergodic theory, notably those of Parry, Petersen, Walters, and Denker, Grillenberger and Sigmund, contain substantial amounts of material on topological dynamics.

I was introduced to topological dynamics by Professor Walter Gottschalk over thirty years ago.

Iwthe course of writing this book, I

have received help from many individuals.

Among those who made sugges-

tions, answered questions or read parts of the manuscript are Ken Berg, Ethan Coven, Gertrude Ehrlich, Elie Glasner. Jonathan King, the late

Introduction

viii

Doug McMahon, Bill Parry, Jonathan Rosenberg, Jaap van der Woude, and Jim Yorke. Special thanks are due to Nelson Markley, who read several chapters, suggested a number of changes and additions, and supplied firm but constructive criticism. Professor Leopoldo Nachbin invited me to contribute this monograph to the prestigious series "Notas de Matematica." I owe him many thanks for his encouragement and patience during its preparation. The technical typists at the University of Maryland, Kristi Aho, Virginia Sauber, and Stephanie Smith capably produced the camera ready manuscript and cheerfully endured my many changes and corrections. Jim Hummel helped with a typographical problem.

I would also like to thank

the editorial staff at North Holland Press for their cooperation.

I have two further acknowledgements. The first is to the profound influence of Robert Ellis. More than any other individual, Bob is responsible for the development of the subject of topological dynamics. Among h i s many contributions which appear in this book are the enveloping semigroup, the joint continuity theorem, and several of the structure theorems for minimal flows. But of greater than o r equal importance to his specific contributions is Bob's insistence that topological dynamics is a theory, and not merely a collection of techniques.

I hope that this

point of view manifests itself in the present work. Finally, I want to thank my dear friend Barbara Meeker for her love and support. ThAs was expressed by a skillful and judicious combination of encouragement and prodding, which provided me with the self confidence to complete this work. J . Auslander College Park, MD January, 1988

USA

Bibliography 1.

J. Auslander, L. Green, and F. Hahn, Flows on.homogeneousspaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, 1963.

2.

I . U . Bronstein, Extensions of minimal transformation groups, Sitjthoff and Noordhoff, 1979, (Russian edition, 1975).

3.

M. Denker, C. Grillenberger, K. Sigmund, Eraodic theory on compact spaces, Lecture Notes in Mathematics, vol. 527, Sprin’ger-Verlag, 1976.

4.

R. Ellis, Lectures in topological dynamics, Benjamin, 1969.

5.

H. Furstenberg, Recurrence in ernodic theory and combinatorial number theory, Princeton Univ. Press, 1981.

6.

S. Glasner, Proximal flows, Lecture Notes in Mathematics, vol. 517,

Springer-Verlag, 1976. 7.

W.H. Gottschalk and G.A. Hedlund, Topological dynamics, Amer. Math. SOC. Colloq. Publ. vol. 36, 1955.

8.

V.V. Nemytskii, Topological problems in the theory of dynamical systems, Uspehi. Math. Nauk. 5 (19491, Amer. Math. SOC.Translation, ~

no. 103 (1954). 9.

V. V. Nemytskii and V. V. Stepanov, Qua1 itat ive theory of differential eauat ions, Princeton Univ. Press, 1960 (Russian edit ion, 1949).

10

W. Parry, Topics in ernodic theory, Cambridge Univ. Press, 1981.

11

K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.

12

W. A. Veech, Topological ‘dynamics,Bull. Amer. Math.

SOC. 83, 775-830

(1977). 13.

P. Walters, Lectures in ergodic theory, Lecture Notes in Mathematics, vol. 458, Springer-Verlag. 1975.

14.

J. C.S.P. van der Woude, Topological dynamix, Mathematische Centrum, Amsterdam, 1982.

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xi

Contents Introduction

V

Bibliography

ix 1

Chapter 1

Flows and Minimal S e t s

Chapter 2

E q u i c o n t i n u o u s Flows

35

Chapter 3

The E n v e l o p i n g Semigroup o f a T r a n s f o r m a t i o n Group, I

49

Chapter 4

J o i n t C o n t i n u i t y Theorems

57

Chapter 5

D i s t a l Flows

65

Chapter 6

The E n v e l o p i n g Semigroup, I1

83

Chapter 7

The F u r s t e n b e r g S t r u c t u r e Theorem f o r Distal Minimal Flows

95

Chapter 8

U n i v e r s a l Minimal Flows and Ambits

115

Chapter 9

The E q u i c o n t i n u o u s S t r u c t u r e R e l a t i o n and Weakly Mixing Flows

125

Chapter

The A l g e b r a i c Theory of Minimal Flows

137

Chapter

Disjointness

149

Chapter

I n v a r i a n t Measures o n Flows

169

C h a p t e r 13

Kakutani-Bebutov Theorems

183

C h a p t e r 14

G e n e r a l S t r u c t u r e Theorems

195

Appendix I

Nets

253

Appendix I1

Uniform S p a c e s

259

This Page Intentionally Left Blank

1

Chapter 1 Flows and Minimal Sets Topological dynamics may be regarded as the study of "long term" or "asymptotic" properties of families of self maps of spaces.

In most

cases of interest, the collection of maps is a group under composition. The natural setting is that of a transformation group. 1.

Definition. A transformation group is a triple

is a topological spac'e, continuous map of

X

x

(X.T,n) where

T is a topological group, and n

X

is a

T to X, satisfying

(i)

n(x,e) = x

(x E X, e the identity of T)

(ii)

n(n(x,s),t) = n(x,st)

(x

E

X, s , t E TI.

A synonym for transformation group is flow, - and we will mostly use the

latter term. X

is called the -phase space and T

the phase group or --

act ing group. -Each t E T defines a continuous map

t (x) = n(x,t).

71

nt

of X

to X

by

If t,s E T, it is immediate that nSnt = nts; in e

,

the identity map of X, so each nt t -1 = nt -1 . homeomorphism of X onto itself, with ( n ) particular ntnt-' =

IZ

With occasional exceptions, we will suppress the map and just write on" x

xt

in X to obtain another point

flow appear as:

We regard t

in place of n(x,t).

the map

(x,t)Hxt

xt

in X.

II

is a

notational y,

in T as "act ng

Thus the axioms for a

is continuous, xe = x, and

(xslt = x(st). We will usually write

(X,T) (or just X,

if the group T is

understood) for a flow. What we have defined as a transformation group could be called a right

chapter I

2

transformation group (or right flow). flow

(p,C,X) or

Of course, we could define a left

(C,X) in the obvious way. p

g(hx) = (ghlx, etc.

If

T can be defined by

tx = xt

:

GxX~X, p(g,x) = gx,

(X,T) is a (right) flow, then a left action of -1

.

For the most part, we will be concerned with right flows. However, we

will sometimes encounter a situation where we have two group actions on a space. A bitransformation group mation group (gxlt = g(xt) every t E TI.

(G,X,T) consists of a right transfor-

(X,T) and a left transformation group

(x

E

X, g E G, t E TI

Thus we may write

( s o every

(C,X) such that

g E G

"commutes" with

gxt unambiguously.

We will always suppose the phase space X of a flow is Hausdorff this is a standing assumption. (Of course, in any constructions we carry out, o r examples we present, the Hausdorff property must be verified; this will usually be trivial.) In fact, the theory which we will develop

is for actions of groups on compact Hausdorff spaces, but we will not assume compactness for the time being. As for the group

T; it can be abelian or non-abelian, although

almost all of our examples are of the former. It is not to be compact,

or act in a compact manner.

While actions of compact groups (indeed,

even of finite groups) are important in many branches of mathematics ( f o r example. parts of topology and differential geometry), in our subject compact group actions are "dynamically trivial." The Justification for this assertion will emerge as we proceed.

In general, we will identify an element t of X

T with the homeomorphism

E

it defines (nt in our original riotation). Thus T may be

regarded as a subgroup of the total homeomorphism group of X. course, it is possible that distinct elements s the same homeomorph'ism: xs = xt

for all x

E

X.

and

Of

t of T define

The action of T

is

Flows and Minimal Sets

called effective if this does not occur for some x E X.

--

3

t

i.e., if

#

e, then xt

This is only a minor technical problem.

is not effective, let F = [t E Tlxt = x for all x E XI. closed (since X group T/F

#

x

If the action Then F

is a

is Hausdorff) normal subgroup of T, The quotient

acts on X by

x(Ft) = xt, and this action is clearly

effect ve. Therefore, we can assume that the action of T

on X

is

effect ive. Moreover, the topology of the group is really not that important. We are interested in the action of a group of homeomorphisms on a topological space, and the given topology on the group is for most purposes irrelevant. For this reason, we will frequently assume that T has the discrete topology.

(Some of our definitions will apparently

depend on the topology of T, but we will show that they are in fact indepenae..t of it. ) The most intensively studied cases have been where the acting group T is Z

or R

respectively). of X

(the additive groups of integers and real numbers, If T = a ,

and cpn(xl = xn.

iterate of cp

cpfx) = xl.

let

(As usual, if

-1 n and cp-” = ( c p 1 1.

Then cp

n > 0, cpn

is a homeomorphism

denotes the nth

Conversely, if cp

is a self

homeomorphism of X, it defines an action of Z as above.

is effective if and only if We will refer to the pair

(pn

f

identity, for a1

(X,cp)

as a cascade.

n

*

(The action

0.1

(Other terms in use

are discrete flow - and discrete dynamical system.) Thus a cascade consists of a homeomorphism and its powers. use the letter T for a homeomorphism

--

(Warn ng:

we will sometimes

in this case we will always

refer to the ”cascade (X,T).”) If T = R,

then the action defines a one-parameter group of

homeomorphisms (9,) of the space X.

and in this case we write cpt(x)

Chapter 1

4

instead of xt.

The axioms appear as cpo(x) = x and

cps(cpt(x)) = cpS+,(x).

Customary terms for an R

system, real flow, and continuous - --

action are dynamical

flow.

It is admittedly somewhat inconsistent for us to write the generating homeomorphism of a Z

action, as well as the action of R,

on the left

However, this corresponds to the usual practice in the subject. The classical example of an R

action is the one defined by the

solutions of autonomous systems of differential equations. Let D be a region in Fin,

= f(x),

and consider the system

where f

:

D+Rn

and f satisfies conditions sufficient to guarantee that solutions exist, are unique, depend continuously on initial conditions, and are defined and remain in D for all real t. define =

f(x)

If xo

E

D and t

E IR,

t(xo ) = u(x 0't) where a(xO,*) denotes the solution of

cp

for which a(x , O ) = xo. Let xo

yo = cpt(xO). Let

0

$(s)

= cps+t(~O)

E

D, s,t E

= u(xo,t+s).

Then

R, $(s)

and let is a solu-

tion satisfying $(O) = yo, so by uniqueness $ ( s ) = a(yo,s) = cp (yo). S

Then 'P,+~(X~)= $(sl = cps(yo) = cps(cpt(xo)). other conditions (continuity of

Thus 'P,+~ = 'Ps'Pt. The

( x O , t ) ~ ctp(xo 1 and cpo = identity)

are obvious, so we do indeed have an R

action on D.

The observation that the solutions of an autonomous system define a flow motivated much of the early work in topological dynamics. It is possible to define and discuss many of the notions in the qualitative theory of differential equations in purely "topological dynamics" terms, but this is not the direction which we w i l l pursue in this monograph. (This point of view is developed in the books "Qualitative Theory of Differential Equations" by V. V. Nemytskii and V. V. Stepanov (Princeton University Press, 1960) and "Stability Theory of Dynamical Systems" by N. P. Bhatia and

C.

P. Szego (Springer-Verlag, 1970).)

Flows and Minimal Sets

5

Before proceeding with the development of the theory, let us briefly mention some variations of the concept of a flow. An obvious modification is to replace the acting group by a semigroup. In the case of H, the additive semigroup of positive integers, this corresponds to the study of the iterates of a continuous (in general non-invertible) map of a space.

The study of the dynamics of maps of a

closed interval is currently an extremely active research area.

(See

2. Nitecki, "Topological dynamics on the interval," Ergodic Theory and

Dynamical Systems, Progress in Mathematics, Vol. 21, Birkhauser, 1979.) If

cp :

X+X

is a continuous map of a compact space, it can be

"converted" to a homeomorphism by an "inverse limit" construction. Let

XL

denote the space of 2-sided sequences of elements of X, with the

product topology, and let

2

denote the subset of 'X

1 = x those sequences x = (x 1 for which cp(x i i+l i easily from the compactness of X that Let

:

i+?

be defined by

is a homeomorphism of

2.

(p(x) = x',

%

consisting of

It follows

(i E Z ) .

is a non-empty compact space.

where xi = cp(xi).

Then

Using this device it is possible to obtain

certain dynamical results for continuous maps by first proving them for homeomorphisms.

(See H. Furstenberg "Recurrence in Ergodic Theory

and Combinatorial Number Theory," Princeton University Press. ) Another object which has been studied is a "partial flow" or "local (semi) dynamical system," which is a generalization of real actions. this case, n

is a "partial" map of XxR

to X.

That is, n

defined on an open subset of XxR. Thus, if x E X, n(x,t) for t

in an interval Ix

cases 0 I ax), they are defined

- (ax,px)

(--OD 6

ax

< p,

5

00,

In

is

is defined

or in some

and the usual axioms for a flow are satisfied insofar as (so

n(n(x,t).s)

= n(x,t+s)

if the left side is

defined). These arise from autonomous systems of differential equations

Chapter 1

6

for which the solutions are not defined for all real

t

("finite escape

time") and also from functional differential equations with time lag. (See N. P. Bhatia and 0. HaJek "Local Semi-Dynamical Systems," Lecture Notes in Mathematics, 90, Springer-Verlag, 1969. Now we proceed with the discussion of flows. Let

(X,T) be a flow,

let K be a subset of X, and B a subset of T. We write Kl3 the set

[xblx E K, b

E

Bl.

is called the orbit of x.

notation O(x)

If x

E

If T = H

for the orbit of x.

for

X, the set xT = {x)T = [xtlt E TI or

IR

we will frequently use the

(In these cases, the term

trajectory is sometimes used. 1 The subset K of X (equivalently KT c K).

is said to be invariant if KT = K Thus a set is invariant if and only if it is a

union of orbits. The proof of the fo lowing proposition is immediate. 2. Proposition.

K be an nvariant set. Then the closure, complement,

( i ) Let

boundary, and interior of K are all invariant sets. ( i i ) If

{K,)

is a family of invariant sets, then u KO, and n KO!

are invariant.

It follows from this proposition that if x

E

X,

the orbit -closure

3 is invariant. There are several standard ways of obtaining new flows from given ones. The first is trivial, but nevertheless very useful. Let be a flow and let 2 be an invariant subset of X interest 2 is also closed). "subflow" of

(X.T).

Then T acts on 2 ;

Secondly, if

(with the same acting group TI,

(Xa,T1 ( a E A )

(X,T)

( i n most cases of we say

(2,T) is a

is a family of flows

then T acts on the product space

Flows and Minimal Sets

7

TI Xa by acting on each coordinate: if x = (xa), then (xt),, = x t. a

a

We write

(nXa,T) for this product flow. a

In later chapters, we will

consider "large" products (i. e. , with uncountably many factors Xa).

In

this case, the product space is not metrizable, even if all the factors are. For this reason, we will not in general assume that the phase space is metrizable in our development of the theory. Another construction, using,equivalence relations on the phase space, will be discussed later in the chapter, in connection with homomorphisms of flows. Now we are ready to define minimal sets, the study of which is the main focus of this monograph.

(X,T) be a flow. A subset M of X is a -minimal set if M

Let

closed. non-empty and invariant, and if M these properties.

or N = 4 . )

N = M

(That is, if N c M

is

has no proper subsets with

with N

closed invariant, then

Note that a non-empty subset M

of X

is minimal if

and only if it is the orbit closure of each of its points. For, if M is mimimal and x E M, non-empty so

= M.

-

its orbit closure xT

is closed invariant and

On the other hand, if M

is not minimal, let

4 * N $ M with N closed invariant. Then, if x -

xT

*

E

N,

c

N

so

M.

Thus every point of a minimal set "generates" it.

Minimal sets are

also called "minimal orbit closures."

It is possible - and this is the case which will be of most interest that

(X,T) is itself minimal (equivalently X =

this case

(X,T) (or X)

3

for all x E XI.

In

is called a minimal transformation group or

minimal flow. ~Since the intersection of closed invariant sets is closed invariant,

Chapter 1

8

we immediately obtain Proposition. Let

3.

(X,T) be a flow and let M I

subsets of X. Then MI = Mz

or

MI

n b$ =

and Mz

be minimal

4.

It is not in general the case that X is a union of minimal sets. However, as we now show, minimal sets always exist when the phase space is compact. 4. Theorem. Let

(X,T) be a flow with compact Hausdorff phase space

X. Then X contains a minimal set. Proof. Let M denote the class of non-empty closed invariant subsets of X. Note that X

{Ma) M

*

E

M,

so

M

f

4.

Partially order A by inclusion. If

is a totally ordered subfamily of

= n

MU

f

4,

M, then by compactness of X,

and clearly it is closed invariant, so

M

* E

M.

Thus by

Zorn’s lemma, 1 contains a minimal element, which is a minimal subset

x.

of

Thus if the phase space of a flow is compact, it always contains a minimal subset.

(We will see later that this is not the case if only

local compactness is assumed.) It is not necessarily the case that a minimal subset of a flow is “interesting.“ minimal set is a fixed point of the flow -x t = x for all 0 0’

t

E

--

An example of a “trivial“ a point x0 such that

T. (Synonyms -- especially when T = R -- are

rest point, equilibrium point and stationary point.) Obviously the --

is a minimal set. Another simple example -- in case

singleton tx,)

T=

Z

and cp

the generating homeomormorphism

--

is a periodic orbit

that is, the orbit of a point xo for which (pp(xo) = xo is the smallest such positive integer) ~(x,) =

ZZQ =

{x,,(p(x,~,

--

(where p > I

so

. . . ,cpP-l(xo~) is minimal.

Similarly, if {pt) defines an action of R on X, and x

E

X is

Flows and Minimal Sets

not a fixed point of the flow, but cp

S

(XI

=

9

x for some

*

s

is called a periodic point, and its orbit O(x) = {cpt(x)) orbit. The minimum T of those

s

> 0 such that

cp

s

(XI

0, then x

is a periodic = x

is called

the period of x, and it is easily seen that O(x) = {cpt(x) (0 I t < and so is homeomorphic to a circle. (Note that the set F of which cp,(x)

= x

is a closed subgroup of

and since F

R

smallest positive element T and F = {ntln

E

f

for

s

R,

T},

F has a

H).)

Fixed points and periodic orbits are sometimes referred to as "trivial" minimal sets.

(Periodic orbits can be defined for arbitrary

acting groups -- see exercise 3.) Throughout the book, we will be presenting examples of "non-trivial"minimal sets, mostly for acting groups Z

and R.

It is perhaps now appropriate to explain our point of view concerning minimal sets. We have referred to fixed points and periodic orbits as "trivial" minimal sets. Of course, there are many circumstances in mathematics in which a group acts on a space and one wishes to determine whether there is a fixed point or a periodic orbit (the latter is a fundamental question in differential equations), and the answers to such questions are often decidedly "non-trivial." However, our main concern in this monograph is with the internal structure of the minimal sets themselves and the relations among them, and from this point of view there is not too much which can be said about fixed points and periodic orbits.

In this connection, the following theorem is of interest. 5. Theorem. Let

(X,T) be a flow with T = Z or IR and X Hausdorff.

Then an orbit is compact if and only if it is periodic. Proof. Obviously, a periodic orbit is compact. Suppose T = IR, usual, the action of R

is denoted by {cp,).

and as

Let x E X with O(x)

Chapter I

10

compact, so we may suppose X = O(x). Kx = [cpt(x) It E K].

and write

Then there is a countable subset C of IR

u cpc(Kx).

that X = O(x) =

Let K = [-1,11,

such

By the Baire category theorem, some cpc(Kx)

ccc

has non-empty interior, so pT(x) R

int(Kx).

E

has non-empty interior. Let z E K with

Kx

Suppose cpt(x)

x for all t

#

onto X defined by d t ) = cpt(x)

*

0. Then the map

is a continuous bijection.

a of

If

(r

were a homeomorphism, the compact space O(x) = X would be homeomorphic with R,

which is impossible. So a is not a homeomorphism, and it

follows easily that there is a sequence cpt (x)+(pT(x).

Since cp,(x)

E

k

s c K with cpt ( x ) = cps(x),

{tk)+m

(or

-m)

with

int(Kx1, there is some tk > 1 and O(x)

so

is periodic. This completes the

k

and we omit the easy proof for T = Z.

proof for T = R,

Now we will develop an important "recursive" concept. Let T be a topological group. A subset A of T there is a compact subset of K of

T = AK

(=

If T

[akla E A, k

= Z

or R ,

relatively dense Let

--

E

is said to be (left) syndetic if

T such that

K1).

a subset of T is syndetic if and only if it is that is, it does not contain arbitrarily large gaps.

(X,T) be a flow, and let x c X.

We say that x

periodic point if for every neighborhood U of x,

is an almost

there is a syndetic

subset A of T such that xA c U. Almost periodicity is a strong form of recurrence

--

the orbit returns

to an arbitrary neighborhood infinitely often. We omit the definition of recurrent for general acting groups T Chapter 7 ) .

For an action {pt) of R,

(see Gottschalk and Hedlund, a point x

is recurrent if and

only if, for every neighborhood U of X and z > 0, there is a t with

It1 > z such that cpt(xl

E

U. The definition is similar for H

E

R

Flows and Minimal Sets

11

act ions. As Gottschalk has remarked, a point is periodic if it returns to

itself every hour on the hour and is almost periodic if it returns to an arbitrary neighborhood every hour within the hour (where the length of the "hour" depends on the neighborhood1. Almost periodic points and minimal sets are intimately related, as the following results show. (X,T) be a flow, with X

6. Lemma. Let

Then, if x

locally compact Hausdorff.

is an almost periodic point, the orbit closure

3

is

compact. Proof. Let U be a compact neighborhood of x, and let

A = [t

Tlxt

E

E

Ul. Since x

is almost periodic, there is a compact

subset K of T such that T = AK. Thus xT = xAK c UK,

which is

compact, so ZT is compact. (X,T) be a flow, with X

7. Theorem. Let

Then x ;;T

E

locally compact Hausdorff.

X is an almost periodic point if and only if the orbit closure

is a compact minimal set.

Proof. Suppose M = 3 is a compact minimal set and let U be a neighborhood of x.

First note that M c UT.

(If not

M

\ UT

is a

By compactness, there is a finite n Uti. Now, if T E T, subset K = itl,. . . ,tn} of T such that M = i=l -1 XT E Uti, for some i with 1 5 i 5 n, so xzti E U. So, if closed invariant proper subset of M.)

u

A

=

[t Ixt

E

U],

syndetic, and x

r t ; '

E

A,

t E

Ati c AK, and T = AK. That is, A is

is almost periodic.

Conversely, suppose x

is an almost periodic point.

By the preceding

lemma, ZT is compact. If ZT is not minimal, it contains a minimal subset M'.

Let U and V be disjoint open sets with x

E

U and

Chapter 1

12

M' c V, and let K be a compact subset of such that WK-l c V.

neighborhood of M'

T with xs

s E

E

T. Let W be a Now M' c 3, so there is an

W. Then xsK-' c WK-' c V and xsK-' n U = $. then T

follows that if A = [tlxt E Ul, arbitrary compact subset of T, A

*

AK.

Since K

It

is an

x is not

is not syndetic, and hence

almost periodic. Theorem 7 has a,number of immediate corollaries, which are not obvious consequences of the definition of almost periodicity. 8.

(X,T) is a flow with X compact Hausdorff, then

Corollary. If

there is an almost periodic point in X. Proof. Every point of a minimal subset of X (X,T) is a flow (with X

9. Corollary. If

then x

E X

is almost periodic. locally compact Hausdorff)

is almost periodic if and only if it is discretely almost

periodic (i.e., almost periodic with respect to the discrete topology of TI.

10.

X

Corollary.

Let

(X,T) be a flow with X compact Hausdorff.

hen

is a (necessarily disjoint) union of minimal subsets i f and only f

every point of X

is almost periodic.

(In this case, we say X

is

pointwise almost periodic. 1 The first part of the proof of theorem 7 also yields the following corollary. 11.

Corollary. Let

(X,T)

be a minimal flow with X compact

Hausdorff, and let U be a non-empty open subset of X. Then there i's a n

finite subset K = {tl,,.., tn)

of T such that X = UK =

u j=l

Ut

j'

O u r next result is an "inheritance" theorem, which relates the action

of a syndetic subgroup of T to the action of T.

Rows and Minim1 Sets

Let

Lemma.

12.

T.

Suppose

Suppose

y E

a.Then

S

is normal, t h e r e are

Since

1

-1

xs i ' + y t

such t h a t

.

S be a normal subgroup of

is an almost p e r i o d i c point f o r t h e flow

xt

xts.+y.

so

X

E

t E T,

Then, i f Proof.

x

be a flow, and let

(X,T)

13

is an almost p e r i o d i c point f o r

x

Since

is

t h e r e is a n e t i

'

E S

(X,S). in

such t h a t

S

such t h a t

xsi't+y,

almost p e r i o d i c , t h e r e are

S

s

.

and s o t h e r e are

yt-lsi''+x

s

{si)

(X,S).

s

i

"' E S such t h a t

i

E

"

S

ys "t-'+x i

ysi"'+xt.

or

Theorem.

13.

Suppose

T.

subgroup of

(X,T)

Then

is a flow and

(X,T)

is a s y n d e t i c normal

S

is pointwise almost p e r i o d i c i f and only i f

(X,S) is pointwise almost periodic. Proof.

Suppose

compact subset of that

y

is

Then

= xk

{ti)

s i E S,

with

follows t h a t ys +x' i

T = SK.

S almost periodic.

t i = s .1 k i ' i

with

T

t h e r e is a net

periodic,

ys j x ' .

is pointwise almost p e r i o d i c .

(X,T)

ki

Then

in

K.

T

(X,T)

and l e t

such t h a t

xk-'

is

S

k k i

-1

be a

K

y

such

E

is pointwise almost

+e,

Write

yti+x.

W e may suppose

and s i n c e

i l

-1 .

Since

x E X

E K,

ki+k

and a l s o ysiki = y t i + x ,

ysiki+x'k

ys k.k-l+x'

E

Let

Let

and t h a t

x = x'k.

so

It

we have

almost p e r i o d i c , s o

x

is

S

almost periodic, by l e m m a 12. Suppose t h a t let

y E

n.

Let

where

si E S,

k E K

such t h a t

is pointwise almost p e r i o d i c .

(X,S)

be a net i n

{t.) 1

1

x s +yk-l. i s

i

with

xti-+y.

x

E

X

ti = siki,

Write

'

E S

Since

(X,S)

such t h a t

is pointwise almost

yk-lsi8-+x,

so

x E

F.

t h a t t h e proof of t h i s implication does not use t h e assumption t h a t

is normal. 1

and

As i n t h e first p a r t of t h e proof, t h e r e is a

k. E K.

p e r i o d i c , t h e r e are

T

Let

(Note

S

Chapter 1

14

Now we present some examples of non-trivial minimal flows. The simplest example of an (infinite) minimal cascade is an irrational rotation of the circle. To be precise, regard the circle K as the set of complex numbers of absolute value one, and let u (n f 01, phism cp

a = exp(2nie1, where 0

so

with an

#

1

is irrational. Let the homeomor-

be defined by ( p ( z ) = uz. We show that the orbit of

: K+K

the complex number 1 K.

E K

(that is, the set of powers {a"))

is dense in

Once this is proved, it will follow that every orbit is dense (since

n n if z E K, cp ( 2 ) = a z )

proving minimality of

(K,(p).

Let @ be a limit point of the (infinite) set {an)n=1,2,,

and let

..

> 0. Then there are non-zero positive integers n and k such that

c

&

Ian-@[ < and < 2' - Thus k Z H ~ z is an isometry, it follows that

- un I

<

n n+k

For some m, the points u , a E

n+mk

,. ..,u

> 0, the set {an)

Since the map

E.

lun+2k-un+kI =

and, in general, Iun+(m+l)k - ,n+mk

E,

Thus, for any

<

I

<

(m=1,2,. . . 1.

E

"wind" around the circle.

(n > 0 )

is

E

dense in K. This

completes the proof. Closely related to this example is the example of the irrational (real) flow on the torus. Let the two-dimensional torus K2 be represented as the plane R2

modulo the integer lattice points Z2. m

Thus we regard

(x,y) = (x',y')

if and only if

(x-x',y-y')

Let 1.1 be an irrational number, and define the flow

(p

E ZL.

: KxR-+K

by

cp((x,y),t) = (pt(x,y)= (x+pt,y+t). Thus the orbits of this flow ("flow lines") are parallel lines with slope p. (0,O) is dense. Let

t

0

+ n = y'

such that

(mod 11,

(x',y'l

E K

2

We show that the orbit of

, and let

E

> 0. Let t = y', 0

so

for all integers n. Now, choose an integer n

IptO-x'+pnI <

E

(mod 1).

The choice of such an n

is

possible because of the minimality of the irrational rotation of the

Flows and Minimal Sets

15

circle, which we have just proved. Then

whose distance from (0,O)

on K2

(x',y')

is less than

Then the orbit of

E.

is dense.

It is easy to show that all orbits are dense, as in the case of the cascade

(K,cp)

above. Actually,, these are examples of equicontinuous

flows, and we will prove in the next chapter that for such flows minimality is equivalent to the existence of a dense orbit.

(This is not -

the case for an arbitrary flow.) Similarly, the (real) flow defined on K where

aP

is irrational, is minimal.

flow defined above, then

2

by

In fact, if p =

a

P- and cpt

is the

A rich source of examples is

$t = pPt.

be a finite set of cardinality

provided by the symbolic systems. Let A which we write as {0,1,2,.. . ,p-l}.

p > 1,

$,(x,y) = fx+at,y+@t),

Let

R denote the collec-

tion of two-sided infinite sequences w = (w(n)),

with o(n) E A,

I

define a metric on s1 by d(w,o') =

and

(an equivalent

n=-m metric is given by

D(w,w') =

1 -, n+1

negative integer such that w(n)

#

where n

or w(-n)

w'(n)

z

given the discrete topology, then 51 = A , Tychonoff's theorem.

(In fact, R

Note that the points w

and w'

agree on some large "central block"

is the smallest non-

*

w'(-n)).

If A

is

which is compact by

is homeomorphic to the Cantor set.) of R

-- that

are close together if they is, if d n ) = w'(n),

for

In1 I N.

A homeomorphism u of R to itself is defined by d w ) ( n ) = o(n+l) (n E Z). The map u (s1,u)

is called the shift homeomorphism and the cascade

is the shift dwamical system on

p smbols. These cascades are

Chapter 1

16

also called symbolic systems. We introduce some suggestive terminology and notation. The set A

is

called the alphabet, and a word (or a block) is a finite sequence of "letters" of A a word in w

(for example, if p = 2, 01101 is a word).

is a word of the form o(m)w(m+l)

. . . o(m+r)

r 2 0. If a and b are words, we can form the word

If w

E

Q

for some

ab

in the

obvious way, by juxtaposit on. We may also speak of (left or right) infinite words.

Of course any w E Q

is a (two-sided) infinite

word. Note that if w , w ' E Q,

then w' E

if and only if every word in

is also a word in w .

w'

In general, we are interested in closed invariant sets of the shift system, in particular minimal subsets, rather than the "full shift" (Q,a). To obtain a minimal subset of

Q,

to construct an almost periodic point

o

It is easy to see that w E R if every word in w

it is sufficient by theorem 7 (so

o(w) is a minimal set.)

is an almost periodic point if and only

occurs "syndetically often." That is, if a

word in w , there is an N > 0 (depending on a) such that if

a

is a subword of

(Note that in w

o

w(n)w(n+l).

--

n E Z,

. .w(n+N).

is a recurrent point of w

occurs again

is a

if every word which occurs

and therefore infinitely often.)

The construction of almost periodic points in A'

which are not

periodic (equivalently of non-trivial minimal subsets of

z

A )

is by no

means trivial. One of the earliest examples is due to Marston Morse. Let

p = 2, so

a point of AN

A = {0,1}. We first define a one-sided sequence (i. e. ,

(where N = {0,1,... } . I

Write

0' = 1,

a = ala2...am is a word, define a' = al'a2' . . . a

m

'.

1' = 0 and if

(For example, if

a = 01101. a' = 10010.)To define the "Morse sequence," we define

Flows and Minimal Sets

17

inductively a sequence of words a where each a n n

al = 0. and if a n

fact the first half) of an+l. Let defined, let an+l = a a '. n n

is a subword (in has been

Thus a2 = 01, a3 = 0110, a4 = 01101001,

etc. The "limit word" is a right infinite word w = 0110100110010110. . . Another way of constructing the one-sided Morse sequence is as follows. If b

is a word, let b

the length of b) 1+10.

*

denote the word (of length twice

obtained from b by the "substitution" 0401,

(For example, if b = 01101, then

b = al = 0, and inductively let 1

*

b = ak, for k k

*

S

n.

*

bn+l - bn* . *

.

(b 1' = (b')

it is easy to see that

For,

bn = a n'

Now suppose, inductively, that bn+l

[4'

I .

Let

--

*

*

= a = n ) ' = b b ' = a a'= = bn-l*(bn-l* n n n n

It follows that the limit word

substitution 0+01,

In fact

Then bn = an = an-lan-l' ,

(an-lan-l') = an-1an-1 = any1 a n+l'

*

b = 0110100110. )

w

bn

reproduces itself under the

1-+10.

From this latter characterization, if follows that every finite word which occurs in w

For

occurs syndetically often.

0 occurs syndetically often since the sequence is made up of

pairs 01 and

10. But the sequence reproduces itself under the

substitution 0+01

and

l+

01 also occurs syndetically

10 so

often. For the same reason, the initial words 0110, 01101001, etc. occur syndetically often. But this sequence of initial words (in our notation bn)

include all words of w

as subwords, so all words occur

syndetically.

To obtain an almost periodic point of A' "reflection.

we extend

w

by

That is, if n < 0, define w(n) = w(-n-1).

the two-sided infinite sequence w =

Thus o

J.

. .. 100101100110100110010110. . . ,

where the vertical arrow indicates the 0th position o(0). Equivalently,

(if, when b = bl...b

m

is a word,

6

denotes the

is

Chapter 1

18

"reverse" of b.

6=

bmbm-l...bl) then w

is the "limit" of the words

c a a where the Oth position is the initial letter of a n n n' the point

w E A

=a that a2n+1

To see that

just defined is indeed an almost periodic point, note for n = 1,2,. . .

2n+1'

(this is an easy induction) and

an argument similar to the one above for the one-sided Morse sequence shows that a 2n+1"2n+1

It follows as above

occurs syndetically in w .

that all words which occur actually occur syndetically. Therefore w

--

an almost periodic point

its orbit closure

is

is called the Morse

Mo

minimal set. It is not difficult to give a combinatorial proof that o periodic, but we present a "dynamical" proof.

is not

If we now write wo

for

the one-sided Morse sequence, then w = wowo.

t

An argument similar to the

80'wo

is also an almost periodic

shows that C =

one given above for w point

(< =

J

. . . 0110100101101001. . . ) . M1.

also a minimal set

that

lim d(d(w),un(<))

Then the orbit closure of

Now, since w(n) = <(n) = 0, (we say that

asvmmptotic) and since d(w) E Mo,

w

<

is

for n 1 0 , i t follows and

<

are positively

on(<) E M1, Mo = MI.

(Recall that

distinct minimal sets are disjoint, and so must be a positive distance apart.) But it is immediate that a finite minimal set cannot contain (distinct) asymptotic points. infinite, and w

Therefore the Morse minimal set is

is not periodic.

Another description of the one-sided Morse sequence is obtained as follows. If n

is a positive integer, write n =

ci2' 1=o

ck = 1.

Let

e(n)

where c = 0 i

(mod 21, and 0 ( 0 ) = 0. Then

=

i =o For, if 0

J)+1

S j

< , ' 2

it is immediate that

(mod 21, so O(j+zJ

= O(J1'.

Since e ( 0 ) = 0 = ~ ( 0 1 ,

19

Flows and Minimal Sets

the result follows by induction. The existence of asymptotic points in the Morse minimal set is characteristic of minimal subflows of the symbolic system. For is a two-element set, A = { O , l } ,

simplicity we suppose A

be an infinite minimal subflow of Q = A

N t 2, there are points wN and ~ ~ ’ (=0 1) and

w (n) = oN’(n)

N

these points is established, let = w

lim wN

that

and

i

and ;’

are in X.

= c’(n)

;(n)

2

z . We will show that for each

wN ’

for

{Ni}

-

lim wN‘ = w’ i

in X such that wN ( 0 ) = 0 ,

1 In

for n t 1, so

5

N.

Once the existence of

be a sequence with exist. Since X

;(O) = 0, ; ‘ ( O )

Also

and let X

= 1, so

lim d(rn(G),o-n(;‘)) nJm

G

N i +co

such

-

is closed, w

*

z‘,

but

= 0. (Thus

and

are positively asymptotic.1 Now suppose for some N

above.

we cannot find points wN

Then, if w , w‘ E X with (say) w ( 0 ) = 0, ~ ‘ ( 0=) 1,

an N 2 1 such that

w(n)

f

w’(n)

for some n with

is, the sequence of values w ( 1 ) , w ( 2 ) , . . .w(N) w E X.

w(n)

and wN ’

Since X

for each m.

as there is

1 I n I N.

determines w ( 0 )

That for

is shift invariant, o(m+l), . . . ,o(m+N) determines Hence X must be finite.

Asymptoticity is a special case of proximality which, together with related notions, will be studied extensively in this book. a flow, then x and y z E X

@.nd a net

{ti)

in X are said to be proximal in T such that

xti+z

If

(X,T) is

if there is a

and yti+z.

If X

is compact, x and y are proximal if and only if for every a E (the unique uniformity of X) Thus if

there is a t E T such that

%

(xt,yt) E a.

X is a compact metric space, x and y are proximal if and

only if inf d(xt,yt) = 0. tET We write P or P(X)

f o r the proximal relation in

Chapter 1

20

X

:

P is a reflexive symmetric T

P = [(x,y)lx and y are proximal].

invariant relation, but is in general not transitive or closed. Note that (if X is compact Hausdorff) P = n [crTla E % I . If x and y are not proximal, they are said to be distal, and the flow (X,T) is called distal if there are no non-trivial proximal pairs (i.e., P = A,

the diagonal).

The analysis of distal flows will be

carried out in chapters 5 and 7 . Other examples of flows arise from consideration of spaces of functions. Let CO(T)

denote the set of bounded continuous real (or

complex) valued continuous functions, provided with the topology of uniform convergence equivalently, the topology defined by the norm llfll = sup If(t)l). taT

If t

E

T, and f

E

CO(T).

ft is defined by

It is immediate that this defines an action

translation: ft(7) = f(tr). of T on CO(T). If T = R,

the additive group of reals ( s o ft(7) = f(t+r))

then

f E C (T) is an almost periodic point if and only if it 1s a Bohr 0

uniformly almost periodic function: if

2:

> 0, there is a relatively

dense (syndetic) subset A of R such that

< 2: for T E A . (Of course, this is the reason for t a use of the term "almost periodic" in arbitrary flows.) It will be shown

sup If(t+t)-f(t)I

in the next chapter that f

E

C (R) 0

has compact orbit closure if and

only if i t is almost periodic. If t

E

T, then T acts as an isometry on CO(T); d(ft,gt) = d(f,g)

Thus, this flow is (where d(f,g) = IIf-gll = sup If(t)-g(t)I). taT "equicontinuous." The formal definition and properties of equicontinuous flows are the subject of the next chapter, and as we will see. equicontinuous flows have a more or less complete classification. Therefore,

Flows and Minimal Sets

21

the flow on C (T) is of limited dynamical interest. A more interesting 0

flow is obtained by endowing the space of bounded continuous functions with the topology of uniform convergence on compact sets. For this purm

pose, suppose T is sigma-compact (T = and define a metric by p(f,g) = sup min[{;z; n

u Kn, where Kn n=1 If(t)

-

is compact)

g(t)l,a}].

Let

n BO(T) be the bounded continuous functions with the topoloky induced by in the case of CO(T),

p.

As

of

T on Bo T I ;

translation of functions defines an action

the flow (BO(T),T) is called the Bebutov system. In

contrast with the flow (CO(T),T) which has "few" compact orbit closures, every (left) uniformly continuous function has compact orbit closure in BO(T) (exercise 12).

Most important, (BO(T),T) has a "universal"

property. For a large class of acting groups (including T = R )

any

flow on a locally compact metric space (subject to an obvious necessary condition on its fixed point set), can be regarded as a subflow of BO(T). The precise statement and proof will be presented in Chapter 13. In any mathematical system, one is interested in the maps which respect the structure of the system. The appropriate maps in topological dynamics are those which are continuous and equfvariant. To be precise, let

(X,T) and

(Y,T) be flows (with the same acting group).

homomorphism from X to Y is a continuous map A n(xt) = n(x)t

(x E X, t

E

:

A

X j Y such that

TI.

If there is a homomorphism

II

from X onto Y, we say that Y

factor of X, and that X is an extension of Y. 14.

Proposition. Let (i)

A :

X+Y

be a homomorphism.

If Xo is a minimal subset of X, then Yo = lt(XO)

is a

minimal subset of Y. (ii)

If Y is minimal and X is compact, then

H

is onto.

is a

Chapter 1

22

( i i i ) If X

is minimal, and n,#

:

X+Y

are homomorphisms which

agree at one point, then n = #. and Yo are minimal subsets of X and Y respectively

(iv) If Xo

such that n(Xo) n Yo

f

then n(Xo) = Yo.

$,

Proof. (i)

(ii)

If yo

If Xo

E

Yo and x0

E Xo

with n(xo) = yo, then

is a minimal subset of X, then n(X)

3

n(Xo)

(t

E

T).

=

Y, by

(i). (iii)

If n(xo) = $(xo),

then n(x,t)

and # agree on a dense set, so

= #(xot)

Thus n

n = 14,.

(iv) follows from (1) and the fact that the minimal subsets of a flow are disjoint or equal. Since our main interest is in minimal flows, it will usually not be necessary to specify that a homomorphism is onto. The meaning of the terms isomorphism, endomorphism, and automorphism should be clear. Almost by definition the properties which will interest us are those which are preserved by isomorphisms. The next result shows that homomorphisms of minimal flows have a "semi-open" property. 15. Theorem. Let U

n:X+Y

be a homomorphism of minimal flows, and let

be a non-empty open set in X. Then n(U)

Proof. Let V

has non-empty interior.

be a closed set in X with V c U and

int V f $. n Since X is minimal, there are t 1' . . . ,tn E T such that Vti = X, so i=1 n n Y = n(X) = n(Vti) = n(V)ti. Therefore n(V)ti has non-empty intei=1 i=1 rior for some ti, so int n(V1 * 4. Since n(U) D n(V1, int n(U) * 4.

u

u

u

FIows and Minimal Sets

Note that if s

:

X+Y

is an onto homomorphism, then

R(n) = R = [(x,~') 6 X x X)n(x) = n(x')l equivalence relation. (x,x') E R and t

E

(That is, R

is a closed

is a closed T

R is not closed, then the quotient space X/R

map

X-+X/R

invariant

T, then (xt,x't) E R.) Conversely, if

lence relation, then the quotient space X/R

by

T

is a closed subset of X x X and if

flow with X compact Hausdorff and R

T acts on X/R

23

(X,T) is a

invariant equiva-

is compact Hausdorff.

(If

will not be Hausdorff.)

(xR)t = (xt)R, for x E X, t

E

T, and the natural

is a homomorphism. Hence these are just two ways of

talking about the same thing, and we will use them interchangeably. A standard method for obtaining such equivalence relations is by means

of compact group extensions. Let

(G,X,T) be a bitransformation group

with X compact Hausdorff and G compact. Then the action of G defines a closed invariant equivalence relation R on X x' = gx

for some g E C.

"orbit space" of G.

We write X/G

(Cx)t = C(xt).

for X/R.

(x,x')

E R

if

Thus T acts on the

In this case X

is said to be a

group extension (or C extension) of (X/C,T).

It is frequently assumed that the action of C is stronnly effective):

x

E

X.

if g E C with g

#

e

free (a synonym is

then gx

*

x for all

(We write e for the identity element of both groups C and

T.) Note that if (X,T) is minimal, then an (effective) G actions is always free. For g E C

defines an automorphism of

gx = x for some x E X, then g

(X,T) and if

is the identity automorphism

(proposition 14, (iii)).

For an example of a group extension, consider the automorphism order 2 of the Morse minimal set Mo "interchange" map

rp

where, as.above, w'

(po

of

obtained by restricting the

of the full shift S2 = { O , l ) '

to

Mo

(cp(o) = w ' ,

is obtained from w by interchanging 0 and

1.)

Chapter 1

24

If o =

,

0

J. . .lOOlOllOlOOl.. .

is the generating point of

is easy to see that oo' = cp(u0)

E

Mo

an automorphism of order 2 of Mo, on Mo

so

Mo,

then it

(proposition 14, (iv))

is

'po

and thus defines an action of Z

2

which commutes with the shift u.

Another important class of homomorphisms are the proximal extensions. The homomorphism 71:X+Y

n(x) = n(x'), R(n) c P(X).

then x and

is proximal if whenever x,x'

x'

E

X with

are proximal. Equivalently,

As we will see in the final chapter, the proximal and group

extensions are used as "building blocks" for a certain class of flows which include many of the known examples. We now present two examples of minimal cascades on which proximal homomorphisms are defined.

In both of these examples the proximal

relation P is a closed equivalence relation, and the homomorphism is the canonical map X+X/P. The first example is a "non-homogeneous" minimal set, which is a modification of a construction of E. E. Floyd.

In the ensuing discussion

"rectangle" will mean a closed rectangle in the plane with sides parallel to the axes.

If B

is a rectangle, B = Ia,a+hl x [b,b+kl, then h ( B )

disjoint rectangles h ( B ) = B0 v B1 v B2, h k 2h 3h Bo = [a,a+glx [b,b+~], B1 = [a+T,a+Tl x [b,b+kl, and 4h k B2 = [a+>,a+hl x [b+-,b+kl. That is, h ( B ) is obtained from B 2

denotes a union of 3

where

by

deleting the "middle fifths" rectangles and then deleting the top and bottom halves respectively of the remaining left and right rectangles. n If K is a disjoint union of rectangles K = Bi, then we define

u

m

h(K) =

i=1

u h(Bi). i=1

Now, let B ( O ) = [O,l] x [ O . l l , B(n+l) = X(B(")).

and define inductively

Flows and Minimal Sets

Thus B(n)

consists of 3"

n (k = 0,1,2,...,3 -1)

25

disjoint rectangles (1)

arranged as follows Bo

,

Bil)

and B2('1

the "left," "center" and "right" rectangles (corresponding to and B2

in the definition of h

above).

(n+l)u B(n+l) u B(n+l) j+3n j+2*3 n'

= Bj

Bo, El,

At the next stage, let

1 from left to right. h(B:"))

are

In general

from left to right. The first two

stages of the construction are shown in figures 1 and 2 (where the rectangles are labeled 0,1,2, and 0,1,2,. . . ,8, Let

X =

n

B(~). Since B(n)

respectively).

is a decreasing sequence of

n=O,1,2,. ..

compact sets, X

is

a compact metric space, which consists of vertical

line segments, some (in fact all but countably many) of which are degenerate (single points).

There is one segment of length 1

every n > 0, infinitely many of length homogeneous

as x

--

if x

E

1/2".

X, its dimension at x

The space X

and, for is non-

is 1 or 0 according

is on a non-degenerate or degenerate segment.

A homeomorphism T of X

is defined by permuting the rectangles

B(n) J .

The permutations B!n) uB!~) (where j+l is understood J J+1 n modulo 3 ) induce a map of X to itself. (Note that n B(n) # 4 jn

and only if j n+1 n

B!~) Jn

n

p

to n B (n) +l. jn

jn (mod 3

)

so

if

the permutations induce a map of

If these sets are non-degenerate vertical line

segments, map the first linearly onto the second.)

To show that the cascade (X,T) is minimal, let to be the unique line segment of length one in X, and let

Xo

be a minimal subset of

X. I t is sufficient to show that to c Xo. Since the orbit of every point of X xo

E

intersects every rectangle Bin)

X0 n to. But if x'

E to, and

there is a point

V is a neighborhood of x'

there

26

Chapter I

I] Fig. I

Fig. 2

El

Flows and Minimal Sets

is a rectangle B!n) J (X,T) is minimal.

c

V,

O(xo) n V

so

*

4.

27

Hence x’

E

Xo, and s o

The homeomorphism T permutes the vertical segments, and it is easy to see that

x and x’ in X are proximal if and only if x and x‘

are on the same vertical segment. Thus the proximal relation P closed equivalence relation, and T quotient space X/P.

is a

induces a homeomorphism on the

The latter space is homeomorphic to a Cantor set.

The factor cascade will be analyzed in the chapter on equicontinuous flows. An interesting biproduct of this example is the construction of a cascade with a locally compact phase space which has no minimal subsets. Before describing this construction, we note a property of the proximal relation in X. such that tive)

(x,x’) E P with x

If

d(T”(x),T”(x’))

>

f

x’

then there is an

E

> 0

for infinitely many (positive and nega-

E

n. This follows from the fact that there are infinitely many

segments of length 1/2 and the homeomorphism T maps the segments

1 i near 1y . Now, let

Y

=

P

\

A

(so

y = (x,x’) E Y

if and only if x

t

x’

and

x and x‘ are on the same vertical segment. Since Y is an open subset of the compact space P, it is locally compact.

Let

$ : Y j Y

be the restriction of the product homeomorphism T x T. To show that

(Y,$) has no minimal subsets, we define a real-valued function u on Y with the following two properties:

00,d y ‘ ) 5 dy), then for some- y ‘ . 00, ~ dy‘) <

(i)

If y,y‘ E Y

(ii)

If y

E

Y,

with y’

E

The existence of such a function u subsets. For, if y If

r

E

Y and

r

=

implies that

00,let

were minimal, we would have y

E

O(y‘)

and dy).

Y has no minimal

y‘ E Y with d y ’ ) < dy). and d y ) I d y ’ ) < u(y1,

Chapter 1

28

a contradiction.

If

y = (x,,x,)

E

Y, let

u(y) = d(T

dy') = dy).

mk

IC,

= dy),

invariant: cr($(y))

(y)+y'

E

Moreover, if

for y

y,y'

E

E

where

lo.

k

is the

are on lo. Note that

Y,

so if

u

y' E Cl(y),

{m k1 is a sequence such that

Y, then dy') < dy).

of generality-, that

k xl),T (x,))

k and T x,)

k

unique integer such that T (x1 ) is 9

k

ImkI-+m

and

To see this suppose,.without loss

Since

!.0 is the unique segment of

length one, and T maps segments linearly mk mk 1 1 d(T (xl),T (x,)) 5 sd(x1,x2) = ~ ( y ) , s o , if y' = (x'1'x'), 2 then

= d(x',x') 1 2

1 -cr(y).

Finally, as was noted above, there is an n nk k E > 0, and a sequence nk+m such that d(T (xl),T (x,)) 2 E , so n k there is a y' E Y with $ (y)+y' (and therefore, as we have just u(y')

5

2

shown dy') < dy)).

Thus properties ( i ) and (ill are satisfied.

Our next example is the "two circle" minimal set, due to Ellis. Let

Y be the circle, regarded as the real numbers modulo 2n, and let Y1 Y2 be two disjoint copies of Y. Points in Y 1 and Y2 will be written (y,l) and (y,2) respectively. Let X = Y u Y 1 2' X will be

and

topologized by specifying an open closed neighborhood base for each point.

If

E

> 0, let

N(y,l,c) = [(y+t,l)lO

S

t < el u [(y+t,2)10 < t

5

el

and

N(y,Z,c) = [ (y+t,1) 10 > t 2 -el u [(y+t,2)10 2 t > -el; N(y,2,e) are open closed basic neighborhoods of respectively. With this topology X

N(y,l,c) and

(y,l) and

(y,Z)

is a compact Hausdorff zero

dimensional space which is first countable but not second countable (and therefore not metrizable). by

Let T

be the homeomorphism of

z(y,j) = (y+1,j) (j=1,2). Then the cascade

(Although each of Y1 and Y2

X defined

(X,r) is minimal.

is invariant under

t,

a neighborhood of

x = (y,j) contains points of both Y1 and Yz and it follows easily

Flows and Minimal Sets

29

from the minimality of the corresponding rotation of the circle that every orbit meets every basic neighborhood.) If y E Y, then and

(y,2) are proximal

tn(y,l)

--

(y,l)

for every E > 0 there is an n such that

and rn(y,2) are both in N(y,l,c). Thus, the map

(j = 1,2), defines a proximal homomorphism of

fy,j1-y

Y onto the rotation of

the circle by one radian. A flow is said to be proximal if it is a proximal extension o,f the

trivial (one point) flow

--

equivalently every pair of points is proximal.

Abelian groups admit no minimal proximal actions (exercise 16). A simple example of a minimal proximal flow is provided by the action

of T = SL(2,R) on X = R v

{m}.

(the group of 2 If x

E

x

2 real matrices with determinant

i]

X and t =

~

It is well known and easily checked that the action of T on X transitive (if x,x'

X, there is a t

E

T with xt = x')

so

is

certainly

are distinct points of X,

(X,T) is minimal. Moreover, if x and x'

then there is a t E T with xt = 0, x't = 1. action of T on X

1)

ax + b E T. Define xt = cx + d'

Thus, to show that the

is proximal, it is sufficient to show that x = 0

lim xtn = lim x't = 0, so

x

and x'

are proximal.

Exercises. 1.

Let

(X,cpt) be a real flow, and let x

E

X.

Then the orbit of x

is a single point, a periodic orbit, or a one-to-one continuous image of R.

In the latter case, the orbit is homeomorphic with R

if and only if

x is not recurrent. There is apparently no known topological characterization of orbits. (Not every continuous one-to-one image of R

is an orbit - for example,

Chapter 1

30

a "figure eight" cannot be an orbit.) 2. Let

be a real flow in the metric space X.

{cp,)

is the set o+(x) = [cpt(x) t

semi-orbit of x of

x

w(x) =

is defined by

The positive The omeRa limit set

2 01.

n 0+ (cpZ(x)). T>-O

(i)

i f and only if there is a sequence

y E w(x)

and

tn+m

lim cpt (XI = y. Hence x n-m n

{tn)

in R

with

is (positively)

recurrent i f and only if x E o(x). (ii)

O+(x) = O+(x) v w(x)

and this union is disjoint if and only if

x e w(x).

(iv) The omega limit set is (positively and negatively) invariant and if t

E

(so

R , w(cpt(x)) = o(x)

we may speak of the omega limit

set of an orbit). (v)

If X

is compact, d x )

is non-empty and connected.

The omega limit set can be defined for cascades in the obvious manner. The properties above {except for the conclusion

d x ) is connected in (v)) 3. Let

hold in this case as well.

be a transformation group and let x E X. The period of

(X,T)

XI.

x

is the set P = [t

x

is said to be a periodic point if

E

Tlxt =

is clearly a subgroup of T,

P

P

locally compact and separable, then x

is syndetic. Show:

is continuous and

bijective and is a homeomorphism if and only if the map e. 1

4.

(i)

Let

(X,p)

space.

Let

be a minimal cascade, with X E

is

is periodic if and only if xT

is compact. (The map of TIP to xT, P t w x t

at

if T

and

twxt

is open

a compact metric

> 0. Show that there is an n > 0 such that, if

Flows and Minimal Sets

x

(ii)

E

X, the set

I(pj(x) 1 ljl

n)

is c

dense in X.

Generalize (i) to an arbitrary minimal flow (X,T). (T is an arbitrary group and X

5. Let

5

31

is a compact Hausdorff space.)

(X,T) be a minimal flow (with X compact) and let p

invariant measure on X.

(That is, p

p(X) = 1 and p(At) = p(A)

for A

be an

is a Bore1 measure on X with

measurable and

t

E

T.) Then, if V

is a non-empty open subset of X, p(V) > 0. 6. The flow

(X,T) is called topologically transitive if every non-

empty open invariant set is dense, and point transitive if it has a dense orbit.

(Points with dense orbits are called "transitive points.") Note

that the "full shift" on p symbols is point transitive. Every point transitive flow is topologically transitive, and if the phase space is second countable the converse holds.

In this case the set

of transitive points is residual.

(X,cp) is said to be totally minimal if (X,cpn)

7. A minimal cascade

is minimal for all integers n (i)

f

0.

Show that a minimal cascade with a connected phase space is totally minimal.

(ii) Suppose

(X,cp) is minimal but not totally minimal.

(a) Let

p

be the smallest positive integer such that 1 is not minimal. Show that p1 is prime.

(b) Show that all minimal subsets of

(c) Let

(X,(ppl) are isomorphic.

X1 be such a minimal subset, and let

(X ,(p ) 1 1

(X,(ppl)

(pl = (pplIxl.

If

is not totally minimal, then the smallest positive

integer p

2

such that

(X,,(p~I is not minimal is a prime

with p2 2 PI' Continuing in this manner, we either obtain a totally

Chapter I

32

minimal cascade p1 5 p2

,.. .

5

(Xn,qn) or a sequence of primes (In the case of the non-homogeneous minimal

set considered in this chapter, all 8. Let

w = w w w

For

syndetically.) by

w(n) =

,

A.

(That is, every finite word in o occurs n = 0,1,2,. .,,

let

.L . O w w l...wnwn+l..,

..OO..

.

position. Let

o*

Show that

=

lim u'~').

w(")

be the bisequence defined

where, as usual, the vertical

arrow denotes the O'th the w("),

i = 3.)

be a one sided almost periodic sequence on the

0 12"'

finite alphabet

p

w

*

w*

be a subsequential limit of

is an almost periodic point of

n.-m 1

A".

Show that any two such points define the same minimal set.

9. Consider the point

J

[O

defined for j > 0 by

in { O , l ) '

w = (w.)

k if j = 3 (3n+l)

z . ) Show that w is an z almost periodic (and not periodic) point of ({0,1) ,u). (This sequence (Extend by reflection to obtain a point in {0,1)

can be constructed "in stages" as follows. Corresponding to write

k = 0,

0 1 ~ 0 1 ~ 0 1 ~ 0 1. .~ .0 1Next, ~ , start "filling in the gaps"

by the above pattern,

so

corresponding to k = 1, we obtain

0 1 ~ 0 1 ~ 0 1 ~ 0 1 ~ 0 1O~. .0. , 1 ~and 0 1 continue this process for

k = 2,3,... . I 10. Let R = {0,1}'

a : R+n x

by

and u: R+n

be the shift homeomorphism. Define

a(xI(2n) = a(x)(2n+l) = x(n)

by the "substitution" 0-00,

1-11].

(so

Let

is obtained from

a(x)

xo

E

R

be a

bisequence which contains all finite blocks and define xn by

xn = ~r(x~-~).Let no =

u

O(xnI.

n=O,1,2,. ..

inductively

Show that the cascade

Flows and Minimal Sets

0

has no minimal sets.

,IT)

Suppose T = R

11.

(i)

or

(The space Ro

33

is not locally compact.)

h.

(X,T) is minimal with X compact.

Suppose

Then X

"positively minimal" (i.e., for every x E X, O+(x) Suppose X

(ii)

is locally compact and

is almost periodic (and so (X,T)

(iii) Suppose x E X

X

is dense).

(X,T) is positively x

Suppose there is a recurrent point

minimal.

is

in X.

Then x

is in fact compact).

is minimal with X

locally compact and every

is positively and negatively recurrent. Then X

is

compact. 12. Suppose the group T

is sigma compact. Show that

f

E

BO(T)

has

compact orbit closure if and only if it is left uniformly continuous. (That is, if c > 0 there is a neighborhood U of e -1 s1 s2 E U

such that

If(sl)-f(s2)I < c . )

implies

13. Show that the conclusion of theorem 15 remains valid if it is only assumed that the set of almost periodic points in X

is dense

(X is

not necessarily minimal) 14.

Let

( Y , I )

be the cascade discussed in this chapter which has no

minimal subsets. Show that every point of Y has a non-empty omega

limit set, but that 15. Let

(X,T) and

(Y,+) has no recurrent points. (Y,T) be flows and let

II :

X+Y

be a

homomorphism. (i)

Show that

x(P(X1) c P(Y).

is minimal, and there is a y E Y such that 0 -1 whenever x , x ' E K (yo), (x,x') E P(X). Show that II is a

(ii) Suppose Y

proximal homomorphism.

Chapter 1

34 16.

(i)

Let

(X,T) be a flow and let

(x,y) E P and also

x,y E X.

Show that if

(x,y) is an almost periodic point of

(XxX,TI then x = y. ( i i ) If the group

T

is abelian, then there are no non-trivial

minimal proximal actions of

T.

35

Chapter 2 Equicontinuous Flows A

flow is called esuicontinuous if the collection of maps defined by

the action of the group is a uniformly equicontinuous family.

(The for-

mal definition is given below.) The equicontinuous flows are dynamically the "simplest" ones; in fact, there is a complete classification of equicontinuous minimal flows. Indeed, a fruitful way of looking at a general minimal flow is to consider to what extent it "differs" from being equicor.tinuous. This will be made precise in later chapters. Our discussion of equicontinuous flows will extend over several chapters. If these flows were our only concern, we could obtain our results more economically. However, our procedure will enable us to introduce certain concepts which will be useful in the study of more general minimal flows. Outlines of alternate proofs of some of the results will be indicated in the exercises. Now for the precise definition. A transformation group equicontinuous if, for any a E ?$, there is a 6 whenever

(x,x')

E

f3,

then

(xt,x't)

E

a, for all t

(This may be phrased more succinctly as: f3 E

?.$

E ?.$ E

(X,T) is

sucn that

T.

if a E I$, there is a

such that f3T c a. 1

If X

is metrizable, with compatible metric d, then. this reduces to

the familiar

"E-6"

definition: if

E

that if d(x,x') < 6, then d(xt,x't).< Note that if the phase space X

> 0, there is a 6 > 0 such E,

for all

t

E

T.

is compact, then equicontinuity is a

topological notion, since a compact space has a unique compatible uniformity. (If the space is not compact, then in fact equicontinuity

Chapter 2

36

depends on the uniformity, and not merely the topology. However, we will not be concerned with this case.) As

is the case throughout this monograph, our main concern is the

analysis of minimal flows. However, in the first part of this chapter, we will consider equicontinuous flows which are not necessarily minimal. Except where otherwise stated, we will assume that the phase spaces of the flows involved are compact Hausdorff. The following proposition is almost obvious. Nevertheless, it is of considerable importance for the study of equicontinuous flows. 1.

Proposition.

1)

If

(X,T) is equicontinuous and if S

is a subgroup of T, then

(X,S) is equicontinuous. 2) If

(X,T) is a flow, with T a compact group, then

equicontinuous. (Hence, i f T, then

S

(X,T) Is

is a subgroup of the compact group

(X,S) is equicontinuous.1

(As usual, the action of S

on X

is Just the restriction of the action

of T.1 Proof. 1) is obvious. If T (x,t)wxt Our

is compact, the defining map

is uniformly continuous, so

(X,T) is equicontinuous.

first substantial result on equicontinuity is that it is

equivalent to a strong form of almost periodicity. The flow (X,T) is said t o be uniformly almost periodic, if, for every a E syndetic subset A of T such that a E A.

(For short, xA

c xu,

s,

there is a

(x,xa) E u for all x E X ahd

for all x

E

X.

2. Theorem. The flow (X,T) is equicontinuous if and only if it is uniformly almost periodic. Proof. Suppose (X,T) is uniformly almost periodic. Let

EquicontinuousFlows

(=?$I,

a E 21

fl

let

3

E 21

such that f3

T with xA c xf3, for all x such that T = AK. (x,x')

E U

Now, let 6

implies (xk, x'k)

E 6

(x,y) E 6, then t-' = ak

c a, and let A

X. Let K

E

E

31

be a compact subset of T

such that if k

P ("6(K u K-')

(for some a

be syndetic in

E A,

K u K-I, then

E

c p"). k

E

Then (xt,xta) E f3, (xta,yta)= (xk-l,yk-l)E f3,

If t

K),

so

E

T and

ta = k-'.

yta,yt) E 6 ,

so

3

(xt,yt) E f3 c a, and thus 6T c a. Therefore, (X,T) is equicontinuius. To prove the converse direction, we require several lemmas. 3. Lemma.

If

(X,T)

is equicontinuous, it is pointwise almost

periodic (that is, every x Proof. Let x U

Let a

E

t E T

such that

E

arbitrary, x

E U

such that PT c a. If y

(xt,y) E Is.

5,

E

X is an almost periodic point).

It is sufficient to show that ?i' is minimal.

X.

and let /3

E

Then (x,yt-')

E

E

?i', there is a

PT c a. Since a

E U

is

and the lemma is proved.

It follows from the above lemma that an equicontinuous flow is minimal if and only if it has a dense orbit. The easy proof of the next lemma is omitted. (Xi,T)(1 = 1 , . . . ,n) is a finite collection of n equicontinuous flows, then the product flow ( II XI,T) is i=1 4. Lemma.

If

equicontinuous. (In fact, an arbitrary product of equicontinuous floh-? is equicontinuous. This is exercise 2.) Our final lemma is a "finite" version of our theorem. 5. Lemma. Let X

',. . . ,xn E X.

xiA

c

xia (i =

(X,T) be equicontinuous, let a

Then there is a syndetic subset A 1,

. . . ,n).

E

21,

of T

and let such that

Chapter 2

38

Proof. Consider the product flow (Xn,T) which, by the preceding

*

21 be defined by Xn (n times). That is,

lemma, is equicontinuous. Let a

a+ = a(n)

= axax.. . x a

[(xi,.. . ,xA,) (yi,.

. . ,yk))

E a*

E

if and only if

(x;,y;)

a,

E

(i = 1,. . . ,n). Now, by our first lemma, x = (xl,. . . ,xn) is an almost

periodic point of (X",T) so there is a syndetic A c T such that xA c xa

.

It follows immediately that xiA c x.a (i = 1

1,.

. . ,n),

and the

lemma is proved. Now we can prove the converse direction of the theorem. Suppose (X,T) is equicontinuous, and let a E 21. and let 6

E 'U

such that 6T c 8.

3

Choose f3 E 21 with 0 c a,

Now, let xl,.

. . ,xn

subset of X which is "6 dense" - that is, if y (y,x,)

E 6,

X, then

E

for some i = 1,.. . ,n. By the last lemma, there is a

syndetic subset A of T with xiA c xis (i = 1 , ..., n). and a and

E A.

be a finite

Then (y,xi)f 6 ,

(xia,ya)E /3

for some i, (1 I i

S

n), (xi,xia)E 6 ,

2

(since 6T c 8).

Now let y E X

Then (y,ya) E 6 f3 c /3

3

c a,

and we

have yA c ya, which proves uniform almost periodicity. 6. Corollary. A factor of an equicontinuous flow is equicontinuous.

Proof. Let

(X,T) and

(Y,T) be flows and let

II :

X+Y

be an

(onto) homomorphism. Suppose (X,T) is equicontinuous, and let f3 Let a

E 'f$

such that

(nxn)(a)c f3.

Since (X,T) is uniformly almost

periodic, there is a syndetic set A c T such that all X E X , aEA. Then (n(x)a,n(x)) (ya,y)ef3 for Y E Y, a E A ,

so

E

E

(nxn)(a)c 8.

(xa,x)E U ,

for

That is,

(Y,T) is uniformly almost periodic,

hence equicontinuous. Now we turn to the systematic construction of equicontinuous minimal flows. To this end, we define a compactification of the topological

Equicontinuous Flows

group T to be a pair and cp G.

:

T+G

(G,cp) where G

is a compact topological group

is a continuous homomorphism such that

In this case, T acts on C by

Moreover, if H

39

is dense in

(g,t)++gcp(t) (the group product).

is a closed subgroup of C,

homogeneous space G/H

cp(T)

then T acts on the

(Hglt = H(gcp(t)).

of left cosets by

It is easy

to see that the flow (C,T) is minimal and equicontinuous. (Equicontinuity follows, for example, from proposition 1.) the identity e

is dense, so the minimality of

The orbit of

(G,T) follows from

lemma 3. It fallows from corollary 6 that the flow

(C/H,T)

is equicon-

tinuous. We will see in the next chapter that all equicontinuous minimal flows arise in this manner. Compactifications of the additive groups Z of integers and R

of

real numbers are called, respectively, monothetic and solenoidal groups. Intrinsic characterizations of these groups are provided by the following theorems (due to Anzai and Kakutani).

7. Theorem. Let G be a compact abelian group. Then the following are equivalent:

(i)

G

(ii)

There is a g

is monothetic.

n

{g 'n=o, 21,.. .

E

G such that the set of powers of g, is dense in G.

(Such an element is called a

topolonical generator of G.) n

(iii) The character group G of G to

is (alRebraicallY) isomorphic

a subgroup of the circle group.

8. Theorem. Let

G be a compact abelian group. Then the following

are equivalent: (i)

G

is solenoidal.

(ii)

G is separable and connected.

CIIupter 2

40 A

(iii) G

is (algebraically) isomorphic to a subgroup of

R.

We omit the proofs of these two theorems except for a brief indication of the implication ( i i i )

;$

Suppose

(i) in theorem 8 .

.. into R.

isomorphism of G


Then if a

1

= exp(iat(x))

(1E GI,

<,

and

<,

C. By the Pontrjagin

is a character of (p(a) E

G such that

It is easily verified that

= exp(iaz(x1).

are defined by A

duality theorem, there is a unique


R

E

is an algebraic

7

x(cp(a)) =

is a continuous

(p

*

homomorphism. If such that

(p(R)

xo((p(a))=

were not dense in G, there would be a

1 for all a

whose range is the complex number T(x,)

f

0, so

exp(iaor(xo))

xo

but

E R

Since

1).

E

G

*

1.

(the character

1

is an isomorphism

T

1 for some a.

f

f

xo

xo(cp(ao))

and

E R,

This contradiction completes the proof.

For the complete proofs of theorems 7 and 8 see volume I. chapter 6 of Hewitt and Ross Abstract Harmonic Analysis, Springer-Verlag.

The first two examples in chapter 1 (the irrational rotation of the circle and the real flow on the two torus) are equicontinuous minimal flows, since the transformations which define them are isometries. Now consider the cascade

2 ( K ,(p)

defined by

where 0 < a, f3 < 1, and na + m/3 P Z for are rationally independent).

Again,

(p

(p(x,y) = (x+a,y+P),

(m,n) f ( 0 , O )( a and 13

is an isometry,

so

2

(K

,(p)

is

equicontinuous, and hence pointwise almost periodic. We show that the orbit of

(0,O) is dense. Since its orbit closure is minimal, there is

N

an N > 0 such that d((O,O),(p (0,O)l < for some integers j and f3

guarantees that

in the cascade flow defined by 2 is dense in K .

(K,(p

r N

= )

e,

E.

Now

N

(p

( 0 , O )= (Na-j,Np-l)

and the rational independence of a and

Np-e is irrational. Thus the orbit of Na-j is a subset of the orbit of

qt(x,y) = (x+rt,y+t), which, since Since

(pN

(0,O)

(0,O) in the (real)

r

is irrational,

"pushes" points along this flow orbit in

Equicontinuoiis Flows

cpN

steps" it follows that the

"E

41

orbit of

is

(0,O)

E

dense in the

n

flow orbit, and hence 2~ dense in .'K {cpJ ( 0 , O ) } j E n

2 is 2& dense in K .

Thus certainly the orbit

Since

is arbitrary, this proves

E

mini ma1 ity . Note that if a and p for some

(m,n)f ( 0 , O ) )

are rationally dependent

2

in this case, the set

2

then the cascade

[(x,yf E K Imx + ny

(K ,cp) E

kZ1

(ma + np = k E Z )

is not minimal.

For,

is a non-trivial closed

invariant subset of K"

It is easily shown by induction that the cascade generated by the of Kn given by cp(xl, . . . , xn) = (xl+al, . . . , x +an) is n

homeomorphism cp

minimal if and only if al,...,anare rationally independent. Another example of an equicontinuous minimal cascade is provided by the dyadic group (also called the "adding machine"). C =

Il {O,l}, i=O,1,2,.. .

where

{0,1) has the discrete topology. Thus the

elements of G are one-sided sequences g = (g0 , g 1'"'

or

1.

Let

We define addition in G

where gi = 0

by componentwise addition modulo

2,

We may also regard G as the collection of "formal

with "carrying." W

power series" l g i a i , where gi = 0

or

1.

If go = (1,0,0 ,... 1

i =O then go

is a topological generator (since the set of multiples of g 0

consists of all elements of the form all

1, with not (xo,xl,...,xn,O,O,...

xi = 0, and this set is certainly dense in GI.

-go = (1,1,1,.. . I .

Note that

From the general considerations mentioned earlier, it

follows that the homeomorphism

t : G j G ,

t(g) = go + g (g E GI,

gives

rise to an equicontinuous minimal cascade. Note that G homeomorphism

t

is homeomorphic to the Cantor set and. in fact, the can be defined in terms of the usual "middle thirds"

construction. At the n-th stage of this construction, we have a closed

42

Chapter 2

set Xn which consists of 2” disjoint segments S(n,m) ( 0 where if 0

5

k

S

2”-l, S(n,k) and S(n,k+2”-’)

terminal thirds, respectively, of S(n-1,k).

X

=

n

n=O,1,.. . {x) =

Then I/J

:

Xn

(where Xo = [ O , l l )

n

n=O,1 , . . . X+X

S(n.mn)

(0 S

m

5

2”-1),

are the initial and

Then the Cantor set

and, if x

mn

S

E

I 2”-1, mn+l

X then

=

n mn(mod 2 1 ) .

is defined by

{$(x))

=

n

n=O,1,.. .

S(n,mn+l(mod 2”)),

and it is easily verified that the cascades (CJ)

and

(X,$) are

isomorphic. Recall that i n chapter 1 we introduced the flow (CO(T),T) where C (TI is the set of real or complex valued bounded continuous functions 0

on T with the uniform topology and T acts on C0 (TI by translation: where ft(z) = f(tr).

(f,t)i-+ft

Clearly, T acts as a group of

isometries, so the flow (CO(T),T) is equicontinuous. Now the space

CO(T) is not locally compact, so we cannot apply our theorems in Chapter 1 directly to establish the relation between almost periodicity and

compact minimal sets in CO(T).

For this purpose we require the

following lemma and its corollary. (Compactness of the phase space is not assumed. 9. Lemma. Let

and let x

E

(X,T) be an equicontinuous flow, where X

X be almost periodic. Let

subset K of T such that xK is Proof. Let 6 correspond to

E

E

&

> 0. Then there is a compact

dense in xT.

in the definition of equicontinuity,

and let A be a syndetlc subset of T such that d(xa,x) < 6 a

E A.

is metric,

for

Let K be a compact subset of T such that T = AK and let

Equicontinuous Flows

t

E

T. Then t = ak, for a 10. Corollary.

Let

metric space, and let x -

xT

43

k E K, and d(xt,xk) = d(xak,xkl <

E A,

X a complete

(X,T) be equicontinuous with E

c.

X be almost periodic. Then the orbit closure

is compact. Proof. It follows immediately from lemma 9 that xT

n)

is totally bounded. Since X

is complete,

n

(and therefore

is compact.

Note that in lemma 9 and corollary 10, the assumption that equicontinuous can be relaxed to

"T is equicontinuous at

(X,T) is

x."

Since C (TI is a complete metric space, the next theorem follows 0

easily from corollary 10, theorem 2, and theorem 7 of Chapter 1. 11.

Theorem. Let

f E CO(T).

(i)

f

(ii)

The orbit closure

(iii)

E

Then the following are equivalent:

i s an almost periodic point of the flow

(iv) TT If T = R,

fl

(CO(T),T).

is compact.

is a compact minimal set.

is a compact equicontinuous minimal set. then, as we noted in Chapter 1,

periodic point if and.only if f

f E C (R) 0

is an almost

is a Bohr almost periodic function.

The equivalence of Bohr almost periodicity and the relative compactness of the translates of f

((i) and (ii) above, respectively) is due to

Bochner. The latter property was taken by Von Neumann as the definition of an almost periodic function on an arbitrary group.

It can be shown

(Hewitt and Ross, volume I, p. 246) that this property is equivalent to the relative compactness of the "left orbit" - that is, the orbit of f with respect to the action defined by

f(r) = f(rt).

Note that the only compact orbit closures in CO(T)

are equicontin-

uous minimal sets. The final result in this chapter gives a sufficient condition for

Chapter 2

44

equicontinuity in terms of the action of the automorphism group. The proof depends on a lemma from general topology which is not quite standard, s o we include it here. Let X be a topological space, and let Y be a metric space. Let cpn

:

(n = 1,2, . . . 1 and let cp : X+Y.

X+Y

We say that cpn

convewes

&Q

cp c

uniformly

neighborhood U bf x and an integer No < c

d(cp(x'),cpn(x'))

for n

every c > 0, we say

x, for every x

E

No.

E

at x

if there is a

n

c

such that if x' E U,

If cpn-+cp

uniformly at x

c

uniformly at x.

Q +cp

easy to check that cpn-+cp at

2

X and

> 0.

Let x

If X

for

is compact, it is

uniformly if and only if cpn+cp

uniformly

X.

12. Lemma. Suppose X

is a Baire space (i.e., the countable

intersection of open dense sets is dense) and let cpn be a sequence of continuous functions from X x

Let E be the set of x

X.

E

cp n ( x ) + c p ( x )

to Y such that

for every

in X €or which rpn3 9 uniformly at

x. Then E is a residual set. Proof. For c > 0, let A ( & ) cpn+cp

uniformly at xo,

E

We show that A ( & )

Clearly A ( & )

I s dense.

Let W

E

X such that

is open and E =

1

n

A(E).

.

k=1,2,. .

be a non-empty open set in X.

If

K = [ z E Wld(cp(z),cpm(z)) < c/5, for all n for some n, Kn contains a non-empty open

is a positive integer, let

n

m

be the set of xo

w

=

uKn, so

U. Let x

E

U and let p

nl. Then

2

set

2

n. Then there is an integer q

2

n

(x)) < c/5 and a point z E U n Kn such that 4 d(cp (x),cp (z)) < c/5 and d(cp (x),cp (z)) < c / 5 . Then d(cp(x),cp (XI) S P P 4 q P d(cp(x),cp (XI) + d(cpq(x),pq(z)) + d(cpq(z),cp(z)) + d((p(z),cp (z)) + for which d(cp(x),(p

4

(z),cp(XI)< P P

d(rp

c

< ~ 1 5 ) . That is, if

(since z

x

E

U,

E

Knv d(cp(z),cp

q

d(cp(x),cp ( X I )<

P

P (z)) and d(cp(z).y, E

for all P

2

( z ) ) are

P n,

so

Equicontinuous Flows

cpn-+cp

E

uniformly at all points of U.

45

Hence W n A(&)

3

U

f

+,

and

the proof is completed. 13. Theorem. Let

(X,T) be a minimal flow with X

Suppose the automorphism group A

of

compact metric.

(X,T) is transitive.

with cp(x) = y. 1 Then

if x,y E X there is a cp E A

(That is,

(X,T) is

equicont inuous. (or more precisely the family of self maps of X

Proof. We regard T

which T defines) as a subset of C(X,X) the continuous maps from X to itself, provided with the topology of uniform convergence. sufficient by the Ascoli-Arzelk theorem, to show that T compact in C(X,X).

Let

{ti)

is relatively

be a sequence in T and let x E X. 0

Then there is a subsequence which we still call

{ti)

such that

lim xOti exists. It follows from the transitivity of A exists for every x E X. Then there is a map

t i +h

(h(x) = lim xt

pointwise

i

lim cp(x)ti = lim cp(xti) = cp(h(x)). X. x

1'

h

If cp

We show that

t +h

1

E

X-+X

:

for x E XI.

By lemma 12, there is an x E X such that Let x E X and let

It is

J

t +h

J

that lim xti such that

E

A, h(cp(x))

uniformly on uniformly at

> 0. Let cp E A such that cp(xl)

let 6 > 0 such that if d(zl,z2) < 6

=

&hen d(cp(zl),cp(z2))

<

= E.

x and Let

U1 be a neighborhood of x1 and j, a positive integer such that if j 1 jo and

y

E

ul, d(yltj,h(yl))

d(cp(y t ),cp(h(yl)) < 1 J j 2 jo. Hence t -+h j

E.

Thus if y

< 6. E

Then d(cp(Y1

U = c p ( U l ) , d(yt

j

1) = for

uniformly on X.

Exercises.

1.

Let

(X,T) be an equicontinuous flow, with

Then there is an equivalent invariant metric. p

(X,d) a metric space.

That is, there is a metric

on X, which induces the same topology as d, such that

Chapter 2

46

2. Let

(Xa ,TI (a E 9) be a family of equicontinuous flows. Then the

product flow (Wa,T) is equicontinuous. a 3. Let

(X,T) be a flow, and let S be a syndetic subgroup of T.

Suppose

(X,S) is equicontinuous. Then

4. Let

(X,T) be an equicontinuous minimal flow, where the group T

(X,T) is equicontinuous.

is abelian. Define a group structure as follows. Let

xo E X, and

(xot)(xos) = xOts (t,sE TI.

define multiplication on the orbit xoT by

Show that this multiplication is well defined and can be extended to all of X, so that X is a compact abelian group with identity xo.

Does

this procedure work if T is not abelian? 5. Later in this book, we will extensively study the regionally proximal

relation Q. This is defined by Q = n[ala if and only if for any neighborhood U of x there are x'

/3 E I$

E

U, y'

V, and t

E

E

E

31. Note that

(x,y) E Q

and V of y, and any T such that

(x't,y't)

E

8.

Show that the flow (X,T) is equicontinuous if and only if Q = A ( s o that

6. If

(X,T) has no non-trivial regionally proximal pairs).

(X,T) is any flow, let R denote the orbit closure relation, so

(x,y) E R

-

if and only if xT =

relation if and only if a) If

F.

Note that R

is an equivalence

(X,T) is pointwise almost periodic.

(X,T) is equicontinuous, then the orbit closure relation is

closed. b) The flow (X,T) is equicontinuous if and only if the orbit closure relation in (XxX,T) is closed (use exercise 5). 7. Suppose

(X.T) is equicontinuous and topologically transitive (every

invariant open set is dense).

Show that

(X,T) is minimal.

Equicontinuous Flows

47

t,al , . . . ,ak be non-zero real numbers and let

8. Let

there is a relatively dense (i.e., syndetic) subset A

> 0. Then

E

of integers such

-

m a 1 < c. i i (Let X 1 , . . . , X k be pairwise disjoint circles with circumferences k lall,.. . , lakl respectively, and let X = u Xi. Define a homeomorphism

that if n

E

there are integers ml,..,,mk and

A,

Int

i=l X+X

(p :

9. If

such that the cascade ( X , ( p ) (X,V)

is a uniform space, and

to itself, then @ a@ =

[(cp,+)l(p,+

and

E Q,,

t

@ = [n

It

(cp(x),+(x))

TI

E

Q,

generated by the family

a, x

E

XI,

where a

E

Y.

is equicontinuous if and only if the

(X,T) E

is a family of maps from X

@

is the uniformity on

Show that the flow collection

is equicontinuous.)

is totally bounded with respect to the above

uniformity. 10. Let

ft

E

C(X)

(X,T)

be a flow, and let f E C ( X ) .

with respect to

(X,T)

compact in C ( X ) . f

E

C(X)

ft(x) = f(xt).

be defined by

E

%.

that if 11.

Then there is an

<

E,

E

is relatively

(Use the Ascoli-Arzeli theorem as well as

X be a compact Hausdorff space, and let E

> 0, and a finite subset

for all

f E F, then

x+a, R (x) = x+B. Find conditions on a and 6 factor of Ra.

T, let

is almost periodic

TI

F of C ( X )

such

(x,y) E a.)

Consider the rotations of the circle defined by

B

E

(X,T) is equicontinuous if and only if all

Show that

are almost periodic.

If(x)-f(y)l

We say f

if the collection [ftlt

the following lemma: Let a

If t

so

a and 13: Ra(x) = that R

13

is a

This Page Intentionally Left Blank

49

Chapter 3 The Enveloping Semigroup of a Transformation Group, I The enveloping semigroup of a flow, which was introduced by Ellis, is

It has proved to be

a kind of compactification of the acting group.

extremely useful in the study of dynamical properties of transformation groups. The enveloping semigroup, as well as certain objects derived from it, will be used throughout the book.

In this chapter, we develop the elementary properties of the enveloping semigroup. We use it t o obtain the structure of equicontinuous minimal flows, and also indicate its connection with the proximal relat ion. If X

is a compact Hausdorff space, then Xx

denotes the collection

of all (not necessarily continuous) maps from X

to itself, provided

with the product topology, or, what is the same thing, the topology of pointwise convergence. By Tychonoff’s theorem, Xx

8,

If 5, E

Hausdorff.

is compact

we write x5, for the value of

E at x

the

“x’th“ coordinate of 5 ) .

Xx also has a semigroup structure defined by composition of maps: i f <,r)

E

Xx

then 5s

E

Xx is given by x(5,r))

=

(x<)(r)), (x E XI.

The semigroup Xx has rather limited continuity properties. that if xcn+x<,

(5,)

X is a net in X , then

for all x

(if x E X, xr)

E

X

is easy to see that

E

so

En+$.

X. Thus if E, +E,

and

n

xr)C;,-+xw$).

Enr)+Er),

If

in Xx

r)

r)

E

Recall

if and only if

X

X ,

then r)5,,-+~5,

is a continuous map, then i t

but in general 5,

nQ+
Thus Xx

is

not a topological semigroup. Now consider a transformation group, which for the moment we write

Chapter 3

50

?

=

(x

E

(where X

(X,T,n)

again as

t [ n It

E

TI

X, t

E

TI).

(recall nt Thus

f

'i:

T acts on Xx

7

and note that

regard

(E,T)

in X

X.

E(X,T)

Let

X . Thus E

is also compact Hausdorff.

(x)
by

(or E ( X )

ts

t

( ( I 1s = TZ

E

X X , t E TI

(s,tf TI),

is also invariant. Thus T acts on E,

so

E,

and we may

as a transformation group. Note that (as a special case of

convergence in

xX 1,

a net

if and only if xt +x< n

{ntn)

for all

We will usually identify t and write t

X

Let

xnt = xt

is defined by

is an invariant subset

?

the closure of

X+X

:

is a subset of

or E) be the closure of Recall that

is compact Hausdorff).

+<

n

x

E

i

converges to t: E E, n

t

n+<,

X.

T with the map n L which it defines,

E

instead of n

in

t

"+<.

Thus T

is regarded as a subset

X (and of X 1,

This is a slight abuse of notation, since it is t possible that s and t are distinct group elements, but nS = I of E

(that is, xs = xt, for all x

E

XI,

but this will usually not present

a problem. Moreover, if

{tn)

is a net in T and

refer to convergence in E

tn+t

(xt,+xt,

t

E

then tn- + t

T,

for all x

E

X).

will

Of course, if

with respect to the topology of T, then also tn+t

in E,

but not conversely (consider the example of the irrational flow on the torus). Now, let

5 , E ~E

and let

{sn) be a net fn T

with s - 9 . n

Then

E. Since E is closed, we have T q E E. Hence X E2 = EE c E, and E is a subsemlgroup of X . E is called the <s,+
and <sn

E

envelopinn seminroup (or Ellis seminroup) of the transformation group (X, TI.

We emphasize that while T

(or more precisely

is a group of

homeomorphisms of X to itself, the elements of E are, in general,

The Enveloping Semigroup of a Transformation Group, I

51

neither one to one nor onto nor continuous. The possession by

E of

these properties under various conditions is correlated with dynamical properties of the flow (X,T), as we shall see.

As a flow (E,T) is point transitive, that is, it has a dense orbit, since 3 = of

T

(=

-., , T)

=

E. In general, (E,T) is not minimal. Minimality

(E,T) also has dynamical implications.

It is easy to verify that E(XxX,T), "naturally" isomorphic with E(X) <,q E

E(X) then the expression

as well as E(E(X),T)

(see exercise 2 ) .

are

Therefore, if

is unambiguous - it has the same


meaning whether it is regarded as a semigroup product in E(X) q

1.

E

E(E(X))

xt +y. n

If y E

E(X).

(X,T) is a flow, and x

Proposition. If

Proof.

<E

acting on

n,

or as

{tn)

then there is a net

Let (a subnet of) t,+c

y = x< E xE. The proof that

xE c

E

X, then xE =

E

in T such that

E. Then xtn+x<.

a

n.

so

is similar.

Thus an advantage of the enveloping semigroup is that it allows us to replace arguments involving limits (y = xp, for some p

E

(y = lim xt ) n n

by equalities

E).

Now we use the enveloping semigroup to determine the structure of equicontinuous minimal flows. To this end, note that if flow, then T

t (or more precisely [ n It

subset of Cu(X,X),

E

(X,T) is any

TI) may be regarded as a

the collections of continuous maps from X

to

itself, with the topology of uniform convergence (as well as a subsetpf

XX , as discussed above).

If

(X,T) is equicontinuous (not necessarily

minimal) the uniform closure clU(T) = G, is a compact subset of C (X,X) by the Arzela-Ascoli theorem. Moreover, using elementary U

properties of uniform convergence, it is easily shown that G

is a

Chapter 3

52

topological group. For instance, if in T with t

-+c,

n

s

E

G

and

isn) are nets

{tn),

(uniformly!), then tns,-+
-+T)

n

maps from TxT-+G and XxT-+X. and

<,?I

Also the -1

(t,s)‘ts

defined respectively by

( x , t ) H x t are uniformly continuous, and so extend to continuous

maps of GxG+G,

and XxG-+X.

Thus G

is a compact topological group

and G acts as a group of homeomorphisms on X. Summarizing, we have proved: 2. Theorem. Let

(X,T) be an equicontinuous flow. Then there is a

compact group G and an action of G on X which extends the action of T. As

we observed in the preceding chapter, if a compact group G

on X, the action of C

acts

is equicontinuous, as is the action of any

subgroup T of G. The theorem just proved shows that all equicontinuous flows arise this way. (X,T) is equicontinuous, its enveloping

Next, we show that when semigroup coincides with G.

For let

j : Cu(X,X)+Xx

be the inclusion

map; clearly j is continuous. If clu and cl denote uniform and P pointwise closure, respectively, then j(clUT) = cl T (equality holds since cl T U

map,

so

G =

is compact).

E and

jlC is a homeomorpsism.

equicontinuous, and {tn) and only if tn+<

P That is, j ( G ) = E. But j is the inclusion Thus, if

is a net in T, then tn+<

(X,T) is pointwise if

uniformly.

We have therefore proved one direction of the following theorem. 3.

Theorem. A flow is equicontinuous if and only if its enveloping

semigroup is a group of homeomorphisms. The sufficiency, which is much deeper, wil chapter.

be proved in the next

The Enveloping Serniqroup of a Transformation Group, I

An equivalent formulation of this result is:

53

let G be a group of

homeomorphisms of the compact Hausdorff space X. Suppose’ G regarded

as a

subset of Xx, is compact. Then G acts equicontinuously on X.

4. Lemma. Let

(X,T) be a flow, with T abelian.

(1)

If p E E(X), t E T, pt = tp.

(2)

If

(X,T) is equicontinuous, E(X)

Proof. Let

{si) be a net in T with si-+p.

tsi+tp, sit-+pt (since t E T

Then tsi = s.t, 1

is continuous), and

tp = pt. This

The proof of ( 2 ) follows from (1) and the fact that all

proves (1). p E E(X)

is abelian.

are continuous, when

5. Lemma.

If

(X,T) is equicontinuous.

(X,T) is an equicontinuous flow, then the flow

(E,T)

is minimal. Proof.

(E,T), as a subflow of

X (X ,T) is equicontinuous, hence

pointwise almost periodic. Since (E,T) is also point transitive, it must be minimal.

(An easy direct proof of this lemma is also possible.) (X,T) is a flow, and x E X, the map

Note that if

p ~ x pdefines

a homomorphism of the flow (E,T) onto the subflow (3,T) of ( s o if

(X,T)

(X,T) is minimal, this map defines a homomorphism of E onto

XI. Now we are ready for the theorem on the structure of equicontinuous minimal flows. 6. Theorem. Let x E

X.

Let F = Fx = [p E E(X) Ixp =

of E(X).

XI.

Then F

is a closed subgroup

T acts on the space of left cosets [Fqlq E El, by

(Fq)t = F(qt) (t If T

(X,Tl be an equicontinuous minimal flow, and let

E

TI, and the flow (E/F,T) 1s isomorphic with

is abelian, F = {e}

and

(X,T) is isomorphic with

(X,T).

(E,T).

Chapter 3

54

Proof. Most of the assertions are obvious (or follow easily from the If T

previous discussion). xtp = xpt = xt, so

E

F, t

E

T, then

is the identity on xT. Since p

p

(X,T) and (Y,T) be flows and let

7. Theorem. Let

is continuous,

= X.

is the identity on

p

is abelian, p

71 :

X+Y

be an

onto homomorphism. Then there is a unique continuous semigroup homomorphism

8 : E(X)+E(Y)

(x E X, p

n(xp) = n(x)e(p)

such that

We write (just for this proof) TX and Ty

Proof.

E(X)).

E

for the maps defined

T on X and Y respectively, and endow TX and Ty with the Y topology and uniformity inherited from Xx and Y . Then the map

by

TX+Ty

8 :

defined by O(t) = t

is uniformly continuous (this is an

immediate consequence of the definition of flow homomorphism).

Thus 8

has a continuous extension, still called 8, to a continuous map of

E(X)

Since n(xt) = r(x)t = n(x)e t),

to E(Y).

(x E X, p

E

E(X)),

and n(xpq) = n(xp)0(q) = n(x)0(p)0(q).

n(xpq) = n(x)O(pq),

so,

TX is dense in E(X), If

I[

:

X+Y

the map e

E

is onto), 8(p)8(q) = 8(pq).

(since II

so the uniqueness of

notationally, and just write yp Thus we regard E(X)

this case, it is possible that p y

Also

0

Finally

is Immediate.

is a homomorphism of flows, we will frequently suppress

y E Y, p E E(X).

all

then ~ ( x p )= n(x)e(p)

1

#

p2

(instead of ye(p))

as acting on in E(X)

but

for

Y. Of course, in ypl = yp2,

for

Y (which is the case if and only if 8(p1) = 8(pz)).

The enveloping semigroup is particularly useful in studying proximality.

If

(X,T) is a flow recall (chapter 1) that

proximal if there is a net

xt + z n

and yt + z . n

{tn)

in T and z

x,y

E

X are

in X such that

The close connection of the enveloping semigroup

and proximality will be intensively studied in a later chapter. At this

The Enveloping Semigroup of a Transformation Group, I

55

point, we prove a simple proposition indicating this relation. (This is another example of how the enveloping semigroup enables us to replace

limits by equalities.) 8. Proposition. The points x and y

in X are proximal if and only

if xp = yp, for some p E E(X).

{tn)

Proof. Suppose x and y are proximal. Let let z E X such that xtn+z will still denote by

{tn),

and ytn+z. such that

tn+p

be a net in T and

Choose a subnet, which we E

E(X).

Then clearly

xp = z = yp. Conversely, if z = xp = yp, for some p E E, then there is a net

and x

itn)

in T, with t,+p,

so

xtn+xp

= z,

and ytn+z,

and y are proximal.

Exercises. Let

1.

(X,T) be a flow, with T abelian. Then the following are

equivalent: a) E(X)

is abelian.

b) All elements of E(X)

are continuous.

c) The action of T on every minimal subset of X 2. Let

in the following sense. If {tn)

E(X),

tn+F,

(X,T) be a flow. Show that E((E(X),T))

E

E(X)

if and only if tn+t

semigroup product q< E

E(E(x))

3. Let

is isomorphic with

is a net in T, then

E E(E(X)),

is equal to qg

and, if q E E(X),

the

the action of

on q E E(x).

(X,T) and

homomorphisms. Then to E(Y).

in E(X)

is equicontinuous.

(Y,T) be flows, and let K ~ , K: X+Y ~ A

1

and a2

be (onto)

induce the same homomorphism of E(X)

This Page Intentionally Left Blank

Chapter 4 Joint Continuity Theorems This chapter is devoted to proving the theorem, due to Ellis, that a transformation group is equicontinuous if and only if its enveloping semigroup is a group of homeomorphisms, as well as several related theorems. The proof given here is essentially due to J.P. Troallic (Bull. SOC. Math. France 107 (19791, 127-137). If Y and 2 are topological spaces, C(Y,Z) will denote the to 2 provided with the topology

collection of continuous maps from Y

of pointwise convergence. Unless the contrary is stated, all topological notions in C(Y,Z) will be with regard to this topology.

If

(Z,?$)

is

denotes the uniformity on C(Y,Z) defined (Y,Z) [ ; l a E 31, where (f,g) E if and only if

a uniform space, then U by the base

(f(y),g(y)) E a, for all y E Y. We write

C (Y,Z) for this uniform U

space. Here (as elsewhere) we will frequently assume, without explicit mention, that an element of a uniformity is symmetric. 1.

Lemma. Let Y be a compact Hausdorff space, and let

Z

be a metric

space. Then (1)

A

A pointwise separable subspace

of C(Y,Z)

is uniformly

separable. (2) Let

H E C(Y,Z) and let f E

subset D of H such that f E (3) Let

subset of

fi.

Then there is a countable

6.

B be a non-empty subset of C(Y,Z) is a Baire space. Let

V

E

?l(y,z).

such that every closed Then there is an

f E B such that f Q B\V[fl. Proof.

(1) We will construct metrizable spaces

and.

2,

with

2

Chapter 4

58

separable, and a subset

such that, if A and

of C ( ? , z )

given the relative topology from Cu(Y,Z) then A Let

and

and CU(v.F)

are

respectively,

are homeomorphic.

D be a countable pointwise dense subset of

N

2

Y = Iff(yfffEAIyE Y] c ZA and

=

u f(Y) c

A.

Let

?

Note that

2.

is

fEA metrizable (since it is homeomorphic to its image in the metrizable space

ZD via the projection map

separable metric (since each f(Y) pointwise dense in A).

If g

= [ n lg E

@I

C

U

E

is a compact metric space and

A,

let

n

g

:

P+z

is

D

is

be the projection

A].

(I[

(?,s) is a separable metrizable space (since i

its subspace

so

2

((f(y))fEA) = g(y)), and let g It is easily checked that 2 and A are homeomorphic.

onto the g’th coordinate Y

A

and that

(f(y))fEAH(f(y))fED),

is separable. Since A

2

is separable metric),

is homeomorphic with

i, A

is separable.

(2) If h

(where d

E

H, and m

is the metric on 2 ) .

For fixed m

and n, the set

m,n} hEH is an open cover of Ym, which has a finite subcover

{‘h,

{Rh,m,n1heH . m, n f

and n are positive integers, let

E

Then D =

u

H is a countable subset of H and m,n m ~ n

E. (3)

Suppose first that B

is countable. Then

is pointwise

Let

separable, and therefore uniformly separable, by (1). countable uniformly dense subset. Then if

E

> 0,

c

Dc

u SCl2(f)

be a (where

fED

S (f) denotes the closed &

in Cu(Y,Z)).

g

c

u

S,/Z(f) feD

ball about

E

Now each Sc/Z(f) A

g.

Since

i?

A

k

f with respect to the metric

is pointwise closed and

is a Baire space there is an fo E D and

Joint Continuity Theorems

a non-empty (pointwise) open set such that Then if f

E w n

B, clearly f

e'

59

*

$I

w n

c Sc/2(fo) n

i.

B\SEo.

Now, let B be an arbitrary subset of C(Y,Z) and suppose the conclusion fails. Then there is an c > 0 such that f E B.

all that

f

E

By ( 2 ) , there is a countable subset Df

fJf.

D1

inductively by

defined, let Dn+l =

Ed

E

B\Sc(f),

of B\Sc(f)

Choose f E B and define a sequence D (n = 1,2, n

subsets of B

d E

f

= {f},

u

u

D\SEo,

for all

such

...

of

and, if D1,...,Dn have been

Db. Then D = Dn bEDn n=1,2,.. .

c Dn(B\Sc(d)) c

for

d E D.

is countable and

But this contradicts

the first part of the proof.

2. Lemma. Let X be a locally compact Hausdorff space, Y a compact Hausdorff space, and Z continuous, and let V

a uniform space. Let U

E

cp :

X+C(Y,Z)

be

Then there is a dense open set

(Y,Z)'

X such that if x E U, then cp-l(V[cp(x)I) Proof. We first suppose that Z

U

in

is a neighborhood of x.

is a metric space. Then we may suppose

that there is an c > 0 such that

V

< c, for all y

= [(f,,f,)ld(f,(y),f,(y))

U =

u

int cp

-1

(ScI2('p(a))). Then U

E

Yl.

Let

is open, and if

x

E

U,

acX cp-l(Sc(cp(x))) is a neighborhood of

x. To show that U is dense in X,

let o be a non-empty relatively compact open set in X. such that

cp(b)

e'

Let

(lemma 1, ( 3 ) ) . Since cp

(~(w)-S,/~(cp(b))

b E w is

~~

continuous, b so

0-cp

-1

(ScI2(cp(b))).

b E w n U. Therefore U

Hence b E int cp-1(Sc,2(cp(b)))

is dense in X.

Now consider the general case. The uniform structure by a family M

of pseudometrics on Z .

associated metric space, let and let

rm : C(Y,Z)+C(Y,Zm)

c U,

II

m

:

Z+Zm

For

m

E

A,

%

is defined

let Zm be the

be the canonical projection,

be defined by rm(f) = xmof. Now if

Chupter 4

60

there is a finite set ml, . . . , m E M such that (writing P YY,Z) for Zm , ri for rm , etc.) there are V, E 25 with L I i i i

P

n

(rixri)

-1

(V,) c V.

Let

'pi

= rioq

X-+C(Y,Z).

The,n

'pi

is

i=1

continuous, so applying the first part of the proof, there is an open set

Ui,

which is dense in X, such that

f

ui,

x e

is a cp~l(~i[(pi(x)~)

D

neighborhood of

x

E

3.

x.

U, cp-l(V[cp(x)l)

Then U =

U i=1 i

is open and dense in X, and if

is a neighborhood of x

is compact in the pointwise topology

(that is, in the topology induced on G

G on Y

Proof.

from C(Y,Y)).

Let

V

E ?1

(Y,Y)'

hgo E U.

By lemma 2, V[hgo]

gho, and it follows that VIgol

we regard

cp

a a map from G

Therefore G = cp(G)

G+C(Y,Y),

:

the

be a dense open set in G

and let U

satisfying the conclusion of lemma 2. Now let such that

Then the action

is equlcontinuous.

In lemma 2, take X = G, 2 = Y and cp

inclusion map.

of

X.

Theorem. Let G be a group of homeomorphisms of the compact

Hausdorff space Y, and suppose G

of

E

go

E

G

and choose h E G

is a (pointwise!) neighborhood

is a neighborhood of go.

to Cu(Y,Y),

then

'p

Thus, if

is continuous.

is compact in the uniform topology, and so by

Ascoli's theorem G acts equicontlnuously.

4. Theorem. Let

(X,T) be a transformation group, with X compact

Hausdorff, and suppose the enveloping semigroup E(X) homeomorphisms. Then Proof. Take C = E(X)

(X,T)

is a group of

is equicontinuous.

in theorem 3. Then G acts equicontinuously

and so does its subgroup T. In theorem 4, it is essential that E(X)

be a group (that is, all

elements are invertible), and not merely a semigroup of continuous maps.

Joint Continuity Theorems

X as the one point compactifica-

A simple example is provided by taking

tion of the reals ( s o X

is homeomorphic to the circle) and

the

(p

(p(x) = x+l for x real and the point

homeomorphism of X defined by at infinity w

61

a fixed point.

(X,(p)

The cascade

is certainly not

equicontinuous. However, the enveloping semigroup consists of the powers of

together with the constant map

(p

(x E X).

XHU

On the other hand, if we require minimality, the assumption that all are continuous is sufficient for equicontinuity. For

elements of E(X)

this we need a lemma, which says that if a flow is "pointwise equicontinuous" it is equicontinuous. We omit the simple proof.

5. Lemma. Let

(X,T) .be a transformation group, with

Hausdorff. Suppose, for every a neighborhood V

t

all

E

E

\,

X compact

and every x E X, there is a

of x, such that if y E V,

then

(xt,yt) E a, for

T. Then (X,T) is equicontinuous.

6. Theorem. Let

(X,T) be a minimal flow, and suppose that all

elements of E(X)

are continuous. Then

Proof. Let

(p :

X+C('E(X),X)

(x E X, p E E(X)). andlet

U E %

V = [ (f,g)I (f(p),g(p)) in X

hood of x.

So,

(E(X),X)

x s 0

E

hood of xos, so

x's

E

Wo

and

so E

El. By lemma 2, there is a dense

such that if x

E

U, W = (p-'(V[(p(x)])

if x'

U.

is continuous. Now, let

cp

p

E

E

a, all

W, ((p(x),(p(x'))

thing, (xp,x'p) E a, for all such that

(p(x)(p) = xp

be defined by

Since E(X) c C(X,X),

V=ae'U

open set U

(X,T) is equicontinuous.

Let Wo =

-1

Wos

p E E. cp

-1

V, or what is the same

E

Let

xo E X, and let s E T

(V[(p(x,s)l).

Then Wo

is a neighborhood of

(x st,x'st) E a for all 0

a neighborhood of xo such that

t

is a neighbor-

E

T.

xo,

so if

is a neighbor-

-1 x' E Wos ,

That is, W* = W s-l 0

if x' E W*, (xt,x't) E a, for all

is

Chapter 4

62 T E

T. It follows from lemma 5 that

(X,T) is equicontinuous.

7. Theorem. Let T be a locally compact Hausdorff space with a group

structure for which group multiplication is separately continuous, and suppose T acts on the compact Hausdorff space X continuous manner.

(That is. the defining map x

in a separately :

XxT+X

separately continuous.) Then the action of T on X

is

is jointly

cont inuous. Proof. Let imply that

cp : cp

T-+C(X,X)

be defined by cp(t)(x)

=

xt. The hypotheses

is continuous. It is sufficient to show that cp

continuous when it is regarded as a map from T to Cu(X,X).

is This is

accomplished, using lemma 3, as in the proof of theorem 7. In theorem 7, we may assume that the space X is locally compact Hausdorff. For, if Xu = X u {d is the one point compactification of X, we may extend the action of T to Xw by defining ot = w (t

E

TI.

Then theorem 7 can be applied to the action of T on Xu. The hypotheses of theorem 6 cannot be weakened to "all elements of a minimal right ideal are continuous." For consider the minimal flow

is the circle, and T is the total homeomorphism group.

(X,T) where X

The unique minimal right ideal in E(X,T)

consists of the collection of

constant maps, but the flow (X,T) is not equicontinuous (in fact, i t is a proximal flow

-

all pairs x and y in X are proximal).

8. Theorem. Let X be a compact Hausdorff space which has a group

structure and suppose that multiplication is separately continuous. Then

X is a topological group. Proof. Let X act on itself (on the right) by

( x , y ) ~ x y . By

assumption, this action is separately continuous. I t follows from theorem 7 that the action is jointly continuous. Thus multiplication in

Joint Continuity Theorems

X

63

is jointly continuous. Continuity of inversion follows easily, using

the compactness of X, from the joint continuity of multiplication. The conclusion of theorem 8 still holds if it is only assumed that X is locally compact. Using the remark following theorem 7, it is shown,

just as in the proof of theorem 8, that multiplication in X

is jointly

continuous. The proof that inversion is continuous is somewhat more involved. A proof was given by Ellis (Proc. Amer. Math. Soc. 8 (19571, 372-373). Theorems 3, 4, 7, and 8 are also due to Ellis (Duke Math. Jour.

24 (19571, 119-125). Other proofs of Ellis' theorems are by Namioka (Pacific Jour. Math. (Acta Math.

(19741, 515-531) and de Leeuw and Glicksberg

10s (19611, 63-97).

The latter proof is reproduced in the

book Weakly almost periodic functions

on seminroups, by

R.B. Burckel

(Gordon and Breech, 1970). Theorem 6 is a "folk theorem." The author has not seen it in print.

This Page Intentionally Left Blank

65

Chapter 5 Distal Flows In this chapter, we begin the systematic study of distal flows, which were mentioned briefly in chapter 1.

These are generalizations of

equicontinuous flows, and, as we shall see, these two classes of flows have some common properties. Chapter 7 will be devoted to the striking structure theorem for distal minimal flows, due to H. Furstenberg. Recall that a flow (X,T) pairs.

That is, if

is distal if it has no non-trivial proximal

x,y E X with x

#

y, then there is an a E

(depending on x and y, of course) such that

t

E

T.

If

and only if

(xt,yt) Q a, for all

(X,d) is a (compact) metric space, then inf d(xt,yt) > 0 whenever x teT

#

%

(X,T) is distal if

y.

It is easy to see that equicontlnuous flows are distal. For, suppose (X,T)

is equicontinuous, and x,y E X are proximal.

let 6 E

3

such that 6T c a.

Let

t

(x,y) = (xt,yt)t-' E 6~ c a. Since a E

E

Let a

E

I$,

and

T such that (xt,yt) E 6,

I$

so

is arbitrary, it follows

that x = y. A simple example of a non-equlcontinuous distal flow is provided by a

disk rotated at different rates around a common center. Precisely, consider the homeomorphism

(p

of the disk D

in the plane given by

(p(r,e) = (r,e+r) ((r,e) are polar coordinates). Clearly the cascade (D,(p)

is distal.

(If two points are on different circles, their orbits

remain on these circles, and on a given circle, see that

(D,(p)

multiple of

II

(p

is a rotation.) To

is not equicontinuous, choose r to be a rational and let n

J

be a sequence of integers tending to

infinity such that nJr is an even multiple of

II,

say

nJr

= 2 mJ- n *

Chapter 5

66

Let

E .

=

n n,

and let r = r+c j

j'

so

(r e ) + ( r , e ) ,

J'

but

J~

cpnJ(r,e)= (r,8+2m n) and (pnj(r , e l = (r 6+(2m.+l)n). Hence the J J j' J cascade (D,cp)

is not equicontinuous. The construction of minimal

distal flows which are not equicontinuous is more difficult, and will be undertaken later in the chapter. Before continuing the discussion of distal flows, we will consider the proximal relation in some detai;.

The results obtained will also be used

in later chapters. The proximal relation in or

P(X) o r Px).

(X,T) will be denoted by

That is, P = [(x,y)

E

P (or P(X,T)

XxXIx and y are proximall.

P is obviously a reflexive symmetric and T invariant relation ((xt,yt) E P whenever (x,y) E P I ,

but, as we shall see in later

chapters is not in general transitive or closed. If x E X, the proximal set of y E X such that

a distal point.

of x, denoted by P(x)

(x,y) E P. If P(x) = {x),

is the

or XP

then x

is called

(So the flow (X,T) is distal if all points are distal

points; equivalently if P = A. 1 Minimal flows with distal points ("point distal flows") will be considered in a later chapter. Although the proof is simple, the following proposition is,very useful.

It may be paraphrased as:

"proximality and almost periodicity

(in the product space) are incompatible."

1.

Proposition. Let

(X,T) be a flow.

periodic point in XxX, with x Proof. Suppose

y, then x and y

are distal.

(x,y) is an almost periodic point and that x

are proximal. Then there is a net that

If (x,y) is an almost

(x,y)ti+(z,z).

{ti)

in T and a point

z

and y E

X such

Since (x,y)T = (z,z)T is a minimal subset of

Distal Flows

XxX, (x,y) E (z,z)T. But

A,

so

67

is obviously a subset of the diagonal

x = y,

In the next proof (and also in the discussion of universal minimal flows) we will use the notion of an almost periodic set. a flow. Then a subset A whenever D

Let

of X is called an almost periodic set if,

is a set whose cardinality equals that of A,

with range z = A,

(X,T) be

and z

E

XD

then z is an almost periodic point of the flow

D (X ,TI. It is clear that this definition depends only on the subset A Naively, the point z

may be thought of as A

of X.

"spread out" to a point

in a product space. 2. Lemma. Let

(X,T) be a flow, and let A

be an almost periodic set

in X. Then there is a maximal (with respect to inclusion) almost periodic set B such that A c B. Proof. Let 5 denote the collection of almost periodic sets in X which contain A.

Since a neighborhood in a product space depends on

only finitely many coordinates, 5

is of finite character, and we may

apply Zorn's lemma to obtain a maximal element (actually, we apply Tukey!s lemma; see Kelley, General TODO~ORY). 3.

Theorem. Let

(X,T) be a flow and let x

E

X.

Then there is an

almost periodic point x* which is proximal to x. Proof. If x

is almost periodic, take x* = x.

maximal almost periodic subset of X, denotes the cardinality of A)

and let z

If not, let A E

with range z = A.

X I*'

be a

(where !A]

Consider

w = (z,x)E XtA'xX. Note that there is an almost periodic point of the form w* = (z,x*) E 3. For if (zf',x*)E periodic, then z"

E

with

(z".xf')

almost

5, which is a minimal set and, if z"tn+z

Chapter 5

68

then ( a subnet of) x"tn-+x*

and

is an almost periodic point, so

A

(z",x")t +(z,x*). n

u {x*)

periodic set. By maximality of A

for some net

(sn)

in T, we have

(x,x*) E P. Now z

= range z u {x*)

we must have x* E A.

appears as one of the coordinates of

x*

Now

z.

-+x* n

XIS

Since

(z,x*)

is an almost That is to say,

(z,x)sn+(z,x*)

and xs +x*, n

is almost periodic, and therefore x*

so

is almost

periodic, and the proof is completed. Every known proof of this theorem requires the use of a "large" product space (another proof will be given in the next chapter).

It

would be interesting to find a direct proof. Proposition 1 and theorem 3 have the following useful corollary.

4. Corollary. Let (i)

(X,T) be a flow. Then

If the product flow

(XxX,T) is pointwise almost periodic,

(X,T) is distal. (ii)

If x

(iii) I f

E

is a distal point, it is an almost periodic point.

X

(X,T) is distal, it is pointwise almost periodic.

Now, returning to our discussion of distal flows per se, we have the following simple lemma. 5. Lemma. - Let

(Xi,T) ( i E 9 ) be a family of distal flows (with the

same acting group TI. Then

(TD(

i

i

,TI is distal if and only if every

(Xi,T) is distal. Proof. Suppose some yj

f

yi.

(X T)

J'

Let x,x' E TMi

is not distal. Let

(y.,y'.) E P(X ) with J J be such that xj = yj, xj = yj and xi = xi

i

for i E 4 with

i

#

j. Then clearly

(x,x')

E

P(X).

The converse is

obvious. The next theorem provides several conditions which are equivalent to distality.

Distal Flows

6. Theorem. Let

69

(X,T) be a transformation group. Then the following

are equivalent: (i)

(X,T) is distal.

(ii)

(Xa,T) is distal, for all cardinal numbers a

(iii) (Xa,T) is distal, for some cardinal number

2 1.

a 2 1. a 2 1.

(iv)

(Xa,T) is pointwise almost periodic, for all

(v)

(P,T) is pointwise almost periodic, for some a

(vi) The enveloping semigroup E(X)

h

2.

is a group.

Proof. The equivalence of ( 1 1 , (ii) and (iii) follows from lemma 5, and these conditions imply (iv) by corollary 4, (iii). implies (v).

If (v) holds, then

pointwise almost periodic, and distal. Thus (i Suppose E(X)

(XxX,T), as a factor of

so,

by corollary 4 , ( I ) ,

(Xa,T) is

(X,T) is

- (v) are equivalent. is a group, and let

of chapter 3) xp = yp, for some p (yp1p-l = y, and X

almost periodic. If almost periodicity of

(x,y) E P. Then (by proposition 7

E

E(X),

so

x = (xp)p-' =

is distal.

Finally, suppose (iv) holds. p

E

E(X),

X (X ,TI, e

Thus e = pq, for some q inverse, and E

Obviously (iv)

E

E.

X Then the flow (X ,T) is pointwise p E

E

a, so

by the assumed pointwise

= pE

(chapter 3, proposition 1).

Thus every element of E has a right

is a group. This completes the proof.

7. Corollary. (a) A distal flow is equicontinuous if and only if all elements of its enveloping semigroup are continuous maps. (b) A distal flow is minimal if and only if it is point transitive (has a dense orbit). (c) A factor of a distal flow is distal. Proof.

(b) is an immediate consequence of the pointwise almost

Chapter 5

70

periodicity of distal flows. To prove (a),use theorem 4 in the “Joint continuity theorems” chapter. If Y is a factor of the distal flow X, then YxY is a factor of the pointwise almost periodic flow X x X . Obviously YxY

is pointwise almost periodic so

Y

is distal.

Theorem 6 underlines some of the similarities of distal and equicontinuous flows - both are pointwise almost periodic and, for both classes of flows, the enveloping semigroup is a group. Partly for these reasons, it was conjectured at one time that distal minimal flows were necessarily

equicontinuous. Counterexamples will be presented at the end of the chapter. However, there is a deep connection between distal and equicontinuous minimal flows. The Furstenberg structure thereom, which we will state and prove in the next chapter, asserts that any distal minimal flow arises from an equicontinuous flow by a (possibly transfinite) sequence of “almost periodic“ o r “equicontinuous”extensions. In particular, a distal minimal flow always has a nontrivial equicontinuous factor. We now “relativize“the notion of distal. Let n homomorphism of flows. Then n whenever r(xl 1 = n(x2),

E

X-+Y be a

is said to be distal provided that

with x1

(Equivalently, f (x,,x2)

:

f

x

2’

P, with x1

then x1 and x are distal 2

*

x2, then n ( x , l

* n(x2)).

In this case, we also say that X is a distal extension of Y. Note that X

is a distal extension of the trivial (one point) flow if and

only if X is a distal flow. 8.

Proposition. A distal extension of a distal flow is distal.

Proof. Let n

:

X+Y

be a distal hofnomorphism with

Suppose (x1,x2) E Px. Then (n(x1),n(x2)) n(xl) = n(x,),

and since n

E

Y distal.

Py. Since Y is distal

is distal, it follows that

x1 = x2.

Distal Flows

71

A relatively simple way to obtain distal extensions is by means of Cartesian products. Let and let

(Y,T) and

X onto Y is a

X = YxZ. Then clearly the projection of

distal homomorphism. periodic.

If Y

2 distal,

(Z,T) be flows, with

is minimal, then X

is pointwise almost

(This is proved most easily using the enveloping semigroup

although it is also possible to give a proof using almost periodic sets.

A more general result will be given in theorem 15 of chapter 7.) Thus any minimal subset of X

is also a distal extension of Y.

An important subclass of the distal extensions are the group extensions, which were defined in chapter 1. bitransformation groups

(G,X,T)

is the orbit space, on which T

Recall that these arise from is compact, and that Y = X/G

where G acts by

(Gxlt = Uxt) (t

also that the group extension is said to be free if g implies gx

f

x f o r all

E

E

G

TI. Recall

with g

e

f

x E X.

9. Proposition. A free group extension is distal.

Proof. Let Y = W G be the natural map. some g E G.

If

as in the above discussion and let Thus n ( x ) = dx')

so

X+X/G

=

Y

if and only if x' = gx for

(x,x') = (x,gx) is proximal, there is a net {tn)

T and z E X such that xtn+z, g(xtn)+gz,

R :

(gx)t +z. n

But also

a

(gx)tn =

gz = z. By freeness, g = e, and x' = x.

In fact, as we shall see in a later chapter, group extensions are "almost periodic" extensi ons. Now we present a criterion for minimality of a class of group extensions of minimal cascades. Let

(Y,T) be a minimal cascade ( s o

homeomorphism of

Y and not a group).

group, and let cp

:

Y+G

Let

C

be a continuous map.

T

is a

be a compact abelian If X = Y x G ,

a homeomorphism S of X by S(y,g) = (T(y),p(y)g).

we define

Obviously, the

Chapter 5

72

projection

( y , g ) ~ ydefines a group extension of Y. The cascade

skew product.

(X,S) is called a

The following theorem, which gives a necessary and sufficient condi(X,S), involves the character group of G.

tion for the minimality of

Let K be the unit circle in the complex plane, K = [ z E CCIlzl = 11. Then G, the character group of G, is the multiplicative group of A

continuous homomorphisms of

G to K

(if yl,r2

(rl;yl)(g)= yl(g)r2(g)).

is defined by

E

,.

G then' r l r 2 E G

The identity of G

(the

n

"trivial character"), denoted by ;r = 1, f o r all

g

E

is the

E

G with r ( g ) = 1

G.

(Y,T) be a minimal cascade and suppose

10. Theorem. Let

(X,S) are as above. Then the cascade

r

E G

with

r

and

(X,S) is minimal if and only i f

the functional equation f(T(y)) = g(cp(y))f(y) solution f : Y+K,

G, cp

has no continuous

1

f

Proof. Suppose the functional equation has a solution f f o r some

continuous S invariant function on X.

If

would have t o be constant. Buf, if g

G such that z(g)

F(y,g)

f

F(y,e), so

(h

E

G),

(X,S).

M

(so

f

1, then

(X,S) is not minimal.

Conversely, suppose subset of X

E

(X,S) were minimal, F

#

(X,S) is not minimal.

X).

Let

M be a minimal

Now, if G acts on X by h(y,g) = (y.hg)

then hS = Sh, and C acts as a group of automorphisms of So,

hM n M = 4.

if h

E

G, hM

is a minimal subset of X and hM = M or

If H = [h E ClhM = MI,

G. Moreover, H

#

C. For, let

n(y,g) = y. Since Y

11

:

then H

X+Y

is a closed subgroup of

be the projection

is minimal, n(M) = Y, so, if y

E

Y. (y,g) E M,

Distal Flows

If H = G, (y,g) E M, for all g

for some g E G.

that M = YxG = X and

,. E G,

let

f E C(Y) (y,g') g'g-'

r(g'g

so

where

(y,g) E M.

-1

1

=

1, r(g) = r(g').

so

(so

f(y) = r(g).

f(yn)-+f(y).

f(yn) = r(gn)), Since

nM

fl

4, g'g-lM

lim f(yn)

let (a subnet of)

exists.

= r((p(y))f(y),

If

(yn,gn)-+(y,g)

M,

E

so

(y,g) E M, S(y,g) = (T(y),(p(y)g)

f(T(y1) = r((p(y)g) = r(cp(y))r(g)

M, and

be a net in Y with yn-+y.

{y } n

r is continuous, r(gn)+r(g),

Finally, if

=

Thus f is well defined.

is compact, we may suppose that

(yn,gn) E M

Note that if also

(g'g-')M

so

To show that f is continuous, let Since K

G, and it follows

1 such that r(h) = 1, for h E H. Define

f

by f(y) = r(g),

H,

E

with

E

(X,S) is minimal, contrary to hypothesis. Now,

M, g'g-l(y,g) = (y,g'),

E

13

and f

E

M

so

is a solution of

the functional equation. We apply the theorem just proved to show the minimality of a class of skew products on

Kk+1,

the

(k+l) torus, namely

k-1 z)wk)

T(z,wl,. . . ,wk) = (az,(p(z)wl,(p(/3z)w2,.. . ,p(P where a,@ E K and

(p :

K+K

extension of K.

as a Kk

must choose a,P and

(p

are to be chosen. Here we regard

Since the character group of Kk

so

k is Z ,

Let

w

E

*

..

f

0 for every

with integral coefficients. Choose @ E K not a root of

unity and let N = [ l < n(1) < n(2) < such that

if

(0,. ,O).

K be non-algebraic. That is, p(w)

polynomial p

I@n(i)-wl

< T1 ,

for i L 1.

such that for some constant c,

n

., . 1

be a sequence of integers

Choose a not a root of unity

la -11 I

C

-,2

for n

in a subsequence

n of N.

we

that the functional equation f(az) =

(p(~)~~(p(f3z)~~. . .(p(pkz)mkf(z) has no continuous solution f : K+K,

(mo,ml,.. . , \ I

Kk+ 1

(We will show below how to obtain such a). Still call this

Chapter 5

74

subsequence N.

Let

rn

and define H(z) =

n anz . n=-cu

C

Now

C lan I

converges uniformly on K, and H(z)

- -H(z) = zanzn = cp(z1

=

Ca-,z-”

= H(z),

so

S

cC-12 <

rn,

so

the series

n

is continuous. Moreover,

H(z)

is real valued. We define

exp(2niH(z)). If f

Now, suppose the functional equation has a solution f. degree

e,

then f(z) =

ze

exp[2zig(z)l,

e

= aexp[2ni(g(az)-g(z))l,

uous. f(az)/f(z)

where g : K-+W

is of

is contin-

2nie and (writing a = e 1

there is a solution if and only if there is an integer u such that k eQ+g(az)-g(z) = moH(z)+mlH(f3z)+. . .+mkH(i3 z ) + u . Let g ( z ) have the

-

n C(an-l)bnzn, Fourier series g(z) ,..Cbnz , SO g(az)-g(z) jn n m.H(PJz) = E m anS z , and comparing coefficients for n # 0 we obtain J n J n n ( a -l)bn = an(mo+mlf3 +. . .+mkgnk) = anp(pn) (where p(A) =

k

C miAi 1.

i =o

n Now, if n E N, an = a -1 Hence C(bn12 =

#

* 0.

0, and we have bn = p(pn)-+p(w)

m.

This is a contradiction, since the sum of the squares of the Fourier

coefficients of a continuous function always converges.

To construct a satisfying the required properties, proceed as follows. It is sufficient to find an irrational number a such that

la

-

mml3’ m(i) r(i) I

and some integer r(i).

for some subsequence { m ( 1 ) )

in [ O , l l of

{n( i I ) ,

(The equivalence of this with the.origina1

Distal Flows

problem is an elementary exercise).

Put

75

m(1) = n(l),

and let

Il(j)

be the closed interval

m(1)-1

u

F1 =

Let

j=l

and let

Choose m(2) = n(i21

Il(j),

12(k) =

[m k - - - +1

k

such that

2

1

mo < -m(1) 3’

, 1 5 k 5 m(2)-1,

and

m(2) m(2)-1

F

u

=

IZ(k).

Note that each

k=l of the form

12(k).

I 1 (j) contains at least two intervals

Proceeding in this way, choose m(i) 2

mo < -

original sequence) so that

(from the

1

m( i-1)

2’

m

contains at least two

Ii-1

the same cardinality as

(r). Then

n Fi

is a subset of

[O,.ll with

i=l

[0,1], and so this intersection contains an

irrational number a. Obviously a has the required properties. The cascades just constructed are in general not equicontinuous. This

is a consequence of a more general theorem on coalescence of equicontinuous flows (see the exercises to this chapter). For now, we note that the cascade on K‘,

T(z,w) = (az,zw) ( a is not a root of unity) is

distal, minimal, and not equicontinuous. Distality and minimality follow as before. k

f

k (The functional equation in this case is f(az) = z f(z),

0; its nonsolution is proved by a simple Fourier series argument, or

by an elementary argument involving degree).

To see that the cascade

(K2,T) is not equicontinuous. note that, f o r n > 0, n(n-1)

Tn (z,w) = ( anz , a

Tn ( z , w ) n

znw), n(n-1)

= ( anzn,-a

w).

and let Now

z = exp($). n

(zn,1)+(l,w),

Then but

Chapter 5

76

ITn(1,w)-Tn(zn,w) I 2 2, so the cascade is certainly not equicontinuous.

Now we turn to a discussion of (real) flows on three dimensional nilmanifolds. These are homogeneous spaces of nilpotent Lie groups. We present here an ab initio treatment, which does not depend on the theory of Lie groups.

We begin by discussing coset flows in general.

G be a topological group, and let H be a closed subgroup of

Let

G. Consider the right coset space G/H = [Hglg E GI,

with the

uniformity defined by the collection 21 = [orvIV E 31,

(where 3

neighborhood system of the identity e a

V

= [(Ha,Hb)la,b E

G, b-’a E V).

and

Note that a neighborhood base of

Hg

is given by the collection [HVglV E 91.

in G/H

We will usually suppose that

K

where

in GI,

is the

is compact).

H is a syndetic subgroup of C

In this case, the coset space G/H

Note that G acts transitively on C/H

by

(G =

HK,

is compact.

(Hg,g‘)H(Hglg’ =

H(gg’) (g,g‘) E G I . The next theorem gives criteria for the action of C on G/H

to be

distal. Theorem. Let

11.

H be a syndetic subgroup of the topological group

G. Then the following are equivalent. (a) (G/H,G) is distal. (b) H =

n HUH

(where, as above, 3 denotes the neighborhood system

UEY

of e

in GI.

(c) If g 4 H, e 4

Proof. (i)

m.

The proof is carried out by noting that each of the statements

- (xii) below

is equivalent to the one following.

(i)

(G/H,C) is distal.

(ill

Whenever Hgl

f

HgZ, and g3 E G, there is a U E 3 ’

such that

Distal Flows

77

G not both Hglg E Hug3 and Hg g E Hug3. 2 -1 (iii) If g g g E G such that glgz d H, there is a U € 9 such 1' 2' 3

if g

E

that if g

(iv)

G not both glg E Hug3 and g g E Hug3. 2 -1 If g1,g2 E G with g1g2 e H, there is a U E

-1 -1 gl HU n g2 HU =

g g-'HU 12

A

9 with

E

9 such that

+.

-1 d H, there is a U If g1,g2 E G with g1g2

(v)

E

HU = 4.

(vi)

If g d H, there is a U

E

9

(vii)

If g d H, there is a U

E

9 such that g d HUH.

(viii) 'H c

such that

gHU

A

HU = @.

u (HUH)'. UE9

n HUH. UEB

(ix)

H

(XI

H =

(xi)

If g d H, there is a U

E

9 with g d HUH.

(xii) If g d H, there is a U

E

3 such that

3

n HUH U€9

(xiii) If g

[1

e H, e

:]

(this is (b)).

t

U n HgH =

m.

Now, let G denote the group of upper triangular 3x3 G =

0

lal,a2,a3E R]

1: 0

+.

(G

matrices,

is the "Heisenberg group"),

and let

H be the closed subgroup of G whose entries are integers. Then G and H are homeomorphic with lR3

1

syndetic in G.

of those

0

1:

and P3

respectively and H

is

(It is easily checked that G = HK, when K consists with 0 Sk. 5 1.) We will sometimes write 1

0

(al,a2,a3) for an element of C. (al+bl.a2+b2,a3+b3+alb2) ( s o

e = (O,O,O), and

Note that

(a a a ) (bl,b2,b3) = 1' 2' 3

G and H are not abelian, the identity

(al,a2,a3I-l= (-al,-aZ,-a 3+a1a21.

We apply 11 to show that

(C/H,G) is distal. by showing

Chapter 5

78

e = (O,O,O) d

HgH

f o r all

g Q H.

a = (a1,a2,a31, b = (bl,b2,b3),

Let

g = (g1,g2,g3) with a,b E H. Then agb = (al+gl+bl,aZ+g2+b2,a3+b3+g3+alg2+alb2+g1b2).Now

i = 1,2,3. If gl Q Z,

f o r at least one

to

0

(since al+gl E Z ) ,

then g3

ct

1 2 3

E G,

12. Lemma. G = H(h(Ra)),

H,

is never

we write Ra = [al,al+l)x[a ,a +l)x[a 2

for every a

2

a +l). 3' 3

E G.

g = (gl,g2,g3)E C.

hl = al-g1+a2, h2

If h

with k E Z

E

1 be defined by h(g) = (g,,g2,g3+2g1g2).

If a = (a ,a ,a )

B.

= k+g3

If g1,g2

so e P HgH.

Let A : G 4 C

Proof. Let

b +g

+g

gi d Z

then al+gl+bl is never close

and similarly if g2 Q Z.

Z, and a +b +a g +a b

close to 0 ,

g P H, so

-

Then there are 6 ,6 , 6 E [0,1) with 1 2 3 1 a2-g2+62, h3 = a3+63+-2 (a 1+61)(a 2+62 1-83-h1 g2 all in

= (hl,h2,h3), then

hg = (a +6 ,a +6 a +6 a ( ' + +6 )(a2+62)) E A(Ra). 1 1 2 2'3 3 2 1 1 Now let cp

:

GxlR-+G

be defined by

1 1 2 (alt,a2t,(a --a a It++ a t 1. 3212 2 1 2

cpa(t)pa(s)

= pa(t+s),

so,

A routine computation yields

f o r each a E G, pa

Note also that cp ( 1 ) = (a a a 1, a 1' 2' 3 subgroup of C

so

:

is a homomorphism.

R+C

9, defines the one parameter

"through" a.

13. Lemma. Let G = H'p(UxlR+)

cp(a,t) = 'pa(t) =

U be a non-empty open subset of G.

(where R+ =

Proof. Note that

Then

[O,m)).

h-l(gl.g2.g3) = (gl,g2,g3-plg2), 1 so

h-1cp(x,t). = (xlt,x2t,( X ~1- ~ X ~ X ~ )and ~ ) ,h-ldUxR+) =

r(ylt,y2t,y3t)iy = (y1,y2,y3) E A non-empty open subset of C. that there is an a E C with Then G = H(X(Ra))

c Hv(UxR+).

-1

(u), t. 2 0 1 .

NOW

A-'(u)

is a

It follows from the above representation

Ra

c A-'p(UxR+),

SO

h(Ra)

C

'p(UxlR+).

Distal Flows

If a

E

79

G, we define an action of R on C/H by

Since Ipa(t)lt

E

R]

is a subgroup of C,

(Hglt = Hgcp (t). a

the flow defined by

is

cp

a

distal. 14.

Theorem. For all a

E

C outside of a first category set, the flow

defined by the one parameter subgroup cp a

is distal, minimal, and not

equicont inuous. Proof. We have just observed that these flows are distal. prove that cpa defines a minimal orbit. To this end, let UI,Uz,..

Therefore, to

low it is sufficient to find a dense be a countable base for G, and let

E~ = [a E GI~~,(IR+I n H U = ~ 41. we show that En is closed and has no interior.

If a

E

,:E

open in G and cp

pa(t)

E

HU,

for some t

R+.

E

Since Hun

is

is continuous it follows that EE is open, and En

is closed. Now, suppose En contains a non-empty open set U. lemma, G = Hcp(UxR+).

so

Un c Hcp(UxR+)

and Hun n cp(UxlR+)

Equivalently, there is a d E U c En such that contradicts d

E

By the previous

4.

f

qd(R+I n Hun

f

4.

This

En. m

Thus En

u

is closed with no interior and so

En n=1

^Hun

category set. If a E (uEnIc = nEz, cpa(!R+l

and it follows that the 9, orbit of the coset H This proves minimality of the flow defined by

*

4,

is a first for every n,

is dense in C/H.

(pa.

Finally, we show that these minimal flows cannot be equicontinuous. In fact, the space G/H

cannot support

equicontinuous minimal flow

with an acting abelian group. For, if the flow were minimal equicontinuous, the coset space G/H group.

would have the structure of a compact abelian

However, the fundamental group z1(C/H)

non-abelian group H

(since H

is isomorphic with the

is discrete, the natural map of G

to

Chapter 5

80

G/H

is a covering map, with fiber homeomorphic to

H, and so

Since a topological

regarded as the group of covering transformations). group has an abelian fundamental group, G/H

H may be

cannot support a group

structure, and the proof is complete. These examples were constructed by L. Auslander, F. Hahn, and

L. Markus, and were the first examples of distal minimal flows which are not equicontinuous. Their work in this general area is cdntained in the monograph, Flows on homoveneous sDaces, (Annals of Math. Studies, No. 53, Princeton University Press, 1963). An interesting historical sidelight is that at the time they were not aware of the notion of distal, but were

trying to construct minimal actions of the real line on nilmanifolds. They were able to produce dense orbits, but could not prove minimality. W.H. Gottschalk suggested that these examples might be distal (and therefore necessarily minimal). The examples of skew products on the torus, which are actually simpler than the flow on nilmanifolds were shown to be distal, minimal, and nonequicontinuous later (by Furstenberg), although they had been considered many years earlier in the ergodic theory literature. The particular skew products on

Kk+1

discussed in this chapter were

first considered by Parry and Walters (Compositio Math., 22 (1970, 283-288) who used them t o obtain non-coalescent distal minimal flows

(exercise 4 ) . Exercises. 1.

The flow (X.T) is distal if and only if the flow (E(X),T)

is

minimal.

2. Let

(X,T) be a distal flow, let u E 'I$, and let F be a finite

subset of X. Then there is a syndetic subset A

of T such that

Distal Flows

xA c xu, for x 3. Let

81

F

E

be a syndetic subgroup of T.

(X,T) be a flow, and let S

Then x,y E X are proximal with respect to the action of T if they are proximal with respect to S .

Thus

(X,T) is distal if and

only if

(X,S) is distal.

4.

(X,T) be a distal minimal flow. If x

F

X

Let

= [p E E(X)Ixp =

with cp(x) = y

XI.

E

X, let

Then there is an endomorphism

if and only if Fx c F

Y’

and

if and only

(p

(p

of

(X,T)

is an automorphism if

and only if Fx = F

Y‘

5. A minimal flow is said to be coalescent if every endomorphism is an automorphism. a) Equicontinuous minimal flows are always coalescent.

(Define a

quasi-ordering - a transitive, but not necessarily anti-symmetric relation) on X by x 5 y elements of E(X) xo.

If

(p

if Fx CF

Y‘

Use the continuity of

to prove the existence of a maximal element

is an endomorphism and y0 = cp(xo),

then Fx = 0

F , YO

so

cp

is an automorphism.

b) Distal minimal flows are not necessarily coalescent. Consider the homeomorphism S

of the infinite torus defined by

where a l p , and

(p

in this chapter.

(Then the cascade

are as in the skew product example presented (Km,S) is the inverse limit

of the skew products on finite dimensional tori, so distal and minimal. 1 Let R

:

Km+Km

R(z,w1,w2,.. . ) = (@z,w2,w3,.. . 1.

(Km,S) is

be defined by

Then R

is an endomorphism of

(Km,S) which is obviously not an automorphism.

It follows also

Chapter 5

82

that

(Km,S) is not equicontinuous, and therefore (for all

sufficiently large n) the distal minimal cascades on Kn

are

not equicontinuous. 6. Show that the cascade

a root of unity)

(K2 , S )

is minimal.

defined by

S(z,w) = (az,zw) ( a not

Determine the regionally proximal relation

in this example. 7. Let

(X,T) be a distal minimal flow which is not equicontinuous.

Show that the enveloping semigroup E(X) theorem 13 of chapter 2 . )

is not metrizable.

(Hint: use

83

Chapter 6 The Enveloping Semigroup, I1 In order to investigate further the properties of the enveloping semigroup, it is convenient to abstract the situation. For this purpose, we consider for the first part of this chapter an abstract semigroup E (with no topology).

A right ideal in E

is a non-empty subset

IE c I.

I such that

A

minimal rinht ideal is one which does not properly contain a right ideal. Note that a right ideal is also a semigroup. Moreover, if

I is a

K is a right ideal in the semigroup I,

minimal right ideal in E, and

K = I.

then it is easy to see that

An idempotent in a semigroup E

is an element

u

E with. u2 = u.

E

The next three lemmas are purely algebraic in character.

1.

I be a semigroup without proper right ideals. Let

Lemma. Let

J(1) = J

be the set of idempotents in

I, for all p

I.

Suppose J

*

#.

(i)

PI

(ii)

up = p, for u E J, p E I.

(iii)

If u

(iv)

If u E J

then Iu is a group with identity u.

(v)

If p E I

then there is a unique u E J

(vi)

Let

=

E

u,v E J

pr = v

(vii)

J and p

E

E

Then

I.

I with pu = u, then p

and let

p

E

E

with

J.

pu = p.

Iu. Then there is an r

E

I with

and rp = u.

I = U IU. UEJ

(viii) If u,v E J, with u Proof.

(ii)

(i)

(pI)I = PI

Let u

E

J.

2

c

PI

f

v, then Iu n Iv

and PI c I

Then by (i),

UI = I,

=

#.

so by assumption,

so

PI = I.

p = uq, for some

Ctiapter 6

84

q

E

I. Then up (iii)

=

uuq = uq = p.

p2 = pp = p(up) = ( p u ~ p= up = p.

(iv) If p,q

E

Iu, then p = ru, f o r some r

= p; similarly q = qu and

then pu = up = p, so

u acts as an identity for

with pq = u, by (1).

inverse of p

Iu.

(v) Let p

Iu.

Then pu = pqp = vp = p.

E

ru2 = ru

=

If p

Then pqu = u

Let q E I

J.

Also

with pq = v.

E

J with pw = p.

qp = u. Then pu = pw, qpu = qpw. But

Iu,

E

Let

q

u

=

qp.

u E J.

so E

is the

= q.)

Put

u2 = q(pq)p = qvp = qp = u

Suppose there were another w

Iu,

E

and qu

(In fact, it is easy to see that qu

I and v

E

pu

so

pq = puqu = pqu E Iu. Also, if p

there is a q E I in

I

E

Iu with

qp = u, so we have u = uu =

uw = w. (vi) Let q

be the inverse of p

r = qv. Then pr = pqv = uv = v

in the group

Iu, pq = u

and rprp = rvp = rp so

and let

rp E J.

But

rpu = rp so by (v) rp = u. Properties (vii) and (viii) follow immediately from ( v ) . Note that the lemma applies to a minimal right ideal

For, suppose K c I

semigroup E.

=

is a right ideal in I,

so

KI c K.

(KI)E c KI and KI is a right ideal in E with KI c I1 c I,

Then

KI

I of a

so

I. Since KI c K c I, we have K = I. I be a minimal right ideal of the semigroup E,

2. Lemma. Let

suppose J(1)

f

$.

Suppose p E E , g,r E I

satisfy pq = pr.

and Then

q = r. Proof. Let

s = qp E

such that sv = s

is a group).

IE c I. Then sq = qpq

and let

a E I such that

= qpr =

sr. Let v

as = v

(recall that

Then q = vq = asq = asr = vr = r.

Now we define an equivalence relation in J(E).

the set of

E

J(1)

Iv

The Enveloping Semigroup, II

if uv = u and

idempotents of E by u-v

85

vu = v.

(If u-v and v-w,

then uv = u, vw = v, and uw = (uvlw = u(vw) = uv = u and similarly w u = w,

3.

u-w. Reflexivity and symmetry are obvious.)

so

I1 and I2

Lemma. Let

and let

u1

J(I1).

E

be minimal right ideals of the semigroup E

Then there is a unique u 2

E

J(12)

and ul12 c IIE c 11,

u I 1 2

such that

u1-'2'

Proof. u I E c u I 1 2

12

so

is a subideal of

I 1 and hence u I = I1.

Thus there is a u2 E I2

Note that u2

is unique.

(If u1u2 = u1p2, p2

p2 = u2. 1 So

u1u2 = ul, ulu; = (u u )u = u1u2, and again by lemma 2, 1 2 2

2

u2 = u2 and u2 E J(12). u2v1 = u2. lemma 1 )

12,

E

then by lemma 2,

Similarly, there is a v1 E J(I1)

u1 = u1u2 = ulu2v1 = ulvl. But

So

with u u 1 2 = ul.

u v

= v1

with ((ii)

of

v1 = u1 and the proof is completed.

so

We wish to apply these lemmas to the enveloping semigroup of a (Now, as before, we write E for E(X) = E(X,T)

transformation group.

the enveloping semigroup of

(X.T).)

Our first task is to show that E

contains minimal right ideals. This is accomplished by the following theorem which also provides a dynamical characterization of the minimal right ideals.

4. Theorem. A subset I of E is a minimal right ideal if and only if

I is a minimal subset of the transformation group (E,T). (Hence minimal right ideals exist, and they are closed subsets of E . )

Proof. Let closed and

I be a minimal subset of the flow (E,T). Then I is Then IE = IT c

IT c 1.

ideal. Let Q

*

K

c

f

= I,

and

I with K a right ideal, let p

-

Then q E I = pT and there is a net (a subnet of) tn+<

C

E

{tn)

E. Then q = p<

E

I E

is a right

K and q

in T with ptn+q. KE c K,

so

E

I.

Let

I c K and hence

Chapter 6

a6

I = K. Thus every minimal subset of the flow (E,T) is a minimal right ideal of the semigroup E. Conversely, suppose

I

ideal of E. Then, if p E I , pE c IE c I, and Since left multiplication in E closed. If p

E

and

so

pE = I.

(pE)E c pE, so

is continuous, it follows that

then I = pE =

I,

is a minimal right

I

I

is

is a minimal subset of

the flow (E,T).

5. Corollary. If E(X),

F is a non-empty closed T invariant subset of

(equivalently, if F

is a closed ideal in E(X))

then F

contains a minimal right deal. Next, we show that all minimal right ideals in E(X)

contain

idempotents. This is a consequence of a more general result. 6. Lemma. Let

F be a compact TI space which has a semigroup struc-

ture for which left multiplication is continuous and closed (that is, if S

is closed, then XS

i dernpotent

is closed, for x E F ) .

Then F contains an

.

Proof. Let C denote the class of non-empty closed subsets S of F 2

such that S c S . If

An application of Zorn's lemma shows that there is a

minimal

S

SS E 1.

Since SS c Sz c S. SS = S by minimality of S. Let p

E C.

such that s p = s ,

s E

S, then (sS)(sS)c SS and sS is closed, so E

and let L = [a E Slsa = sl. Then p E L, and

is a non-empty closed subset of S. If a,b E L, sab = sb = s , 2 L c L. Thus L = S and, in particular s E L so

s2 = s .

S L

so

Then s is

an idempotent (of course, it follows that the minimal element S of C is a singleton set).

7. Corollary. Let F be a compact Hausdorff space with a semigroup structure such that left multiplication is continuous. Then F contains an idempotent.

The Enveloping Semigroup, II

8. Corollary.

If I

87

is a minimal right ideal in E(X,T),

then

I

contains an idempotent. Of course, in this chapter, we are only concerned with the case that

F is Hausdorff. However, the full force of lemma 6 will be used in a later chapter. In any case, our lemmas apply to the enveloping semigroup. E(X) contains minimal.right ideals, and every minimal right ideal contains idempotents ( "minimal idempotents"1. Now we pursue further the relationship of the enveloping semigroup with proximality, which we touched on the first enveloping semigroup chapter. 9. Theorem.

Let

(X,T) be a flow, and let x,y E X.

Then the

following are equivalent. (i)

x and y are proximal.

(ii)

xp = yp, for some p E E(X).

(iii) There is a minimal right ideal

I in E(X) such that xp = yp.

for all p E I. Proof. The equivalence of ( i ) and (ii) has been proved earlier. Obviously ( i i i ) implies (ii). By hypothesis F

*

$,

Suppose ( i i ) holds, and let F = [p E Etxp = y p l . and it is clear that F is a closed ideal. By

corollary 5, F contains a minimal right ideal. This completes the proof. Now suppose E contains exactly one minimal right ideal (x,y)

E

P and

Suppose

(y,z) E P. By the theorem just proved, xp = yp, and

also yp = zp, for p E I , P

I.

so

is an equivalence relation.

In fact, we have

xp = zp (p E I . )

Hence, in this case,

Chapter 6

88

10. Theorem. The proximal relation is an equivalence relation if and

only if the enveloping semigroup has a unique minimal right ideal Proof.

Sufficiency has been proved in the preceding discussion. Suppose

P is an equivalence relation and let I1 and I2 be minimal right Let u1 E J(I1), u2 E J(Iz)

ideals in E.

x

X. Then (x,xul)E P, (x,xu2) E P s o

E

with u1

(xu1,xu2) E P.

(xu1,xu2)u2 = (xu u xu ) = (xu ,xu2), so 1 2’ 2 1

have

.+u2,

(XxX,T). Hence xu = xu2’ Since x

arbitrary ul = u2

E

n

I2

Corollary. If P

Proof.

u

1

E

and

Suppose I1

J(I1), u2

This completes the proof.

I2

are minimal right ideals in E(XI,

with u1

u2

*

and let

x E X.

(xu2,xu1) = (xu2,xu1u21 = lim (x,xu1) s j’ where

T such that s.-+u2. J

X is

is closed, it is an equivalence relation.

and

J(12)

E

E

I 1 = I2 (recall that two minimal

and

subsets o f a flow are disjoint o r equal). 11.

Now also we

(xu1.xu2) is an almost

periodic point of

I1

and let

Since P

Then {sj)

let

(x.xul) E P

is a net in

is closed (and invariant) we have

But as we have noted in the proof o f theorem 10,

(xu2,xu1) E P.

(xu ,xul) is an almost periodic point, so 2 arbitrary u2 = ul. Hence

I 1 = I2

and

xu2 = xul, and since x E(X)

is

contains a unique minimal

right ideal. The conclusion now follows from theorem 10.

12. Theorem. Let

(X,T) be a flow and let

I be a minimal right ideal

in E(X). (i)

If x

E

X, then XI

( i i ) Let x E X. (a) x

(b) x

is a minimal subset of X.

The the following are equivalent:

is an almost periodic point. E

XI.

(c) xu = x.

for some u

E

J(1).

The Enveloping Semigroup, II

Proof.

is a homomorphic image of t h e minimal f l o w

( i ) XI

t h e map

(1,T)

via

p+xp.

(ii)

(b)

XI =

and

x

*

( a ) and ( c )

( a ) f o l l o w from ( i ) .

.$

If

x

is almost

3 is minimal, and X I is a minimal s u b s e t of

periodic,

If

89

F = [p

let

E XI,

s u b s e t of

Hence ( a )

x E XI.

1 and

FL c F.

+

XI.

F

Then

is a c l o s e d non-empty

c o n t a i n s a n idempotent, and (b) +

F

Hence

so

(b).

Ilxp =

E

xE =

(C).

Let

Theorem.

13.

(i)

If

x E X

(ii)

If

x E X,

(X,T) and

be a flow. v

E(X),

is a n idempotent i n

then

t h e r e is a n almost p e r i o d i c p o i n t

x’

( x , x v ) E P. such t h a t

( x , x ‘ ) E P. (iii) If

(X,T)

(x,y) E P

is minimal t h e n

minimal r i g h t i d e a l

I

E(X)

in

if and o n l y i f t h e r e is a

and a

v E J(1)

such t h a t

y = xv. Proof.

(i)

( i i ) Let

( x v ~ v= xv2 = xv,

( i i i ) If

ideal i n

E.

( x , x v ) E P.

be a minimal r i g h t i d e a l i n

I

x’ = xv E X I ,

Then

so

so

( x , y ) E P,

Let

v

E

xv

is a n almost p e r i o d i c p o i n t and

then

J(1)

v E J(1).

and l e t

E,

xp = yp,

for all

yv = y.

such t h a t

p E I Then

( x , x ’ ) E P.

a minimal r i g h t

y = yv = xv.

Recall (from t h e c h a p t e r on d i s t a l f l o w s ) t h a t a d i s t a l p o i n t i n a flow is a point

x

such t h a t i f

y

f

x,

then

x

and

are not

y

proximal. 14.

Corollary.

point. 15.

with

Then

Theorem.

x

Let

(X,T)

be a flow, and l e t

x E X

be a d i s t a l

is an almost p e r i o d i c p o i n t .

Let

(X,T)

and

x a d i s t a l p o i n t and y

(Y,T)

be flows and l e t

an almost p e r i o d i c p o i n t .

x

E

X, y E Y

Then

(x,y)

Chapter 6

90

is an almost periodic point of the product flow

Proof. Let v

n : XxY-+Y

be a minimal idempotent in E(Y)

be the projection map and let

of lemma 6, so there is an idempotent = (x,y)w. Then

z'

n((x,y)w) = R(x',Y')

= z and

y'.

=

Since x

yv = y. Let be the

satisfies the conditions

w in

8

-1

(v).

is an almost periodic point and

(theorem 12 ( i i ) , and theorem 13 (i)).

z'

such that

E(XxY)+E(Y)

8 :

induced semigroup homomorphism. Then O-l(v)

(x,x') E P(X).

(XxY,T).

Let

z' = (x',y')

(z,z') E

P(XxY)

Now y = yv = n(x,y)8(w) = ( z , z ' ) E P(XxY1,

Moreover, since

is a distal point, x' = x and therefore

z = (x,y) is an almost periodic point.

(X,T) be a distal flow, and (Y,T) be pointwise

16. Corollary. Let

(XxY,T) is pointwise almost periodic.

almost periodic. Then

We conclude this chapter with an application of the result that a point in a flow is proximal with an almost per odic point (theorem 13, (ii))

to combinatorial number theory. We will present Furstenberg's

proof of a theorem of N. Hindman on partitions of the set of positive integers. First we require one more general lemma. 17. Lemma. Let

(X,T) be a flow, let x

E

X and let y be an almost

periodic point, with x and y proximal. Let y. Then there is a Proof. Let

E(X)

t

E

U

be a neighborhood of

T such that xt and yt are in U.

I be a minimal right ideal in the enveloping semigroup

such that xp = yp for p E I

such that yu = y.

so

u be an idempotent in I

and let

xu = yu = y. Let

{t

J

}

be a net in T with

t -+us J

Then for J 2 j xt and yt are in U. 0' J J Now for our combinatorial application. A set D of positive integers

is called an IP set if there is a sequence p p p

1' 2' 3'"'

integers such that

of positive

D consists of all finite sums of the form

The Enveloping Semigroup, 11

pi +pi +. . .+pi 1 2 k

i l < i2 <

with

...

< ik.

distinct but the subscripts il,...,ik

91

(The p

need not be

J

are distinct. Thus D

additive semigroup if and only if each p

is an

is repeated infinitely often.)

i

IP sets is exhibited

The relation between topological dynamics and by the following definition and theorem. Definition. A set

S of positive integers is called a central set if

there is a cascade

(X,T), a point

x

E

X, an almost periodic point y

proximal to x, and a neighborhood U of

s

= [n

y such that

> OIT"(~) E UI.

IP set.

18. Theorem. Every central set contains an

(X,T) be a cascade, x

Proof. Let such that

E

X, y an almost periodic point

x and y are proximal. and let

is a neighborhood of y,

S

so

S =

Tpl(x), Tpl(y) E U1

(Note that

y

E

Tpil+.

+pin(x)

. . +Pin-2 ( T p i n-

=

so

Theorem. Let

H, H =

B1 u

...

that

U1 = U

and let pI

Tp2(~),Tp2(y)E Uz

Inductively, we have

and TPk(~),TPk(y)

p. +. * . T 11 +pin-l(Tpin(x))

E

Uk.

Then

Pil+.. . +Pi T

Pil+.. . +Pin-2 (Ui 1 ) c T (ui

C

...

1 =

(U,

I

n

c Tpil(Ui ) c

n-1

2

B1,B2,.. . ,Bq be a finite partition of

u B Then one of the B 4' J

E S.

U2 = U n T-P1(U1). 1

(lemma 17) and put

n

19.

U

and a sequence {pi}

then pi +. . .+pi 1 k

< ik,

U = U n T-p2(U2). 3 2

and TPk (Uk+l) c Uk

P11+Pi2+. ..

T

...

Uz. 1 Next, choose p2

(lemma 17 again) and let Uk+l c Uk

< i2 <

y

determines the required IP set.) Let

(Thus {p,} such that

i

U], where

E

is a central set. We will define induc-

tively a sequence of open neighborhoods Un of of integers such that if

[nlTn(x)

contains an

IP set

Chapter 6

92

It is only necessary to show that some B

Proof.

set. For this purpose consider the shift on q

. . ,q)z ,

R = {l,Z,,

and m(n) = w(n+l)

of fl such that for n > 0, <(n) = i r)

E

R

symbols

for w E R.

E

Rlw(0) = j l

S = [nlun(<)

E

U]

<

Let

be a point

if and only if n E B i ’

be an almost periodic point which is proximal to

then U = [ w

is a neighborhood of

ern(<)

is a central set. Now

un(<)(0) = j if and only if <(n) = j

Sc B

contains a central

J

If q(0) = j,

and

r),

E

<.

and let

U

if and only i f

n E B

if and only i f

so

jr

and the proof is completed.

J

Exercises. Let

1.

(X,T) be a flow, x,y E X.

u [I I I

xq = yq, for all q E

2. Let

Then (x,y)T c P

a minimal right ideal I .

(X,T), (Y,T) be minimal flows.

phism (i.e., if n(x) = n(x’)

then

A :

X+Y

(X,T) be a flow, let x

and a point Let

rxn(P

X

x’

E

N such that x and x‘

P(X,)

with n(x,)

(X,T) be a flow.

such that if F E

I c E(X)

XI = N,

such that

are proximal.

Py. In fact, if (y1.y2) E Py and x1

there is an x2

t

(x,x’) E Px.

A : X j Y be a homomorphism of minimal flows. Then

) =

5. Let

E

(y,y’) E Py

Suppose

X and let N be a minimal subset of

Then there is a minimal right ideal

xT.

4.

E

a proximal homomor-

(x,x’) E Px).

and x,x’ E X with n(x) = y, n(x’) = y‘. Then 3 . Let -

if and only i f

T such that

E

X with n(x,) = yl,

= y2.

A proximal

is a finite subset of A, (xt,yt) E a for all x.y

in X

is a subset

and a E 11 E

F. Show that

for some (equivalently, all) x

of X

then there is a

proximal set if and only if there is a minimal right ideal such that A c xJ(1)

A

E A.

A

I

is a

in E(X)

The Enveloping Semigroup, II

6. Let

(X,T) and

periodic.

Let

(Y,T) be flows, with

n : X+Y

93

(Y,T) pointwise almost

be a distal homomorphism.

Then

(X,T) is

point wi se almost periodic.

7. Let

I

u,v E J(1).

be a semigroup satisfying the conditions of lemma 1 and let Then the groups

Iu and

Iv are isomorphic.

This Page Intentionally Left Blank

95'

Chapter 7 The Furstenberg Structure Theorem for Distal Minimal Flows We begin this chapter by considering the relativization of the concept of an equicontinuous flow. Let

II :

X+Y

is called equicontinuous if, for every a that if t

E

T.

(xl,x2)E (For short:

relation defined by

be a homomorphism. Then E

'l$

there is a p

E

'l$

'T[

such

and R(x,) = R(x,),

then (x t,x2t) E a, for all 1 "pT n R(n) c a" where, as usual, R ( R ) is the R,

R(n) = [(x 1,x2)IR(x,)

= dx2)l.)

Obviously, a

flow is equicontinuous if and only if it is an equicontinuous extension of the trivial flow. (An equicontinuous extension is also called an almost periodic extension.)

It is easily checked that an equicontinuous extension is a distal extension, so if Y

is distal and

is distal. However, it is

R :

X-+Y

is equicontinuous, then X

the case that an equicontinuous extension

of an equicontinuous flow is equicontinuous. The skew product flow on the torus

(z,wlH ( a z , zw) provides an example.

the projection

(Z,W)HZ

It is immediate that

is an equicontinuous extension of the

equicontinuous cascade z w a z on the circle, but, as we saw in the preceding chapter, the cascade on the torus is not equicontinuous. Indeed, this phenomenon (that an equicontinuous extension of a distal minimal flow is distal, but not in general equicontinuous) is the point of departure for the analysis of the structure of distal minimal flows. The theorem of Furstenberg, which will be stated below, says essentially that any distal minimal flow is realize3 as a sequence (in general transfinite) of equicontinuous extensions. In order to state the theorem precisely, we need one more definition. A projective system of minimal flows is

a collection of minimal flows

Chapter 7

96

(Xx ,T) indexed by ordinal numbers h family of homomorphisms, (i)

If v < h <

( i i ) If p I e

Q

(p,

5 8,

(for some ordinal e l ,

for v < h I

Xh+Xv,

:

then

8

and a

satisfying

A1)

= (p,ph.

p :

is a limit ordinal, then X

the Cartesian product flow

(xx 1

A

S 8

(

is the minimal subset of Ir lT Xh,T) consisting of all h

ll Xh with xu = (pv(xh) ( v < A < p ) and (pr : X +Xh is c1 h

x =

E

pro.iective limit) of the flows Xh ( A < p). Another way of looking at this is to consider a minimal flow X = Xe, and a family of closed invariant equivalence relations Rh

X, = X ( A I e l . R

Re

which satisfy (i)

Rh, and ( i i i ) if h

=

in

Ax, ( i i ) if v < h

is a limit ordinal, then

RA =

S 8,

n Rp.

If

h
XA = X/RA and the homomorphisms

(ph

v

are the canonical maps, obviously

this collection of flows and homomorphisms constitutes a projective system. Now we are ready to state the Furstenberg structure theorem.

1. Theorem. Let K :

X+Y

(X,T) and (Y,T) be distal minimal flows and let

be a homomorphism. Then there is a projective system of

minimal flows (Xh,T) ( A

X < e, if

A+ 1 (px

:

Xx+l+Xh

5 0)

with Xe = X, Xo = Y such that if

is an equicontinuous extension.

In particular,

(X,T) is distal minimal. there is a projective system as above, with

Xo = 1, the trivial flow. The proof we present here, due to Bronstein, is a simplification of Furstenberg’s proof, (Amer. J. Math., 8s (19631, 477-5151, and is under the assumption that the phase space is metrizable.

(The metrizability

assumption is only used in the last stage of the proof. )

In the course

of the proof, a number of results are obtained (some of which are not

The Furstenberg Structure Theorem

97

restricted to distal flows) which are of independent interest.

(Other

proofs of the structure theorem have been given by Ellis (Lectures ToDolonical Dynamics, Benjamin, 1969) and Namioka, (Math. Systems Theorq

S

( 19721,

193-2091.1

An essential concept in the proof is the relativized regionally proximal relation. If

X+Y

I :

n

proximal relation Q(n) =

is a homomorphism, the n-regionally

DT n R(n).

Note that

(x.x') E,Q(I)

if

VEZIX

and only if for every a (x,xo) E a, (x',xb) nets {xn), {x;} (xn,x;1)

E

:

X+X/R

Q(R) =

a and

E

{tn)

in T

E

X

=

'7E?-fX

R(n),

E

with x +x, n

x'+x', n

(xntn,x;, tn)+AX.

the quotient map, we write

n

(x0 , X I0)

with

if and only if there are

(xot,x;)t) E a

in X, and and

there are xo,xA

is a closed invariant equivalence relation in

If R n

R(n)

E 'I+

n

Q(R)

X, and

instead of Q(n).

Thus

m.

QE%

The relation between equicontinuous extensions and relative regional proximality is provided by the following proposition.

2. Proposition. The extension

A :

X+=Y

is equicontinuous if and only

if Q(n) = A. Proof. Suppose n and choose f3 E Since a If

I

BT n R(n)

E

5

is equicontinuous and

(x,x')

with f3T n Rfa) c a. Then

'I$ is arbitrary we have

E

Q(n).

(x,x') E BT

@ E

A

a R(I)

%

E

c

G.

x = x'.

is not equicontinuous, then there is an u E a for all

Let

such that

?$. That is, for every 6 there is an

,XI) E f3 n R(I) and t E T such that (x t ,x't 4 a. Regarding B B B B B B B {x t 1 and {x't } as nets, then (subnets of) x t +x, x't +x' and B B 613 B P B B it is immediate that (x,x') E Q(I) with x # x'.

(x

Chapter 7

98

Let

Theorem.

3.

X

be a homomorphism.

and

Y

Then

be d i s t a l minimal flows. and let

11

X-+Y

:

is open.

n

W e first r e q u i r e a l e m m a .

Lemma.

4.

y E Y

such t h a t

Let

Proof.

n

*

(yip n n - l ( y )

E(X)

as a c t i n g on

-1

( y ) p c n-'(y).

n-l(y)p-'

Since a l s o

n

-1

y

E

{x,}

a net

Now l e t

x' E n

and x'p E

in

X

xi

Now, let

xA+x'

Y, x E X

n(xn) = yn

with

E X,

E X

such t h a t

-1

(y)q = n

-1

(y).

qn+q

That i s ,

n(xn) = yn

Let

E E(X)

Thus t h e r e is a

x = x"qn+x"q x = x"q n n' n

with

(y). Then

(y).

That is

*

we have

0,

and t h e lemma is proved. and

{yn)

x +x. n

and

n ( x i ) = yn

n(x') = y = H(x).

so

and let (a subnet o f )

yn.

TI

-1

-1 '-1 (y)p n n (yl

t h a t t h i s is equivalent t o openness of t h e map

-1

x = xop,

so

x"

E

and

and

x +x, n

{yn})

and

( I t is easy t o see

n.)

and let (a subnet o f ) x l q n = xh

qn E E(X)

such t h a t

x'q = x'.

By t h e lemma

n-'(y)

x"q = x.

such t h a t

n(xnl = n(x"q n )

= x

Y with

a net i n

W e show t h a t t h e r e is a subnet (which we s t i l l call

yn+y.

n

(y).

(We regard t h e enveloping

c n -1 ( y ) , s o n -1 ( y ) p = n - l ( y ) ,

To prove t h e theorem, let

-1

and

i . e . , we suppress t h e map

Y,

n ( x ' ) = y, n(x'p1 = n ( x ' ) p = yp = y

n

= n

n-'(y)p

d x ) = y and

= yp.

- see chapter 3. )

t3 : E ( X ) + E ( Y )

Then

0.

so

n n-'(yI,

y = n ( x ) = n(x,p)

Then

semigroup

-1

x E n-'(y)p

= y.

n(x,)

p E E(X)

Under t h e hypotheses of t h e theorem, suppose

If

= n(x'q ) = n(x'1 =

n

so, as noted above,

n

is open.

IT

I t w i l l be shown i n chapter 10 t h a t any d i s t a l homomorphism of minimal flows is open.

5.

Corollary.

Let

X

be a d i s t a l minimal flow, let

cpx : E(X)--+X be defined by

Proof.

E(X) =

a

(px(pl = xp ( p

E

E(X)).

x

E

Then

is an o r b i t c l o s u r e i n t h e d i s t a l flow

X, (p

and let X

(#,TI

is open.

so

The Furstenberg Structure Theorem

E(X)

is d i s t a l minimal.

Thus

99

is a homomorphism o f d i s t a l minimal

'px

flows, and is t h e r e f o r e open. 6.

Lemma.

a E

s.

Then

Let

Proof.

Let

Let

T)

= ( i n t n(V)xint n(V))T.

and

T),

(y ,y') n n with

1Ly.

be a non-empty open subset of

V

T)

Corollary.

7.

be a homomorphism of minimal flows, and l e t

(nxn)(a) E

set which meets Ay c

X+Y

K :

Ay, E

%.

Let

Then K :

8.

Theorem.

(nxn)(aT)

X+Y

such t h a t

VxV c a.

is a non-empty i n v a r i a n t open

YxY.

Since

3 T),

so

Ay

is a minimal s e t ,

(nxn)(aT) E 'I$.

Let

be a homomorphism of minimal flows.

with

YxY

n ( x ) = yn, n(x;l) = y;l, n

T.

T)

t h e diagonal of

be a net i n

in

Then

X

(yn,y;l)-+Ay.

and

Then t h e r e are

(xntn,x;ltn)+AX

xn' x'n

f o r some net

E X

itn}

Consider t h e commutative diagram of minimal flows and

homomorphisms.

Clearly

Proof. nets

{(yn,y;))

B(yn) = p(y;l) x

x' E X n' n

some net

in

and

with {tn)

(nxn)Q(u) c Q ( @ ) . YxY

and

{7n)

(yntn,y;ltn)--+Ay.

n(xn) = y n ~ n .n(x;) in

T.

Note t h a t

@n(XA)= a(xA).

L e t (subnets of)

dx') = y'

a ( xn T-'n)

and

(xntn,xhtn)--,Ax,

Let

and so

in

(y,y') E with

T

Q(P).

Then t h e r e are

(yn,y;l)+(y,y'),

By c o r o l l a r y 7, t h e r e are = yz,,

and

( x n t n , x ' n t n )+Ax

a ( x n ) = pn(xn) = p ( y n t n ) = p(y;lt,) X

-1

~

-

-

-1 +Tx , ~ x z n + x ' ,

so

for =

n(x) = y,

a(xzil). Then (xnT~l(sntnl,X;IT,l(Tntn)) = (x,?) E Q(a) with ( n x n ) ( x , x ' ) = ( y , y ' ) .

=

Chapter 7

100

9. Corollary,

If

n

X

Y

flows and homomorphisms, then f3

is a commutative diagram of minimal

is an equicontinuous extension if and

only if Q(a) c R(n).

Proof. Suppose p

is an equicontinuous extension. If

then

E Q(f3),

(n(xl),n(x2))

so

n(x 1 ) = n(x2)

Conversely, suppose Q ( u ) c R(n), there are

(x,,x2)

(xl,x2) E R(n),

E

so

and let

Q ( a ) with n(xi) = yi

(xl,x2) E Q ( a ) ,

(x1,x2)E R n).

and (y,,y,)

E Q(f3).

Then

(i = 1,2). Then

y1 = n(x ) = n(x2) = y2. 1

Thus Q ( f 3 ) c Ay

and

is equicontinuous. 10. Corollaru. Let

X be a minimal flow and let R and

invariant equivalence relations in X such that canonical homomorphism of X/S

to X/R

S

c R.

be closed

S

Then the

is equicontinuous if and only if

S 3 Q(R).

Proof. This is just a rephrasing of the previous corollary. 11.

Lemma.

Let

(X,T) be a distal minimal flow and let R be a closed

invariant relation in X. of

Let

(x,y) E R and let

U

be a neighborhood

x. Then there is a neighborhood V of y such that, if y1

E

V,

E U such that (xl.yl)E R. 1 (Note: It is not assumed that R is an equivalence relation.1

there is an x

Proof.

It is sufficient to show: if {yn)

yn-+y, then there is a net pn E E(X) with

yn = yp,.

{Xn)

with xn+x

since R

and

with

(xn,yn) E R.

Since the homomorphism p ~ y pof

X is open, and y = ye, there is a net and r +e. n

is a net in X

If xn = xrn, then xn+x

is closed invariant.

{rn) and

on E(X)

Let

E(X) to

with yrn = yn

(xn,yn) = (xrn,ym) E R,

The Furstenbe% Structure Theorem

12. Theorem. Let

101

(X,T) be a distal minimal flow, and let

closed invariant equivalence relation in X.

Then Q(R)

R be a

is a closed

invariant equivalence relation. (x,y),(y,z) E Q ( R ) = r )

Proof. Let

(a n R)T.

U and W be

Let

neighborhoods of x and

z

respectively. Now Q(R)

closed invariant relation

so

by Lemma 11, there is a neighborhood V

y such that if y’

V,

E

there is a z’ E W

with

is certainly a

(y’,~’) E Q(R).

let a E I$, and choose /3 E I$ with P4 c a. Since there are x1 E U, y1 (xltl,yltl)E P.

E

V, t

Since X

1

T with

E

(x ,y ) 1 1

is distal, XxX

E

of

Now,

(x,y) E Q ( R ) ,

R and

is pointwise almost

periodic, and N = (x ,y IT is minimal. Then there is a finite subset 1 1 (x;,y;) E N, (x;,y;)K

K of T such that, for all (x;,yi) E N,

That is, if and

(y;k,yi) E 8,

6E

there is a k E K such that

for some k

is a k E K such that

”f3-dense“ in N.

is

E

K.

(x;k,x;)

<E

In particular, if

(xl
satisfy 6K c 6,

E(X),

(Note that

there

Now, let

and let 0 be a neighborhood of e

such that x10 c U, y10 c V.

E j3,

in E(X)

x10 and y10 are neighbor-

hoods of x1 and y1 respectively, because of the openness of the homomorphism from E(X) there is a z1 E W there is a y2 (y t , z t 1 22 22

E

6

E

y2 = ylEo.

again the

with

(yl.zl) E Q ( R ) .

y10 and z

2

E

From the definition of

W such that (y2,z2)

for some t2 E T. Since 6K c f3,

(y2t2k,z2t2k) E 6, with

From the defining property of V,

to X.)

for k

E

K.

Also, y2

E

E

Q(R),

R, and

we have

ylO, so there is a E0 E (9

Then x2 = xlc0 E x10 c U and

(x,,y,)

E

R. Using

“f3 dense” property of K, there is a ko E K with

(x15 0t 2k0, y15 0t 2k01 E (xltl)Bx(y,tl)f3, (yl~ot2ko,yltl) E f3.

Since also

which means

(x t ,y t 1 1 1 1

E f3,

(x1E0t2k0,xltl) E P, we have

Chapter 7

102

( X1<0t2k0

P

E

Yl
says that

p3.

Recall that x2 = xlco, y2 = ylco, so this

(X2t2k0y2tZko) E p3,

and we have above that

K, so (x2t2k0 'z 2t2k0 ) E P4 c a. 2 Recapitulating, we have (x2,y2) E R, (y2,z2) E R so (x2,z,) E R c R, (y2t2k,z2t2k)

x2 E X1o

U, z 2

C

for all k

P,

E

E

E

W with (x2tZko,z2t2ko)

E

a, which proves that

(x,z)E Q(R). If X

13. Corollary.

is an equicontinuous extension

equivalence relation in X, then X/Q(R) In particular (taking R = XxX) Q

of X/R.

is a closed invariant

is distal minimal and R

is a closed invariant

equivalence relation and the quotient flow X/Q

is the maximal

equicontinuous factor of X. Proof. This follows from corollary 10 and theorem 12. Now let

(X,T) and

(Y,T)

be distal minimal flows, and let

be a homomorphism. For each ordinal number A,

I[

and let Q, = R.

ordinals If h

p

< A.

Suppose Q,

(so

has been defined for all

If A = <+1 define QA = QS+, = Q(Q<) =

n Q,

is a limit ordinal, define QA =

X+Y

we define inductively a

closed invariant equivalence relation QA in X. Write Y = X/R R = R(n)),

:

n

la n Qt)T.

(thus Q, = Q(Qo) = Q(R)).

,
is a successor ordinal, it follows from theorem 12 that Q,

an equivalence relation. Q,

is

is obviously an equivalence relation if A

is a limit ordinal.

If h <

it is clear that Q,

Q Since each QA is closed, it P' follows from general considerations that there is an (least) ordinal number

A >

8

p,

3

for which Qe+l = Qe, and

so

Qe = QA =

nla

n

Q8)T,

for all

8.

If we could assert that Q, = Ax,

for some

8,

the Furstenberg

structure theorem would follow easily, and it is at this point that we

The Furstenberg Structure Theorem

invoke the assumption that the phase space X

103

is metrizable. Let d be

1 and let a = [(x,y)ld(x,y) < n1 n (n = 1,2,.. . 1. Then, if 0 is as above Q, = Q,+l = n (an n Q,)T. n=1,2,.. . Therefore (an n Qe )T is a dense open subset of Q,, for each n, and it follows from the Baire category theorem that n(a n QO)T is dense n

a metr c defining the topology of X,

-

- -

-

Then Q, = n(an n Qe)T c nunT = Px = Ax

in Q,.

Ax,

and X/QO = X.

We now have all the ingredients for the proof of the structure theorem, in case the space X If

is metrizable.

is as in the preceding discussion, then the flows XA = X/QA

0

and the associated homomorphisms define a projective system. (Note that X,

= X/Q,

= X/Ax = X

QA+l = Q(Q,) X/QA+l+X/QA

and Xo = W Q 0 = X/R = Y . )

(A < 0).

Moreover, since

corollary 13 tells us that the homomorphism

(or XA+l+XA)

is an equicontinuous extension.

This completes the proof. The projective system of flows {X,} homomorphisms { qAv ) is sometimes called a “Furstenberg tower.“ Theorem 1 has a valid converse. In fact, if n phism of minimal flows and {XA},

A {q,)

:

X-+Y

and

is a homomor-

is a projective system of flows

and homomorphisms with Xe = X, Xo = Y and

(pi’’

an equicontinuous

extension, then it follows directly by transfinite induction that n

is

distal. Indeed, a relatively straightforward modification of the proof we have presented proves that a distal homomorphism can be characterized as in theorem 1.

(That is, the flows need not be distal; what is

essential is that the homomorphism is distal.) Suppose now that X

is distal minimal and not equicontinuous. Then,

using the notation of the structure theorem Xo = 1, and X1 = X/Q,

= WQ. Hence Xl

For, if X = Xe. X1 = 1,

then Xz,

then

8

the trivial flow,

is equicontinuous. Moreover, X1

>. 1 (since X

is not equicontinuous).

as an equicontinuous extension of XI = 1,

is

#

If

1.

Chapter 7

104

itself equicontinuous. But X, s o X2 = X,,

X, = X/Q,

XI is the maximal equicontinuous factor of

and Qz = Q,.

Thus Q, = Q, = Q for all A ,

and X =

is equicontinuous, which is a contradiction.

Thus we have proved. (X,T) is distal minimal and not equicontinuous.

14. Theorem. Suppose

Then X has a non-trivial equicontinuous factor. Note that the ‘discussion just completed shows: if X and Y are distal minimal flows with metrizable phase spaces, and n non-trivial homomorphism, then Q(n) Q, = R ( n ) , Q, = Q(Q,) Q,

= R(n),

#

R(n).

X+Y

is a

For, if Q(n) = R ( n ) ,

= Q(R(n)) = Q(n) = R(n) = Q,,

for all ordinals A .

:

then

and it follows that

Thus Qe = A = R ( n ) ,

so

n

is an

isomorphism. Moreover, the proof of the Furstenberg theorem in general (i.e., when the phase space is not metrizable) follows from this property. For, suppose whenever X and Y are distal minimal, and n non-trivial homomorphism, then Q(n) f R ( n ) . the ordinal for which Qe+l Q(n) = Qe+l = Qe = R(n),

isomorphic with X.

As

= Qe

and, if

:

X+Y

a

Then, if, as above, 8

n : X+X/Q,,

is

R( n ) = Q,,

and so n must be trivial and X/Qe

is

we have seen, this implies the Furstenberg

theorem. A sketch of the proof of this property in the general case will be given at the end of the chapter. Now we present some applications of the structure theorem. The first, on eigenfunctions of distal minimal flows, is under the assumption that the acting group is abelian. If

(X,T) is a flow, with T abelian, an einenfunction of

a continuous map

cp :

X+K

that, for some character

(X,T) is

(the unit circle in the complex plane), such

r of T, cp(xt)

=

r(t)cp(x)

(x E X, t

E

TI.

Suppose now that X = (X,T) is a distal minimal flow. with T abelian,

The Furstenberg Structure Theorem

105

and let Y be a non-trivial equicontinuous factor (theorem 1 4 ).

As we

have noted, Y may be given the structure of a compact abelian group. Choose y E Y as the identity and let cp

be a non-constant character

0

of Y, so

(t

E

Let

:

T-+K

a(t)

be defined by

= cp(yot)

Recall that the group structure on Y satisfies (y,s)(yot)

TI

yo(St)

cp(yo) = 1.

s,t E TI, so a ( s t l = rp(y 0st) = cp((yos)(yot))

a(s)a(t).

If y E Y and {sn}

yosnt+yt, p(yosnt)+cp(yt)

and also rp(yosnt) = r(sn)r(t),

the equicontinuous flow (Y,T). If by composing cp

with

15. Theorem. Let

T with yosn+y,

is a net i n

Thus cp(yt) = y(t)rp(y)

cp(y0sn)-+cp(y).

= cp(yos)cp(yot)

and

K :

=

then

r(sn) =

is an eigenfunction of

cp

X-+Y

=

is a homomorphism, then,

we obtain

IK

(X,T) be a distal minimal flow, with T abelian.

Then there is a non-constant eigenfunction of

(X,T).

As a corollary to the existence of eigenfunctions, we show that a

simply connected space cannot support a distal minimal flow under the action of an abelian group. For suppose T

is abelian and acts distally

and minimally on the simply connected space X. Let @ tion of the flow (X,T). Since X

@(XI

= exp i$(x),

@(xt)@(x)-'

where $

be an eigenfunc-

is simply connected, we may write

is real valiied and continuous. Then

= exp i($(xt)-$(x)l

and also @(xt)@(x)-'

= .a(t),

character of T. That is, for fixed t E T, exp i($(xt)-$(x)) constant, and since $

is continuous, $(xt)-y?(x)

2 2 a(t2) = $(xt )-$(x) = $(xt )-$(xt)+$(xt)-@(xl

in general, $(xtn)-y?(x)

= a(tn) = na(t1.

have a(t) = 0, and $(xt) = constant, and therefore @

16. Theorem. Let

@(XI.

= a(t).

= a(t)+a(t)

Since @

a is a

Then

= 2a(t),

and,

is bounded, we must

Since (X,T) is minimal, @

is

is constant. Then we have proved.

(X,T) be a distal minimal flow with X non-trivial,

Chapter 7

106

and T abelian. Then the space X is not simply connected. In particular, the n-sphere Sn cannot support a distal minimal flow, if n t 2 . Our next application concerns invariant measures for distal minimal flows. By an invariant measure for the flow (X,T) we mean a Bore1 on X such that p(At) = p(A)

probability measure p

measurable subsets A and t E T. Equivalently, p

for all

may be regarded (via

the Riesz representation theorem) as a positive linear functional on the continuous real valued functions on X with p(1) = 1,

C(X),

that p(ft) = p(f)

(f E C(X), t E TI, where ft

ft(x) = f(xt1.

If

p

pt(f) = p(ft).

Thus

for all t

E

is a measure on X, pt p

E

C(X)

such

is defined by

is the measure defined by

is an invariant measure if and only if pt = p

T.

In chapter 12 we will obtain necessary and sufficient conditions on a group T such that all flows with acting group T

admit invariant

measures. Here we prove the existence of invariant measures for all distal minimal flows. As one would expect, the proof proceeds by "lifting" the invariant measure through the equicontinuous extenslons and

limits guaranteed by the Furstenberg structure theorem. To this end, we need to study the structure of equlcontinuous extensions more intensively. Our main result in this direction is that all fibers of an equicontinuous extension are homeomorphic, and that there is a transitive group of self homeomorphisms of each fiber. In fact, first suppose that Let

I

X-+Y

be a minimal right ideal in E(X)

idempotent in I identity on H(XU) =

H :

H

-1

is just a distal extension. let y

E

Y and let u be an

with yu = y. We first note that u acts as the (y).

yu = y = n(x)

(If x E n-l(y), xu and since

A

1s proximal with

x and

is d stal, xu = x. 1 Now, let

The Furstenberg Structure Theorem

p E 1.

For any extension n -1 (yip c n -1 (yp).

we show that n -1 (yip = n-'(yp), map

p maps

A

-1

(y) to n

-1

107

In this case

( n distal)

from which it follows easily that the

(yp) bijectively (since n

is distal this

map is obviously one-to-one). Let q -1

n

-1

n

E

I such that pq = u. Then n

(ylpq c n-+yp1q c n (y) =

-1

II

-1

(ypq) = n

I

applied to yp we have n

-1

(yplqp = n

For y

E

-1

-1

-1

(y) = n -1

-1

(ylu =

(y), so

(yip = n

-1

(yplqp. Now qp = w

and ypw = yp, so by the same argument (yplw = n

-1

(yp) and n

-1

(yp) = n

-1

(yp)w =

(YIP.

Y, suppose yu = y, and consider p

the above discussion n -1

(yu) = n

(yplq. It follows that n

is also an idempotent in

II

-1

-1

-1

(y)p = n

-1

E

I with yp = y. By

(y), and p defines a bijection of

If we identify two such p1,p2 E I when they define the same -1 map on n (y) (i.e., if xpl = xp2 for all x E n (y)), we obtain a R

(y).

-1

group E (The inverse of the map defined is defined by q E I where Y' pq = u.) Summarizing we have obtained

17. Theorem. Let n

:

X-+Y

be a distal homomorphism of minimal flows,

and let y E Y with yu = y. Then -1 (i) If p E I , p maps II (y) bijectively to

R

-1

(ii) E is a group with identity u, and xE = n Y Y -1 x E n (y). Now suppose that n

(yip = n-l(yp).

-1

(y) for

is an equicontinuous extension. Then it follows

homeomorphism) and in fact that

Ey

E

is continuous (hence a Y is a compact group (see the proofs

easily from the definition that each p

E

of theorems 2 and 3 in the first enveloping semigroup chapter).

Using

these facts and theorem 17, we have 18.

Theorem.

Let R : X+Y

be an equicontinuous homomorphism of

Chapter 7

108

Then E is a compact group of Y -1 homeomorphisms acting transitively on the fibers n (y). If p

minimal flows, and let y E Y.

-1 a (y) homeomorphically onto

maps

19. Lemma. Let

flows and let y TI

-1

m

E

xq

E

:

X+Y

-1

I, p

(yp).

be an equicontinuous extension of minimal

Y. Then there is a unique probability measure on

(y) which is invariant under the action of the group E Y'

Proof. Let

x

n

R

E

m

denote normalized Haar measure on the group E Y' n-1 ( y ) , define a measure m X on n -1 ( y ) by

f(xqp)dm(qp) = mx(f)

(f) = JE f(xqp)dm(p) =

If

(the second equality

JEY Y holds due to the invariance of the Haar measure). transitively on n

-1

we have rnx = m x'

(y),

call this common measure

r

E

E

Y'

fr(x) = f(xr),

p.

Since E acts Y -1 for all x,x' E n ( y ) ;

If we define, for f

E

C(n

-1

a similar computation shows that

(y)) and p(fr) = p(f),

is invariant under E Now suppose 1.1' is any probability Y' -1 Then, if f E C(E 1, measure on n (y) which is invariant under E Y' Y

so

p

p'(fl

=

f(x)dp'(x)

-1 (Yl

=

f (x)d(p'q)(x)

L

W

f (x)d(p'q) (x)dm(q) = E' Y 'n-'[y)

The Furstenberg Structure Theorem

109

= p(f).

p(f)dp'(x) (Y) If

II :

X+Y

is a flow homomorphism and p

probability measure on X, the np

is a T-invariant

will denote the T-invariant

probability measure on Y defined by

np(f) = p(fn) (f

E

C(Y)).

Our

next result shows that in the case of equicontinuous extensions, an invariant measure can be lifted from Y 20. Theorem. Let

n

:

X+Y

to

X.

be an equicontinuous extension of minimal

flows. Suppose there is a T-invariant probability measure Then there is a T-invariant probability measure p

u

on Y.

on X such that

np = u . -1

(y) has a unique E invariant measure, which we call Y as a measure on X which is concentrated on py. We may regard each p Y -1 II (y). We first show that p = pyt (t E T). Note first that p t is Yt Y -1 concentrated on n (yt). Now if p E E then tpt-' E E so if Yt' Y' Proof. Each n

p E Eyt, ( p t)(p) = p (tpt-')(t) = pyt. Thus p t is a probability Y Y Y -1 invariant. By measure (concentrated) on n (yt) which is E Yt uniqueness, p t = pyt. Y Now we define a probability measure p on X by setting

The existence of this integral will follow once we know that the integrand is a continuous function of y, and this matter is attended to below.

Assuming it for now, let

t

E

T.

Then

(pt)(f) = p(ft) =

(ft)du(y) = p t(f)du(y) = (f)du(y) = Sp (f)dut-l(y) = fpy(f)du(yl Y S Y Y = p(f), and so p is the required invariant measure. /p

Chapter 7

110

The continuity assertion mentioned above is equivalent to showing that the map y w p (y E Y) from Y to C(X)* (with the weak * Y topology) is continuous. That is, if {yn) is a net in Y with yn+y,

we want to show p (f)+py(f), n'

suppose p

, and show p' = py.

+p'

f E C(X).

for all

At all events p'

We may

is concentrated

Yn -1

(y). Let q E E and let pn E M with ypn = y , . We may Y' suppose p +u, where u is an idempotent in M with yu = y. (For, n

on

II

if pn+p

E

M, then yp

y and if pu = p, then yu = y.

=

with rp = pr = u, then yr = ypr = yu = y, so rp +rp n x E n

-1

to X

= u.)

(y).

Since

71

yrpn = ypn

If r

E

M

-- yn and

is a distal extension, xu = x, for all

Now the group T, regarded on a family of maps from n

-1

(y)

is an equicontinuous family, and thus it follows from the

Arzeli-Ascoli theorem that identity map of

II

-1

(y).

pnln-l(y1

Let

the inverse map of p In-'(yn1 n

rn E M

r qp = p n' Yn n n

(y)

-1

rnln

.

Now, since

r qp +p'q

Y, n n

Yn n n

I

f(xq)dp' (x) = (p'qlf.

)

is a probability measure concentrated on n

under all q E E Y' 19 that JL' - Py.

weak

Yn

Therefore, p'q = p', -1

(y)

*.

(XI,

r qp (f) = If(xrnqpn)dp

(To see this note that, for f E C ( X ) , p which is close to

(y) is

From the defining

uniformly, it follows that p

-1

n

with pnrn = u, so

and ynrnqpn = yn.

properties of the fiber measures, p

-1 pnln (y)+id

converges uniformly to the

so

p'

which is invariant

It follows from the uniqueness assertion in theorem

Hence the property of possessing an invariant measure is lifted by In equicontinuous extension. Our next result shows that this property is also preserved under projective limits. 21.

Lemma. Suppose X = X

7

is a limit of the minimal flows Xa (a <

r),

The Furstenberg Structure Tharem

and each Xa admits an invariant measure v invariant measure

p

Proof. Write cp

(instead of c p z )

Xa.

If a < 7

a

Then va

extends to a linear functional

ia(l) =

a

for the projection of X = X

onto

7

)

as a subspace of C(X)

(via the

may be regarded as a linear functional of the

norm one on a subspace of C(X).

and, since

Then there is an

a'

for X.

we may regard C(X

map f+fOcpa).

111

va(l) = 1,

By the Hahn-Banach theorem, va

ia on C(X) such that II,;II we have

IIiall

=

1.

Thus

5

a

IIvall,

is a

probability measure, and clearly cp (i 1 = ua' a a

-

Of course, there is no reason to expect any of the measures v a

to be

T-invariant. An invariant measure is obtained as follows. For each a < 7.

let Pa

denote the class of probability measures p

on X such

that cp p

va

is T-invariant on X for all f3 I a. We note that P a - - a Pa, for, if P < a, cpP va = cpPcpava - cpPva, which, as we have

P

-

E

observed, is T-invariant. Clearly, if a < a', sets {Pa)(a
see that each Pa

then Pa, c Pa,

satisfy the finite intersection property.

is weak *

n Pa *

compact, so

so

the

It is easy to

0.

a57

Now, let p E nP,.

We show that pt = p (t E T),

complete the proof. To this end, let Ba

be the subalgebra of C(X)

consisting of functions of the form f o p a (f E C(Xa)).

, subalgebra since Ba c B

whenever a < a',

of limit, it follows immediately that Hence UBa f E C(Xa), p((focpa)t)

UBa

is uniformly dense in C(X). so

p(F) = p(focpa) = cpap(f).

= p((ft)ocpa)

pt(F) = p(F)

for F

follows that pt = p.

and this will

uBa

is a

and, from the definition

separates points of X. If F E Ba, F = focp,

If t

= cpap(ft) = pap(f),

E

for some

T, pt(F) = p(Ft) =

since p E Pa. Therefore

in a uniformly dense subset of C(X)

and it

Chapter 7

112

Now suppose

(X,T) is a distal minimal flow, and let

be a projective system for

(X,T) = (X,,T)

and

(XA,T) ( A

5 0)

(XO,T) the trivial

flow, as in the Furstenberg structure theorem. It follows easily by transfinite induction and the two preceding theorems that all the flows (XA,T) admit an invariant probability measure.

Thus we have proved.

Theorem. A distal minimal flow admits an invariant probability

22.

measure.

It would be interesting to find a direct proof of this theorem (that is, a proof which does not invoke the structure theorem). We conclude this chapter by sketching the proof of the following theorem, which, as we have noted, implies the Furstenberg structure theorem. Theorem. Let

23. H

:

X+Y

(X,T) and

(Y,T) be distal minimal flows, and let

be a homomorphism. Then

Q(H)

*

R(H).

(This theorem, which is due to McMahon and Wu (Proc. Amer. Math. SOC.

82 ( 1981), 283-294), applies to any distal homomorphism.1 If K

is a subgroup of T, and

homomorDhism is a continuous map cp

(x

E

X, k

E

(X,T) and :

X+Y

(Y,T) are flows, a K

such that

cp(xk) = cp(x)k

K).

The main step in the proof of Theorem 23 is the construction of a commutative diagram

where K

is a countable subgroup of T, ( 5 , K ) and

(YK,K) are metric

The Furstenberg Structure Theorem

trivial.

cp,(n-'(y))

are K homomorphisms with nK non-

and $K

minimal flows, nK, cpK,

-1

=

113

($,(y))

(y

E

Y),

and

(cpKxcpK)(Q(n)) c Q(nK).

First we show that the existence of diagram ( * ) implies that

*

Q(n)

R(I).

For suppose

:

is a homomorphism of distal minimal flows with

Consider the diagram ( * I , with the indicated properties.

R(n) = Q(n).

E XK, x xK ,x' K K

Let

X+Y

*

xi and

K (xK 1 = IK (x'). K

'pK(x) = xK, cpK(x') = x'

K and n(x) = n(x'1.

-1

for y

'pK(r-l(y)) = nK ($(y)),

(since (cpKxcpK)(Q(n)) = Q(nK)), R(nK) = Q(nK).

But

E

Y. )

(This is possible since

Then

we have

x,x' E X with

Let

TI

(x,x')

E

R(n) = Q(n),

(xK,xi) E Q(nK,).

That is,

XK and YK are metric spaces, so (by our

discussion of the structure theorem in the metric case) R(nK) This contradiction shows that The flows

(XK,K) and

R(n)

f

E

Q(nK).

(YK,K) are constructed as follows. Let

d

d(x,x') I 1 for all

(For instance, d may be defined by d(x,x') = lf(x)-f(x')l,

X.

f

where

*

Q(n).

be a continuous pseudometric on X, with

x,x'

so

is continuous.1 Let

: X+[O,lI

K = {ki}i=l,2,.. . be a

countable subgroup of T, and define the pseudometric dK on X by m

1

=

dK( x,x'

1

d(xki,x'ki).

*

We define a pseudometric dK on Y as

i=l 2

fo1 lows. Let '2

be the Hausdorff pseudometric on X

and let DK A,B

*

E

denote the collection of non-empty closed sets of X,

X 2 DK (A,B) = max[max d(a,B), max d(b,A)I. acA b€B

dK(Y,y' 1 = DK(n-'(y),rr

-1

equivalence relations ((x,x') E

%

*

YK = Y/%,

Define pK

5<

and

X+AKxYK,

:

X-+AK,

*

Then dK

dk:

if

is defined by

The pseudometrics dK and d* K define

<

on X and Y respectively

if dK(x,x') = 0 and similarly for

and let a :

(y')).

induced by

*K

:

Y+YK

* 5<).

Let

AK = X/%,

be the natural projections.

by 'pK(x) = (a(x),+K(n(x)),

and let XK = (pK(X).

Chapter 7

114

Finally, define nK

XK

:

XK+YK,

by nK(a,y) = y.

and YK are metric, that K acts on XK

and YK,

various maps define the commutative diagram ( * I . -1

nK ($,(y))

= cp,(n-l(yl)

It is immediate that

The property that

is a somewhat involved but straightforward

computation. Note also, that the homomorphism nK ( prov i ded

dK

is non-trivial).

and dK(x,x') > 0. Then a(x) (oL(x'),$~(~)) in XK,

and that the

Let x,x' # OL(X'),

E

is nontrivial

X with n(x) = n ( x ' ) =

and so

(a(x),$(y))

y

#

with nK(uK(x),#K(~)) = nK(aK(~'),#K(~))= $,(y).

Thus, for any pseudometric d on X and countable subgroup K of T -1 -1 we can obtain a diagram of type ( * ) satisfying nK ($K(y)) = (oK(n (y)) (y E Y).

In order to ensure that the K-flows

minimal, and that cpK(Q(n)) c Q(nK),

(XK,K) and

(YK,K) are

it is necessary to choose the

subgroup K more carefully. We omit this construction; it is carried out in the cited paper of McMahon and Wu.

115

Chapter 8 Universal Minimal Flows and Ambits A minimal flow

(M,T) is said to be universal if for any minimal flow

(X,T) there is a homomorphism 7

from

(M,T) to

(X,T).

The first part of this chapter is devoted to showing that there is a unique universal minimal flow.

1.

Theorem.

If T

is a topological group, there exists a universal

minimal flow (M,T), and any two universal minimal flows are isomorphic. Proof. Let

M

= [(X,,T)la

be the collection of (isomorphism

E A]

classes) of all minimal flows with phase group T.

(Note that no logical

difficulties are involved in the definition of A.

If

(X,T) is a mini-

mal flow, then X contains a dense subspace whose cardinality is less than or equal to

(TI, the cardinality of T, so

1x1

in the definition of A we are only considering flows

*

less than or equal to a fixed cardinal.) Let X =

I 2

1x1

(X,T) with

ma, and

*

(X ,TI

let

U

be the product flow. Let

M

*

be any minimal subset of X

(since if n a

:

X +Xa

homomorphism from M

is the

.

Then M

is clearly universal

a'th projection, then nalM

is a

to Xa).

This proves existence. The proof of uniqueness is considerably more involved. We first note that it is sufficient to find a coalescent minimal flow (Z,T) such that

(M.T) is a homomorphic image of

(Z,T).

(Recall that a minimal flow is coalescent if every endomorphism is an autornorphism.)

For, in this case, let

mal flow, and let a Then rf3a : Z+Z

:

Z+M, p

:

M+M',

(M',T) be another universal mini7 :

M'+Z

be homomorphisms.

is an endomorphism. By the assumed coalescence of 2 ,

Chapter 8

116

is an automorphism, and it follows that

ypa

It follows also that

phism.

and M

Z

p

:

M+M'

is an isomor-

are isomorphic, so the universal

minimal flow is coalescent. M,

To show the existence of a coalescent pre-image of

we prove a

more general result . 2. Lemma. Let

(X,T) be any minimal flow. Then there is a cardinal

number a and a minimal subset M of

(Xa,T) such that the flow ( M , T )

is coalescent. Proof.

Recall that in the chapter on distal flows, we introduced the

X, and showed that maximal

notion of an almost periodic subset of

X exist. Let C be a maximal almost

almost periodic subsets of periodic subset of

X and let

E

M.

Now z'

C' = range z' z'

:

C+C'

X"

such that

range z = C and z

= c, f o r each c E C ) .

is one to one (for example z

let z'

z E

Let

M = 3, and

is an almost periodic point, and so

is an almost periodic set.

In fact, C'

is one to one. The proof that C'

is maximal, and

is maximal is similar to

the proof of theorem 3 in the chapter on distal flows, and is omitted.

To see that z' {sn)

is one to one, suppose c1,c2 E C with

is a net in T with

z's + z .

n

one to one, it follows that

Then zc = z 1 c2'

almost periodic set.

If

7

z

7(C)

z

and

= (p(z)),,

for c

is

(z,cp(z))

is an

is an

range cp(z) are both maximal

range z = range cp(z).

is a permutation (bijection) of C, let

induced automorphism of

and since z

C2

so range z u range p ( z )

(MxM,T),

But range

almost periodic sets, so

Let

= z'

l

c

1 = c2. Now, let 9 be an endomorphism of (M,T). Then

almost periodic point of

.

z' C

(Xc,T) ((y*(z)lc = E

C. Since cp(z),

r

*

denote the

Define

r

by

(regarded as a map of C

Universd Minimal Flows and Ambits

to X)

117

range (p(z1 = range z , 7

is one to one and

is a permutation of

4

C and

7 (2) = (p(z). 4

Since ( p [ z ) 4

so

(p

= 7

IM

M,

E

7

(MI

and thus

(p

A

M

f

0,

so

7

*

4

(MI = M.

Now

7 ( z ) = (p[z),

is an automorphism. This completes the proof.

M later in the

We will give another proof of the existence of

chapter, and another proof of uniqueness is sketched in exercise 3 . proof of existence doesn't guarantee that

M

is "non-trivial." Later in

the chapter, we'll show that if the group T T on M

Our

is discrete, the action of

is free.

A flow i s said to be point transitive if

it has a dense orbit.

In

this case, a point whose orbit is dense is called a transitive point. An ambit is a point transitive flow with a distinguished transitive point. Precisely, an ambit (or T ambit) is a triple is a point transitive flow and x E X with 0 Of course, if

(X,T,xO), where = X.

(X,T) is a minimal flow, and xo

is an ambit (a "minimal ambit"). If

(X,T)

E

X,

then

(X,T,xO)

(X,T) is any flow, then

(E(X),T,e) is an ambit (which is, in general, not minimal). We will usually write

(X,xo) for an ambit, when the acting group T

is understood.

If

(X,xo) and

homomorphism

(p

:

(Y,y0) are ambits, an ambit homomorphism is a flow

X-+Y

with (p(xo) = yo.

If such an ambit homomorphism

exists, it is clearly unique - in this case we write Obviously, if (Y,yo) and so

(X,xo) t (Y,yo) 2 (X,xo) then t

(X,x0)

(X,xo) t (Y,yo).

is isomorphic to

is a partial ordering on the collection of

(isomorphism classes) of T ambits. A universal

T ambit is a T ambit

(U,uo) t (X,xo) for every T

T ambit exists, it is unique.

ambit

(U,u,)

(X,xo).

such that Obviously, if a universal

Chapter 8

118

The existence of a universal T ambit can be proved in a manner simi'u be the collection

lar to the proof for universal minimal flows. Let of T ambits

(X,x) E 11,

(X,x). If

#

X = TI Xx. Let XX€U

x, and let U

=

z

be the point of

we write

Xx for X, and let

whose X ' t h

X#

X

coordinate is

5. It is clear that (U,T,z) is a (hence the)

universal T ambit. (X,T) is a flow, then

We observed earlier that if ambit, and if lar,

(X,x,)

is an ambit, then

(E(U),e) 2 (U,uo ,

ambit that Thus U

(U,u,)

and

(X,x,).

2

In particu-

and it follows from uniqueness of the universal (E(U),e) are isomorphic, and may be identified.

is endowed with a semigroup structure, in which the maps <-+q,E

are continuous. We may also regard U E(U))

(E(X1.e)

(E(X),T,e) is an

as acting on any flow X

x< = lim xt n'

where t

n

= et

- if x

+C

n

(via the identification with

E

E

U.

X, and 5 E U

then

The same remarks apply to the

universal minimal flow M. Now we present an alternate construction of the universal ambit, in case the acting group T

is discrete. This is accomplished by means of

the Stone-tech compactification PT of T. Recall that /3T is characterized by the property that any map from T to a compact Hausdorff space has an extension to PT.

Precisely, there is an embedding u of

T onto a dense subspace of PT, such that if X space and cp

-cp

:

PT+X

dT),

so

T+X

:

then there is a (necessarily unique) continuous

such that cp = (pu. (Equivalently, if we identify T T

extension of

is regarded as a dense subspace of PT, then (p (p

= ts

(t

E

with

is an

to a continuous map of PT to X.) This permits us to

define an action of T on PT.

as(t)

is a compact Hausdorff

T). Then as

If s

E

T

let as

:

T+PT

gS

:

PT+BT.

extends to a map

by

If

Universal Minimal Flows and Ambits

5

/3T, write

E

g S ( < ) . Now if

for

<s

s,s’ E

119

( 5 s ) ~ ’=

(since both sides agree on the dense subspace T), so TI.

( < E /3T, s , s ’ E

and

-6 s s ,

T, then gS,gs =

3

(/3T,T) is a flow. Note that

=

T

~ ( S S ’ )

= /3T, so

(/3T,e) is an ambit. In fact, (PT,e) is the universal T-ambit. For if

(X,x )

ambit the map

t H x0t of T to X extends to a homomorphism

PT+X

Therefore

II :

If M

such that n(e) = xo.

/3T+X,

II :

(flT,e)2 (X,xo).

“IM

then

maps M

minimal subsets of pT

(For if

x E PT and

t

onto X.) This also shows that all

are isomorphic.

T with t

E

#

e, then xt

first outline a construction of f3T. from T to

I

Let

= [O,ll.

(f E 9). Then q(T)

r) :

T+19

f

is a subset of the compact space

space, and let p : T+X,

show lim p(t .}

25

(p :

X+I

of

J

I+<.

J

) =

and let

in

That is lim f(t

L i1

BT.

J

)

9

I .

(q(t)If = f(t)

I3, and we

Let us indicate why /3T

be a compact Hausdorff Then there is a net exists, for all

x exists. If not, there are subnets {tlj}

f

{tj}

9. We

E

and

{t.) with p(t )+xl and p ( t Z j ) + x 2 , and x I # x2’ Let J 1.j be continuous with cp(x 1) = 0 and cp(x2 = 1, and define

f E 3 by f(t) = (p(p(t)) (t f(t

<E

(That is, if

x.) For this purpose, we

be defined by

has the required extension property. Let- X

r)(t

is free.

3 denote the set of all maps

Let

define /3T to be the closure of r)(T)

in T with

is a (hence

(X,T) is minimal, and

Next we show that the action of T on PT

{t

M

is any minimal subset of PT, it follows that

the) universal minimal flow for T.

is an

0

= (p(p(t

E

T).

Then eventually

1

. I ) < 8, and f(t 2J ) = (p(p(t2J 1) >

15

does not exist.

3

so

lim f(t 1

J

This is a contradiction.

Therefore x = lim p(t .I

J.

exists, and we define GCS) = x.

For

further details on the construction and properties of the Stone-eech

Chapter 8

120

compactification see Kelley, General To~ology,chapter 5 (where it is carried out for a completely regular Hausdorff space). With this representation of PT, t E T and a net f E 3.

If

ti+E s E

E

BT

if and only if

T, and t.-+< 1

E

is identified with

lim f(ti) exists, for every

PT then <s = lim(f(tis))fEg.

fore, in order to show that the action of T on PT find an f

E

9

such that

lim f(ti) # lim f(tis),

To this end, define two elements t

if

T =

(f(t))fE3,

and

is free, we must (provided s

of T to be

T

There-

#

e).

s equivalent

tsk for some integer k, and let H c T be a set containing

one element from each s equivalence class. Thus T union of the sets Hsn

. . . 1.

If s

is not of finite order,

I

if n is even for H. If the 1 if n is odd k k is m, define f by f(hs 1 = - (hEH, k = 1 , 2, . . . , m-1).

define f on T by order of s

(n = O , k l ,

is the disjoint

Then (in both cases)

f(hsn) =

0

m

Jf(tis)-f(ti)l 2

f.

so

lim f(ti)

#

lim f(tis).

Therefore we have proved: 3.

Theorem. If the group T

is discrete, the action of T on the

universal T ambit, (and hence on the universal minimal flow for TI

is

free. W.A. Veech has proved this theorem in case the group T

is locally

compact (Bull. Amer. Math. SOC.,83, 775-8301, If

(M,T) is the universal minimal flow, then

transitive. It is not known whether

(E(M),T)

is point

(E(M),T,e) is the universal

T

ambit. We conclude this chapter with a sketch of yet another construction of the universal ambit, by means of the maximal ideal space. This approach

will be useful when we consider invariant measures on flows. Let T be a topological group, and let fs

denote the bounded right

Universal Minimal Flows and Ambits

121

(A function f

uniformly continuous real valued functions on T. is right uniformly continuous if, for every

E

hood

V of the identity e in T such that

here

1) 1)

T-+R

:

> 0 there is a neighbort [If-f 11 <

for t

E,

E

V;

denotes the supremum norm and ft is defined by

t

4

f ( s ) = f(st), provide B4

for s,t E T.) Let

*

with the weak

of T on B

by

topology.

t (pt)(fl = p(f 1.

and let B be the weak

Now each pt

In this topology a net

p +p

n

It is easily verified that this

action is jointly continuous. If t (f E 2 3 ) .

denote the dual space of 8 and

for every f E 8. We may define an action

if and only if p (f)-+p(f) n 4

B

E

*

T, define pt by pt(f) closure of the set

= f(t)

[ptlt E TI.

has norm one, and therefore so does each q

It

in B.

follows from Alaoglu's theorem (Hewitt and Ross, Abstract Harmonic Analysis, I, p. 458) that

B

is compact. Moreover, if t,s E T and

f E 8, (pt)s(f) = pt(fsl = fs(t) = f(ts) = pts(f)

so

is clearly point transitive. B

(B,T)

is multiplicative

-

(f,g E 8 ) . From this it follows immediately that

pt(fg) = pt(f)pt(g)

identify B

is an

TI

is called the maximal ideal sDace or

structure space of 8. It is obvious that each pt

is multiplicative. Now if f E 23,

?(q) = q(f),

E

is a compact invariant set, The flow

orbit and its closure B

each q E B

[ptlt

for q

E

we define

?:

E C(B)

by

B. The correspondence f + + ? allows us to

with C(B).

This correspondence is one to one (since linear

functionals separate points), preserves algebraic operations, and separates points (if q , p f E 8, so that

F(p)

#

B with q

E

*

p,

then q(f)

*

p(f1

for some

It follows from the Stone-Weierstrass theorem

F(q)).

[?If E 81 = C(B).

This process can be repeated word for word for any uniformly closed invariant subalgebra B'

t then f E 3'.

for all

of 13.

t

E

(By "invariant" we mean that if f E B ' ,

T.1 Let B'

be the maximal ideal space of

Chapter 8

122

8' - that is, B' pi = pt18,.

is the weak

*

closure of

[piit

E

TI, where

T acts on B',

Just as above, it follows that

(B',T) is point transitive, and the correspondence f -? 8' with C(B').

The map from 8*

to B'*

the flow identifies

is

p++pltB'

defined by

onto (this is a consequence of the Hahn-Banach theorem) and this map restricts to an ambit homomorphism from

(B,T,pe) to

(B',T,pk).

In fact, (B,T,pe) is the universal T ambit.

(X,T,x )

For, let

0

be an ambit. In light of the above discussion, it

is sufficient to show that Let

(X,T,xO) is isomorphic with some

8' = [f E Blf(t) = F(xot),

for some F E C(X)l.

(B',T.p;?).

It is easily

checked that 8' is a uniformly closed subalgebra. Let

B'

be its

maximal ideal space, as constructed above. We first define u : B'+X on the orbit of

p' by e

extended to all of B'.

d p ' ) = xot, and then show that

u can be

t

In fact, if

p E

B',

p = lirn p'

for some net

tn in T, s o

{tn)

lim p' (f) exists for all tn

lirn f(tn) = lim F(xotn) exists in X u(p) =

x'.

Hence u

exists for all

(since C(X)

F

E

f E 8'. C(X).

That is

Then

separates points of XI

lim xOtn = x'

and we define

That is, if f and F are related as above p(f) = F(x'). is a homomorphism from

(B',T,pk)

to

(X,T,xO). The proof

will be completed if we show that u

is one to one and this is almost a

tautology. For if d p ) = u(r)) = x'

then p(f) = F(x') =

this equality holds for all

f

E

B',

we have p =

a(f).

Since

r).

Exercises.

1.

If the group T admits non-distal minimal flows, then there is no

universal minimal ambit. That is, there is no ambit minimal, such that

2.

(M,mol with M

(M,mo) t (X,xo), for every minimal ambit (X,xo).

If (X,T) is any flow, and I is a minimal right ideal in the

Universal Minimal Flows and Ambits

enveloping semigroup E ( X ) , (Hint: if cp cp(p) = r p ,

then the minimal flow

is an endomorphism of

for some r

1.1

E

(1,T) then cp

123

(1,T is coalescent. is o f the form

This provides another proof of the

coalescence (and hence uniqueness) of the universal minimal flow. 3. Let

'G

be a non-empty collection of minimal flows

if C'

is any subcollection of

lT[X'IX'

E 'G'],

then N E C.

of

Xo.

and N

is a minimal subset of

Shoy that there is a unique

That is, Xo

minimal flow Xo.

'G

(X,T) such that

E 'G

and if X

E

'G,

"i7 universal"

then X

is a factor

Examples of such classes C are the equicontinuous and distal

minimal flows.

(The 'G

following property: that p(x)

x,y

universal minimal flows are characterized by the E

Xo,

there is an automorphism cp

of

Xo

such

and y are proximal (J. Auslander, Regular Minimal Sets, I,

Trans. Amer. Math. SOC.,123 (19661, 469-479).

This Page Intentionally Left Blank

125

Chapter 9 The Equicontinuous Structure Relation and Weakly Mixing Flows In earlier chapters, we have determined completely the structure of the equicontinuous minimal flows. These are, in a sense, the simplest minimal flows, so one may inquire whether an arbitrary minimal flow can be "reduced" to an equicontinuous one. That is, can one "divide out"

Of

appropriately so that the resulting factor flow is equicontinuous?

course, the trivial (one point) flow is an equicontinuous factor of any flow, and later in this chapter we consider the weakly mixing flows, which can be characterized as those minimal flows with no non-trivial equicontinuous factor. Our first result is that every flow (not necessarily minimal) has a "maximal" equicontinuous factor, and then we give a reasonably explicit characterization of the equivalence relation which defines this factor.

1. Theorem.

If

(X,T) is any flow, there is a smallest closed

such that the quotient flow invariant equivalence relation S eq (X/S TI is equicontinuous. eq' Proof. Let

S denote the family of closed invariant equivalence

relations S such that

(WS,T) is equicontinuous. Since the one point

flow is equicontinuous X x X Then S*

E

S,

so

S

*

$.

Let

S* = n [ S I S

E

Sl.

is certainly a closed invariant equivalence relation.

Moreover, the flow $* : X+X/S*

(X/S*,T) is equicontinuous. For, if @s : X+X/S,

are the canonical homomorphisms, then (X/S*,T) may be

regarded as a subflow of the product flow

@*(XI-+ [@s(x))s.

lT W S , T SES

via the map

Since subflows and product flows of equicontinuous

flows are equicontinuous, it follows that

(WS*,T) is equicontinuous.

Chapter 9

126

Clearly S*

s*

=

is the smallest such equivalence relation, and therefore

s

eq' is called the equicontinuous structure The equivalence relation S eq relation and the factor flow (X/S T) is called the maximal eq' equicontinuous factor of (X,TI. Obviously, such an "existential" proof as this one cannot tell us much about the equicontinuous structure relation. We can acquire a better understanding by considering the regionally proximal relation. If

(X,T) is a flow, recall that the regionally proximal relation Q,

(or Q)

is defined by

only if there are nets x

-+x, xn '+x'

n

and

Qx = n ( $ 1 ~ {xn},{xn')

E

%I.

Thus, (x,x')

in X, and

Q

if and

in T with

{tn)

(the diagonal of X x XI.

(xntn,xn'tn')+AX

is a closed invariant relation, which contains

see that

E

Ax,

Q

and it is easy to

(X,T) is equicontinuous if and only if Q = Ax.

Thus it would

be reasonable to expect that an equicontinuous flow can be obtained from an arbitrary flow by "dividing out" by the regionally proximal relation. In general, the regionally proximal relation is not an equivalence relation. A simple example, with T = H

is obtained by considering

three concentric invariant circles in the plane which are joined by spirals. Thus it I s necessary at least to divide out by the closed invariant equivalence relation generated by

Q

("closed" so that the

quotient space so obtained is compact Hausdorff, and "invariant" so that the group T acts on the quotient space).

This is indeed

sufficient as the next theorem shows. First we require a lemma.

2. Lemma. Let x.+x. 1

(X,T) be a flow, and let {xi) be a net in X with

Let q E E(X),

Proof. Let

I) E

and suppose x qjx'. i

I$. Then, for

i

1

Then

io, (x,xi)E

(x',xq) E Q. 7,

so

127

The Equicontinuous Structure Relation

3, are

(xq,xiq) E

therefore

(xq,x') E

q. Hence

n

(xq,x') E

a = Q.

VE%

(X,T) be a flow, and let S*

3. Theorem. Let

be the smallest

closed equivalence relation which contains the regionally proximal relation Q.

(That is, the intersection of all closed invariance equivalence

relations containing Q.) Then S* = S eq' 3 S*, since an equicontinuous flow can' t contain eq = S*, it is non-trivial regionally proximal pairs. To show that S eq sufficient to prove that (X/S*,T) is equicontinuous. To this end, we

Proof. Clearly S

prove that its enveloping semigroup E(X/S*)

(See theorem 2 in the "Joint continuity theorems" chapter. 1

phisms. IT:

is a group of homeomor-

X+X/S*

be the canonical homomorphism, and regard the enveloping

semigroup of X as acting on X/S*. We first show that XIS* distal. Suppose

(y,,y,)

E

P(X/S*).

a minimal right ideal of E(X). n(x2) = y2.

I , n(xlu) = n(x2u).

E

X with n(x,)

for all p

E

I,

so if

= yl,

u

2

Hence X/S*

and let q E E(X).

Let

{xi)

with n(xi) = yi, and let (a subnet of) xi+x, subnet of) xiq+x' (x',xq)

E

Q, so

E

so

= n(xlu)

is distal, and it is sufficient to

show that all elements of E(X/S*) are continuous. Let with yi+y

is an

NOW (xiu,xI)E P(X) (i = l , Z ) ,

(since P c Q c S*), n(xi) = n(xiu), (i = 1,2) and y1 = n(x i = n(x u) = n(x2) = yz.

is

Then ylp = y2p, for all p E I,

Let x 1,x2

Then n(xlp) = n(x,p),

idempotent in

in X/S*

Let

X. Then yiq = n(xiq)+n(x').

so

{yi) be a net

be a net in X n(x) = y.

Let (a

By the lemma,

dx') = dxq) = yq. Thus yiq-+yq, and the proof is

completed. Thus the equicontinuous structure relation has been characterized as the closed equivalence relation generated by the regionally proximal relation. As we have noted the regionally proximal relation need not be

128

Chapter 9

an equivalence relation. However, it is a remarkable fact that for a large class of minimal flows (which include those f o r which the acting group is abelian, as well as point distal flows) the regionally proximal relation

an equivalence relation.

(Since it is always closed

invariant, it coincides with the equicontinuous structure relation in these cases.) What we prove here is that f o r a minimal flow (X,T) which possesses an invariant Bore1 probability measure, Q

is an equivalence relation.

We will show in chapter 12 that an invariant measure exists for a large class of acting groups (including abelian groups).

As we showed in the

"Furstenberg structure theorem" chapter, distal minimal flows admit an invariant measure, so our theorem will apply in this case as well.

(Of

course, we also showed directly in that chapter that regionally proximal is an equivalence relation i n distal minimal flows. 1

is an equivalence relation is preceded by a

The proof that Q

sequence of lemmas, some of which may be of independent interest. We proceed somewhat more generally than is necessary for the purposes of this chapter. The added generality is needed for the chapter on disjointness of flows. Let

(Y,T) be minimal flows, and suppose (Y,T) admits

(X,T) and

an invariant measure A.

(In this chapter, we are concerned with the

IY,T) = (X,T).1

case

If N N(x) = [y

is a subset of X x Y and x E

Yl(x,y)

E

E

X, then the "section"

N).

4. L a . Let N be a closed inva,riet subset of X x Y. Then, if x,x'

E

X, A(N(x1) = A(N(x'1).

Proof. Let

c

> 0 and let V be an open set with N(x) c V and

The Equicontinuous Structure Relation A ( V ) < A(N(x)) +

E.

Let

{tn)

be a net in T for which

is easy to see that N(x'tn) c V

for n 2 no.

n

) =

A(N(x'tn))

S A(V)

arbitrary, we have h(N(x'))

< A(N(x)) +

I A(N(x)).

It

x'tn+x.

(This is equivalent to

the upper semicontinuity of the set map XHN(X).)

A(N(x')t

129

=

Hence A(N(x'))

Since c > 0 is

E.

By symmetry h(N(x))

5

A(N(x')),

and the lemma is proved. If N metric

is a closed invariant subset of X

DN on X

by

x

Y, we define a pseudo-

DN(x.x') = A(N(x) A N(x')).

symmetrjc difference: A A B = ( A \ B) u (B \ A);

(Here A

denotes the

it is well known, and

easily proved that DN is a pseudometric.

5. Lemma. The pseudometric DN is T-invariant and continuous. Proof. The T-invariance of DN is an immediate consequence of the assumed T-invariance of the measure A. we first note that if U A(U\N(x)) <

In fact, if N(x0 c U. A(U)

-A(N(x)) = A(U\N(x))

xn+x. Then, if

Let n

DN(x,x ) < n

E,

sufficiently close to x.

for x'

- A(N(x'))

then A(U\N(x')) = A ( U )

<

n N(xn) c U 0'

tion, and A(N(xn)\N(x)) SO

<

c.

Now, let

{x

}

n

=

be a net in X with

> 0, U open in Y with NIX) c U and A(U\N(x))

E

2

DN is continuous,

is an open set in Y, with N(x) c U, and

then A(U\N(x'))

E,

To show that

and A(U\N(xn)) <

I A(U\N(x))

Thus, if xn+x,

E.

< E.

E,

<

E.

by the above observa-

Similarly A(N(x)\N(xn))

< e,

It follows

we have DN(x,x )+O. n

immediately that DN is continuous. Now, let

$

be the equivalence relation defined by

DN

:

(x,x') E

KN

if DN(x,x') = 0. 6. Lemma.

%

is a closed T-invariant equivalence relation which

contains Q. Proof.

The previous lemma implies that

is closed and invariant.

130

Chapter 9

(x,x') f Q, let {xn),{x;l)

If

with xn+x,

x'nt n-+z,

a net in T

for some z E X.

Now

and also D (x , X I ) = DN(xntn,xAtn)+DN(z,z) = 0 , N n n

DN (xn ,x')+DN(x,x') n so we have

and xnt n+ z ,

x'n+x',

{tn)

be nets in X and

DN(x,x') = 0 and

(x,x') E

5.

is the smallest Since the equicontinuous structure relation S eq closed invariant equivalence relation which contains 8 , we immediately obtain

7. Corollary. If N

is a closed invariant subset of X

x

Y,

then

KN.

'eq

Now we specialize to the case Y = X. 8. Theorem. Let

(X,T) be a minimal flow with an invariant Bore1

probability measure h .

Then the equicontinuous structure relation S

eq

coincides with the regionally proximal relation Q. Proof. Since always Q c S it i s sufficient to show that eq' S c Q. Let (x,y) E S and let V be a neighborhood of x. eq eq' Consider the closed invariant subset N of X x X defined by

N =

u

(y,x')T =

u r n .

Hence

x' EV

L

x't n+w,

N = closure (z,w)lytn+ z ,

for some X'E V, and net

Then V c N(y)

closed, V c N(x).

5.

Since V

(x,y) E

S eq'

hood of x, then there is an x' E V and a

t E

yr are in V.

and

is open and N(x)

is

(x,y) E S so (x,y) E eq' For, h(V\N(x)) S h(N(y)-N(x)) 5

A(N(x) A N(y)) = 0, since (x,y) E Summarizing, if

Since V

in T.]

5,

(take tn = el. Now

it follows that V c N(x).

{tn)

and V

is a neighbor-

T such that x ' t

is arbitrary, it follows that

and

(x,y) E Q.

Note that the proof just given shows that in the definition of Q one

The Equicontinuous Structure Relation

131

can take one of the nets in X to be constant. Precisely, we have: 9.

Corollary. Under the hypotheses of the preceding theorem

(x,y) E Q

if and only if there are nets {xn)

with x +x, n

xntn+x

in X and

{tn)

in T

and yt -+x. n

In turn this corollary can be reformulated to yield: 10. Corollary.

If x E X, Q(x) =

n aT(x). "Ilx

That some restriction (either on the group or the action) is necessary in order that Q be an equivalence relation, is shown by the following example, due to McMahon. The space X is generated by the homeomorphisms cp

is the circle, and the group T and @.

The homeomorphism cp

is

a rotation through an irrational angle ( s o certainly (X,T) is minimal). @

-. A

has fixed points at 0, 2

IC,

3H T , and maps the arcs between the fixed

points onto themselves in the same "increasing" manner: where

~ ( 0 >)

0. I t is easily checked that

A

(0,2)

E

#(0) = 0

+

~(01,

n

Q, ( z , n ) E Q, but

( 0 , ~e)Q.

Now we use the technique just developed to investigate the class of flows which are "highly non-equicont inuous. "

The flow

(X,T) is called

weaklv mixing if every non-empty open invariant subset U of X x X dense in X x X*'

is

(as usual, "invariant" is with respect to the product

action (x,y)t = (xt,yt)). mixing, then Q = X x X, so

It is easy to see that if (X,T) is weakly (X,T) has no non-trivial equicontinuous

*'The term "weakly mixing" is used because of an analogous concept in ergodic theory. A measure theoretic system is called weakly mixing if the product system is ergodic. The phrase "topologically ergodic" is sometimes used as a synonym for "topologically transitive" - every invariant open set is dense (so if the phase space is metric this is equivalent to the existence of a dense orbit).

Thus the flow (X,T) is

weakly mixing if and only if the product flow (X x X,T) is ergodic.

Chapter 9

132

factor. In fact, in the class of minimal flows which admit an invariant measure, this property imp1ies weak mixing. For, suppose

is minimal, not weak mixing, and let A

(X,T)

invariant measure. Let U be an open invariant subset of X

U

4 and N = fi

f

*X

x X.

is an open set V with x

Let x E

N(x) = X, i t follows that N Then V7

n

W c N(x71,

E

X.

X

x

XI.

Let

A(N(xT)\N(x))

so

(X,T)

1

7 E

A((V7

X, with

x

is minimal, there

*

Now W = X \ N(x)

V c N(x). =

Since

be an

0

(for if

T such that V-c n W n

W)\N(x))

#

0.

=

h(Vz n W) > 0. Hence D (X,XT) > 0. N be the equivalence relation defined by DN Again, let

5

if and only if DN(x,x’) = 0). Since DN is nontrivial,

((x,x’) E then of

f

and

X x X,

is a non-trivial equicontinuous factor

(X/S,T)

( X , TI.

Thus weak mixing is equivalent to having no non-trivial equicontinuous factor. The next results show that proximality abounds in weak mixing minimal flows. 11. Theorem. Let metric. Suppose

(X,T) be a weakly mixing minimal flow with X admits a regular Bore1 probability measure A .

(X,T)

A be a countable subset of X.

Let

proximal with every a Proof. If x

E

E

Then there is a z

which is

E X

A.

X, then X

=

Q(x) =

n

so

that aT(x)

is

“Ilx open and dense in X. Let {ai}i=1,2 be a countable base for the uniformity 11,

B

ij

and let A = {a1,a2,.. . } .

is open and dense in X,

dense in X I .

so

Let B

n (oliT)(a

1=

1, . i

If z

f

n BiJ’

i,J

clearly (z,a.) E J

ij

= (aiT)(a 1.

J

Then

n BiJ * 9 (in fact 1,J P for J = 1,2,..

12. Corollary. Under the hypotheses of the previous theorem,

..

The Equicontinuous Structure Relntwn

P

2

= X x X,

and P(x)

is dense in X

for every x

133

E

X.

We summarize the characterizations of weakly mixing minimal flows in the following theorem. The equivalence of the statements have either been proved, or follow easily from our discussion. 13. Theorem. Let

(X,T)

be a minimal flow, with

admits an invariant measure A.

X metric, which

Then the following are equivalent:

a)

(X,T)

b)

( X , T) has no non-trivial equicontinuous factor (i.e.,

s

eq

=

is weakly mixing.

x

x XI.

c)

(X,T) has no non-trivial distal factor.

d)

Q = X x X .

e) The proximal relation P f) P

2

is dense in X x X.

= x x x .

P(x)

s dense in X

for every x E X .

h) P(x)

s dense in X

for some x E X .

g)

At present there is no general structure theorem for weakly mixing minimal flows, although certain subclasses have been intensively studied. An example of a weakly mixing flow will be given in the final chapter. Our final result in this chapter is that for a minimal flow with an invariant measure regional proximality can be effected "in any direction." That is, if

then there are nets {x;} {yn},

{xn} , {yn) are nets with xn+x,

(x,y) E Q and

and

respectively, and a net

{y;}

"arbitrarily close" to

{tn}

in T such that

yn-+y {xn}

and

(x;tn,y;tn)4A.

The precise statement follows. 14. Theorem. Let

(X,T)

be a minimal flow with an invariant

probability measure and let neighborhoods W1

of x and

(x,y) E Q. Let

W2

a

E

?$.

Then there are

of y such that if U1 and U2

are

Chapter 9

134

and W2,

non-empty open subsets of W 1

x'

U,, y'

E

E

respectively, then there are

and t E T such that

U2

(x't,y't) E a.

For the proof, we first require a lemma.

Let V1

15. Lemma.

Q(V,)

is open.

Then Q ( V , )

If x1 E V1,

Proof.

V2.

and V2 x

be open subsets of

X and suppose that

c (V, x V2)T.

Q(V,)

construct N

as in theorem 8, using

That is +z,

x't +w, n

for some X'E V2, and some net

Then, as in the proof of theorem 8, if y1 {y,) x Vz c N.

Q(V,)

x

Since x1

V2 c l V l x V2)T.

E

V1

open subset of X.

is open by assumption so we may x

U )T n W

x

2

Let let

n

:

*

We show that i f

U1

and U2

x

V 2 IT.

W be a non-empty

are open sets in X

respectively, then

$.

X-+X/Q

T with

T E

Q(V 2 1 c IV,

(x,y) E Q and let

which are sufficiently close to x and y. (U,

T.]

Q(xl) = S (xl), eq is arbitrary, it follows easily that

Now Q(V,)

Now we prove theorem 14. Let

in

{tn)

E

iterate the above process to obtain Q(V,)

z

x1 and

be the canonical homomorphism, let

zt

E

int n(W)

and let

V

z = n(x) = n(y),

ff

be an open neighborhood of

*

such that V T c int n(W). Let

W1 and W2

n(Wi) c V

Let

*

be open sets in X

(i = 1,2). Let

U1 c W1

and

with x E W1, y E Wz

U c W2 2

and

be non-empty open sets.

n(U 1) ( i = 1 , Z ) . Then n(V,) and Q(Vi) = i ( i = 1,Zl are open. By lemma 15, Q(V1)xQ(V2) c (VlxVZ)T so

V i = Ui n n-'(int

n-'n(Vi)

(Q(V,lxQ(V,))T

c

(U1XU2)T n W *

$,

c

(U1XUZ)T.

Therefore, to conclude that

it is sufficient to show (Q(Vl)xQ(V2)lT

nW

#

$.

Now

The Equicontinuous Structure Relation

135

n(Q(V.)r) c V * z c int n(W) (j = 1,2), so there are x'. E Q(V.1 such J J J -1 that n(x'.z) E int n(W) (j = 1.2). Thus there are x; E I xi)), J -1 :x E I (n(x;)) such that x;r, X*T 2 E W. But x; E Q(x;) c Q(V,) and similarly xa

E

QCV,),

so

(Q(V,)xQ(V,))T

n

W

*

4.

The techniques and many of the results in this chapter, in particular the use of the pseudometric DN and the equivalence relation

%,

due to McMahon (Trans. Amer. Math. SOC. 236 (19781, 225-237).

This is

are

also the case for the results in the disjointness chapter concerning flows with an invariant measure. Exercises. 1.

Determine the maximal equicontinuous factor of the minimal flow which

is the orbit closure of the almost periodic point in {O,l}'

defined in

exercise 9 of chapter 1.

2. Let P be a family of flows (with fixed acting group TI satisfying: (i)

the trivial (one-point) flow is in P.

(ii)

P is productive (any product of flows in P is also in P).

(iii) P is hereditary (a closed invariant subflow of any flow in P

is in P I . Then if (X,T) is any flow, X has a "maximal P factor.

'I

That is,

there is a smallest closed invariant equivalence relation R such that (X/R,T) is in P. 3.

Note that "distal" is a class of flows to which the previous exercise

applies. Show that the distal structure relation is the smallest closed invariant equivalence relation which contains the proximal relation P. (In general, P

is not an equivalence relation for minimal flows, and

even if P is an equivalence relation, it need not be closed. 1 4.

A flow

(X,T) is mixing if given U and V non-empty open sets in

chapter 9

136

X, there is a compact subset K of T such that Ut n V t

E

S.

T

\

f

4,

for all

K. Show that a mixing flow is weakly mixing.

In the case

(X,T) is minimal, give a proof of theorem 3 , which uses

theorem 8 in the “Furstenberg structure theorem” chapter. 6. Show (under the hypotheses of theorem 13) that if

(X,T) is weakly

mixing, with T abelian, then P is not an equivalence relation. More generally, show that the enveloping semigroup E(X)

contains infinitely

many minimal right ideals. 7. Let

(X,T) be a (not necessarily minima 1 weakly mixing flow, with T

abelian. Then any finite product

(X x X x

..

x

X,T) is topologically

ergodic (hence weakly mixing). 8.

Let

Show that

(X,T) be a proximal minimal flow (all pairs are proximal). (X,T) is weakly mixing.

(This problem is more difficult than

the other problems - see Glasner, Proximal Flows, chapter 11.

Note that

for a proximal minimal flow, T must be non-abelian, and the flow (X,T) need not admit an invariant measure, chapter does not apply. 1

so

that the theory developed in this

137

Chapter 10 The Algebraic Theory of Minimal Flows We begin this chapter by reviewing and amplifying some of the contents of the chapter "Universal minimal flows and ambits." We will assume throughout this chapter that the acting group T M = (M,'f)

Let

is discrete.

be the universal minimal flow for T.

minimal right ideal in the enveloping semigroup E ( M ) , i n E M,

the map

p w m p defines a homomorphism of

I

defining property of M,

is isomorphic with M,

If I

is a

then for any

I onto M.

By the

and since M

is

is an isomorphism. Hence we can carry over

coalescent, the map p-mp

I to M.

the semigroup structure of

With this identification, M

can

be regarded as a semigroup, and we can endow it with the structure of a minimal right ideal, as developed in earlier chapters.

In particular, M

contains idempotents -- we write J(M) for the set of idempotents in M and fix u E J ( M ) . Thus it is meaningful to write and p E M.

That is, each ti

defines, ti+p

E

I c E(M),

ti+p,

where

{ti)

is a net in T

is identified with the element of

and

#

it

I is identified with M as discussed

above. If X

is a factor of X'

then, as we have noted a number of times,

we can regard the enveloping semigroup of X' K : X'+X

then n(x')p'

= n(x'p')

as acting on X

for x' E X'

and p' E E ( X ' ) .

Using this convention, together with the identification of M minimal right ideal in its enveloping semigroup E ( M ) , (M,T) p E M,

J

where

{t

J

}

is a net in T

with a

we can think of

as acting on any minimal flow (X,T). That is, if x then xp = lim xt

-- if

E

X, and

such that

Chapter 10

138

tJ+p. xM =

(X,T) is a minimal flow and x

(It follows that if =

E

X, then

X.)

Similarly, the universal T ambit (X,T,x): if x

BT, then x< = lim xt

where J. pT. Of course, since the universal minimal flow M is a E

X and

(@T,T,e) acts on any ambit

E

t.+< E J minimal subset of the flow (pT,T) this shows again that M acts on X. Several arbitrary choices have been made in the above discussion universal minimal set M, a semigroup structure for M choosing a minimal right ideal phism from I to MI

I in E(M)

--

a

(obtained by

and specifying a homomorin M.

and an idempotent u

These will be fixed

throughout the chapter. Let G = Mu. chapter, C

From Lemma 1 (iv) of the second enveloping semigroup

is a group with identity u. G can be identified with the

group of flow automorphisms of M

--

if a

E

G, cp(p) = ap

(p E M)

is

obviously an automorphism of M, and it can be shown that all automorphisms are of this form (see exercise 1). Note that if p a

E

G

Let

and w

E

E

M, p has a unique decomposition as p = aw where

J(M).

In fact, if pw = p.

then p = (pu)w.

(X,x) be a minimal ambit. We will assume throughout the chapter

that xu = x where u

is the idempotent in M

which was fixed above.

The ~rouz,of I! with baseDoint x is defined by

B ( X , x ) = [ a E Glxa =

XI

It is immediate that B(X,x) is a subgroup of G. Thus to each minimal ambit subgroup of G.

If x'

x'p = x. Then if a

E

E

(X,x) with xu = x, we have associated a

X with x'u = x',

P(X,x), x'f3ap-l = x'

then there is p E G with and !?(X,x')

=

pG(X,x)p-l.

That is, a change of basepoint gives rise to a conjugate subgroup of C.

It turns out that the subgroups P ( X . x ) are very useful in the study of proximal and distal extensions.

Recall thatxa homomorphism I I : X+Y

The Algebraic Theory of Minimal Flows

of minimal flows is proximal if whenever

x,x'

then x and x'

71

x,x'

E

are proximal, and that

X with x

and n(x) = n(x'),

x'

#

In these cases, we also say that

X

E

139

X with n(x) = ~(x'),

is distal if whenever

then x and x'

are distal.

is a proximal (or distal) extension

of Y. Note that a homomorphism is both proximal and distal if and only if it is an isomorphism.

Of course, an extension need not be either proximal

or distal. of C determines

Now we consider to what extent the subgroup B(X,x)

the ambit. Obviously isomorphic ambits define the same subgroup, and if X:

(X,x)-+(Y,y)

is a homomorphism then B(X,x)

is a subgroup of

Our first results show that the ambit is determined "modulo a

B(Y,y).

proximal extension." 1.

Theorem. Let

R :

(X,x)-+ (Y,y) be a homomorphism of minimal

ambits. Then B(X,x) = k?(y,y)

is a proximal

if and only if TI

extension.

Proof. If B(X,x) is a proper subset of B(Y,y), p

Then n(xf3) = yp = y = dx),

B(Y,y)\B(X,x).

E

(x,xp) e P, so

n

Then x'u

x6 = x'u,

xs 2.

f

x<,

so

sg-l

but

E

(x',~'')

X with

~[(x'u) = K(x"u).

Let

&

71

is not

P and n(x'f

S.<

E

=

G with

Then y6 = n(xa) = ~(x'u) = 71(x"u) = R(x<) = y< so

-1

Theorem. Let (i)

x"u

#

x< = x"u.

y6C-l = y and 6<

xf3 # x, and

is not proximal. Conversely, suppose

proximal. Then there are x', X"

x(x").

let

B(Y,y).

E

s!

f

X"U,

B(X,X).

(X,x) and

D(X,x) c B(Y,y) extension X'

On the other hand, since x'u

(Y,y) be minimal ambits. Then

if and only if Y

of X.

Is

a factor of a proximal

Chapter 10

140

(ii) B(X,x) = B(Y,y)

if and only if there is a minimal flow X'

which is a proximal extension of both X and Y. Proof. If

z = (x,y) E X x Y

if a

is proximal and

x'

E

X'

with x'u = x'

and

x, then 9(X',x') = 9(X,x) c B(Y,y). If B(X,x) c B(X,y),

=

n(x')

X'+X

TL:

B(x,x),

E

and let X' =

x'

so

then zu = z

B(Y,y),

z.

Now if a

za = z

E C,

is a proximal extension of X. i f and only'if

a

E

let

if and only

If B(X,x) =

B(X,x) = B(Y,y),

so

X'

is a proximal extension of both X and Y. The next few results show that in the case of distality, the group determines the flow. Theorem. Let

3.

(X,x) and

(Y,y) be minimal ambits, with Y

distal. Then B(X,x) c B(Y,y) if and only if there is a homomorphism with n ( x ) = y.

n : X+Y

Proof. By theorem 2, there is a proximal extension X' that Y

is a factor of X'.

That

Y

of

X such

is actually a factor of X

is a

consequence of the following lemma. 4. Lemma. Let

(X,T), (Y,T) and

(X',T)

be minimal flows with

and 9 : X'+Y

(Y,T) distal. Suppose u : X'+X

with u proximal. Then there I s a homomorphism nu =

71:

X-+Y

such that

*.

Proof.

If x

E

X, define n ( x ) = I/J(x')

dx") = x then (x',~'')

$(x")

are homomorphisms

since Y

E

P(X')

so

is distal. Thus

H

where dx') = x.

(#(x'),#(x''))

E

If also

P(Y) and *(x') =

is well defined, and the proof is

completed. 5. Corollary. Let

Then

(X,x) and

(X,x) and

(Y,y) be distal minimal ambits.

(Y,y) are isomorphic if and only if B(X,x) = B(Y,y).

The Akebraic Theory of Minimal Flows

141

The next two theorems concern distal homomorphisms of minimal flows and ambits.

In the chapter on distal flows, we introduced the notion of

an almost periodic &

-- this is a subset A of X such that if

XIA' with range z

z E

A, then z

=

is an almost periodic point of the

(XIA',T).

flow

n : X+Y

6. Theorem. Let

be a homomorphism of minimal flows.

Then the following are pairwise equivalent: (i)

n

is distal.

(ii)

n-l(y)

(iii) If y

n (ivl

-1

is an almost periodic set, for every y E

Y and v

(y)v = n

If y

E

If x E n

R

-1

and it follows that R

-1

M, n -1 (yp) = n-'(y)p.

E

is distal. Let

(y),

(x,xv) E P, so

Now

such that yv = y, then

E J(M)

(y).

Y and p

Proof. Suppose yv = y.

-1

y

Y and v

E

~ ( x v )= yv = y,

(x,x')

that

so

xv = x. That is xv = x, n-'(y)

E

with

J(M)

R(X) = y

for all x

=

E

n

R(XV).

-1

(y),

is an almost periodic set. Conversely, if

x, x'

(y) is an almost periodic set, then if

then

Y.

E

is almost periodic, so

(x,x')

CZ

E R

-1

(y) with x

P. Then n

#

x',

is distal,

and (i) and ( i i ) are equivalent. Now if n-'(y)v

R

is distal and yv = y

c n-'(y)

paragraph that

and if xv = x so

it is clear that n

-1

x

(y E Y,v E j ( M ) )

n-'(y),

E

-1

II

(y)v = n

then

it follows as in the preceding -1

(y). If n-'(y)

= n-'(y)v

then

(y) is an almost periodic set. Hence (i), (ii),

and (iii) are equivalent. Suppose

K

is distal, and let y E Y,

idempotents in M rp = v

with yw = y,

pv = p,

p E

M. Let v and w be

and let

(Enveloping Semigroup, 11, lemma 1, (vi)).

r

E M

with pr = w,

Chapter I0

142

NOW

n n

-1 -1

always n

(ylpr c x (y)w = n

-1

-1

(y)p c n-l(yp),

(yp)r c n

-1

(ypr) = n

-1

(yp)r =

11

-1

-1

(y)w =

-1 (yw) = n (y). By (iii),

Thus

(y).

(iii) again (applied to yp), -1

-1

n

-1 (y), so all the inclusions above are equalities; in

particular n

H

so

(ypl = n

-1

(YIP,

-1 n

we have

( y p ) ~= n -1

-1

(yp)v = n

-1 (yp)rp = n (YIP.

-1

By

(yp), and so

which is (iv).

Clearly (iv) implies (iii) (take p = v

J(M)

E

with yv

y)

=

and the

proof is completed. 7. Theorem. Let flows. Then n

n

:

(X,x)+ (Y,y) be a homomorphism of minimal

is distal if and only if n

-1

(yp) = xg(Y,y)p,

for all

p E M. yu = y. Suppose n

Proof. Recall we are assuming xu = x, so distal. Then always xg(Y,y) c n ( i i i ) of theorem 6, x'u =

x',

x' = xqu and .qu E 9(Y,y). then by theorem 6 (iv), suppose n

a1.a2

E

*

.

xD(Y,y)p,

F(Y,y).

If

Let

8.

II

so if

-1

= xq, H

-1

(yp) = n-'(y)p for all

v

E J(M)

n

-1

for q

(y) then by

E

M,

(y) = xO(Y,y).

p E M,

such that

E

= xg(Y,y)p.

p E M such that

y

= yp.

If p

E

M,

Conversely,

and let x1

*

then

*

-1

Then

-

xa2p,

pv = p,

then

x2 with

*

=

(y

with

(x1, x21 is an almost periodic point, and it

is distal.

be a distal homomorphism of minimal

Theorem. Let K : X+Y

flows. Then H

x'

and we have x1 = x y p , x2

(xl,xz)v = (xl,x2), so follows that

(y) and if x'

Therefore,

-1 (yp) = xB(Y,y)p,

n(x,) = n(x2) = y R -1 (yp) =

H

-1

is

is open and all fibers n

-1

(y) have the same

cardinality. Proof. Let y E Y, xo y,+y.

Let

q E M

n

E

n

-1

(y) and let

with yqn = yn. Let

{y,)

be a net in Y with

(a subnet of) q +q

n

so

The Atebraic Theory of Minimal Flows

yq = y. x'q = x0.

(y)q = n-'(y).

Q

-1

from H

one-to-one so card(n ( y) 1 =

xn+x0

Let and

x' n(x

E

n-l(y)

such that

n ) = n(x'q n )

=

= yq

n

is open.

TI

y, y' E Y and let p

defines a map

card ( n-'

H

Then, if xn = x'qn,

yn. This proves that Let

-1

By theorem 6,

143

-1 (y))

E

M

(y) to card(n

5

with yp = y'. -1 T[

(y').

-1 (y')).

Then xo ++ XOP

Since

s distal, u

H

is

By symmetry

card( H-' ( y' ) 1.

Theorem 8 shows that distal homomorphisms have properties similar to those of covering maps.

Another such property is given in the following

theorem, which generalizes theorem 3. 9. Theorem. Let

(X,x), (Y,y) and

xu = x, yu = y, zu = z . be homomorphisms with <:

(X,x)+(Y,y)

91,

Let

( 2 , ~ )be minimal ambits with

H : (X,x - + ( Z , z )

and

$ : (Y,y)+(Z,z)

distal. Then there is a homomorphism H =

such that

$5

if and only if B(X,x) c B(Y,y).

We omit the proof (see J. Auslander, Homomorphisms of minimal transformation groups Topology, 9 (19701, 195-203.) Now we turn to the question of determining which subgroups of C the groups of some minimal ambit. It turns out that with a topology (the

"t

C

are

can be endowed

topology") and that the subgroups which

correspond to minimal ambits are precisely the subgroups which are closed in this topology.

For this purpose, we need to consider the induced flow on the closed subsets of a space.

is a compact Hausdorff space, let '2

If X

the collection of non-empty subsets of X.

A base for a topology on

is defined as follows. If U1,.. . ,Un are open sets in X, let .. n x {Ul,.. .,Un} = [ A E 2 I A c Ui, and A n U * $, i = 1,..., n]. i i=l

u

collection of all sets {Ul, ..., Un}

denote

'2

The

defines a compact Hausdorff topology

Chapter 10

144

on '2

X 2 , provided with the Hausdorff

(the Hausdorff toDolorrv).

topology, is often referred to as a "hyperspace".

{A 1 J

A net

a E A,

aJ

E

A

in 2'

converges to A

there is an a

j

and a

j

If X

E

A

J

such that

f

'2

if and only if, for every

a.+a,

and if whenever

J

{a,}, then a E A. J d, then so is '2 - a compatible

is a cluster point of the net

is metrizable with metric

metric (the Hausdorff metric) is defined by

h(A,B) = sup[d(x,B) I x E A].

where

As a general reference, see E. Michael, Topolonies on spaces of subsets, Trans. Amer. Math. SOC.2 (19511, 151-182, or Willard, General Topolonu, section 39E.

(X,T) is a flow, there is a naturally defined flow on 2X ,

Now, if if A

X

2 ,

E

and t E T, At = [atla E A].

This flow will always be

denoted by

(2X ,TI. The minimal subsets of

(2X ,TI ( s o called quasi-

factors of

X) will be studied later in the book in connection with

d i sjo i ntness. As was mentioned at the beginning of this chapter, the universal

ambit

(PT,T,e) acts on any flow.

-- if

<E

A ' < .

(3T and A

If tn+<

E

then x

A such that a t +x. nn general

A< =

If A c X

X

2 , E

the action of

<

on A

acts on

(ZX,T)

will be denoted by

if and only if there is a net

{an)

in

This "circle" notation is necessary since in

[a<(a E A] (A

<'A

In particular PT

T

is a proper subset of

.<'A

not necessarily closed) define Ao< =

above criterion for x E <'A

pioc;. Clearly

still holds.

Now we can define the -c-topology on G. This is accomplished by defining a closure operator. If A c G = Mu, define

the

The Algebraic Theory of Minimal Flows 0

cl A = Aou n G = (A u)u. 7

Note that

then g E cltA

T with tn+u

is a closed subset of M

Aou

(in

but is not in general a subset of G.

the usual topology of M) If g E G,

145

if and only if whenever {tn}

then there is a net

in A

{an)

with

is a net in

lim antn = g

(convergence in MI.

To see the clt is indeed a closure operator, note that if A c G,

A = Au c A0u n G'= cl (A), Also

t 0

cltA c clt(cltA) = (Aou n G)Ou n G c (A0u)Ou n G = A u n G = cltA,

cl (cl A) = cltA. Finally, if A,B c G,

so

clearly cl A c cltB.

and if A c B c G,

t

t

clt(A u B) =

(A u B)Ou n G = (Aou n G) u (Bou n G) = cltA u cltB. 10. Theorem, The t-topology on G

is compact and T1.

Proof. First note that if A c G, xu c clrA closure of A {Ai)

E

t

t

By compactness of M,

i

u c cltAi = Ai.

show it is T1, let

11.

Theorem.

closed subgroup of Proof.

p

E

G(X,x).

and f3

{A.1 }

E

G

with the finite

there is a p

E

nxi.

Then for every

i,

T-topology is compact.

To

Then clt{a) = {a}'u n G = {a).

(X,x) is a minimal ambit, then B(X,x)

is a

t

G.

It is sufficient to show that if {an) and {tn) and a t + p n n

Now xant n-+xg and also xantn = xtn+xu

are nets in

E G,

then

= x,

so

xp =

x

B(X,x).

12. Theorem. Let A be a t-closed subgroup of the minimal flow defined by Aou ( X = (A0u)T). X

Now let

has the finite intersection property

This proves that the

a E G.

If

closed subsets of

and T respectively with tn+u

B(X,x)

0

-0

intersection property. Then

pu

denotes the

since Tu c A u n G = A u n G = cl A.

in M)

be a collection of

in M.

-

(where A

G,

and let

Then B(T,Aou) = A,

X

be and

is maximal f o r this property -- that is, if (X,x) is a minimal ambit

Chapter 10

146

with B(X,x) = A,

then X

is a proximal extension of

0

0

Proof. We show A a = A u 0

(a E G)

0

X.

if and only if a

0

E

A.

0

A a = A u , then a = u a E A o a = A u , so

a c A u n G = A .

If

If ~ E A ,

then A0a = Aa-loa c (A0a-1 ) 0a = A0u and similarly A0a-1 c Aou Therefore B(X,Aou) = A.

Aou c Aoa c Aou.

is a factor of X ,

we show that the map 0

0

defined. Suppose A p = A q there are nets {an) n n

(p,q E MI.

(p E MI. is well

Aopc-,xp

Then p = up

E

A0p

{sn) in T with sn+q

in A and

and also xa s = xsn+xq, n n

Then xu s +xp n n

a s +p.

To show that X

(X,x) be a minimal ambit with O(X,x) = A.

Now let

so

so

= A

0

q,

so

and xp = xq.

Thus we have established a one to one correspondence between r-closed subgroups of G and "proximal equivalence classes" of minimal ambits. Exercises. 1.

a) Let

(X,T) be a minimal flow. Show that every homomorphism from

(M,T) to

follows that every automorphism of p w g p , for some g b) Let cp

E

(M,T) is of the form

G.)

be an endomorphism of the minimal flow (X,T). Show that

cp

is "induced" by an automorphism of M.

That is, if

is a homomorphism then there is an automorphism a of that

(It

(X,T) 1s of the form p ~ x p ,for some x E X.

cpo =

Q:

M+X

M such

ua.

2. Discuss the effect of changing the choice of idempotent on the group of a minimal flow. That is, if v

O'(X,x')

= [a'

E

G'lx'a'

=

x'l.

E

J(M),

G' = MV

How is P(X',x')

and x' E Xv,

let

related to B(X,x)

(where xu = x)? 3. Let A c G.

4. Let

Show that c17A = (Aou)u.

(X,x) be a distal minimal ambit, and let

B be a 7-closed

The Algebraic Theory of Minimal Flows

147

subgroup of G with B(X,x) c B. Show that there is distal minimal ambit

(Y,y) with B(Y,y) = B and

5. A minimal flow which

(Y,y) a factor of

(X,T) is called regular if for every x,y

(x,y) is an almost periodic point in

E

X for

(XxX,T) there is an

endomorphism (equivalently an automorphism) cp

of

(X,T) such that

(Compare exercise 3 of chapter 8 . )

cp(xf = y. (i)

(X,x).

Suppose

(X,T) is regular. Show that every a

automorphism cp

a

of X by

sufficient to show: if p,q

cpa(xp) = xap

E

G

defines an

(It is

(P E MI.

E

M

E

X with xu = x,

with xp = xq

then

xap = xaq. 1 (X,T) is regular and x

If

(ii)

a normal subgroup of G.

X be a minimal flow.

6. Let

flow XR

then XR (i)

is

What about the converse?

X is a regular minimal

A regularizer of

X is a factor of XR

(with the same acting group) such that

and such that if Z

show B(X,x)

is a regular minimal flow which has X as a factor,

is a factor of Z .

Show that the regularizer exists and is unique ( s o we may speak of

the regularizer of

a minimal flow).

(Note that a regular

minimal flow is coalescent. A construction of the regularizer is obtained as follows. Let set, and let z (ii)

Let

E

A c X be a maximal almost periodic

XIA' with range z = A.

I be a minimal right ideal in E(X).

Then XR = Then

I

z.1

(regarded as

a minimal flow) is the regularizer of X.

(iii) If G ( X , x ) = A,

7. Suppose

II:

XjY

Then, if n(x) = n(x'), peU[III

what is the group of the regularizer? is a proximal homomorphism of minimal flows. xp = x'p

for all

a minimal right ideal in E(X)I.

Chapter I0

148

a.

Let

(X,x) be a minimal ambit with xu = x, and

i)

If

(the normalizer of A) (ii)

show that there is a y E N(A)

is an automorphism of X,

(p

If

E

N(A)

show that

let A = G(X,x).

such that (p(x) = xy. y

induces an automorphism (as in (i))

on a proximal extension of X. 9. Let

(X,x)+(Y,y)

TI:

be a homomorphism of minimal ambits. Then

is distal if and only if

H

is open and

TI

-1

(y) = xB(Y,y).

10. Here is an alternate description of the

p

E

G, then left multiplication by p

phism of M.

Let

c

M

let TK =

K c C,

and i f

rP

x

a) Show cltK = [ y E Glry c

rK

r-topology on G.

( rt -+Pr)

M be the graph of

u"plP

E

FK]

If

defines an automor-

P,

I'

P

= [(r,l3r)Ir E MI,

K1. (where

TK

denote the closure of

in M x MI.

Use this characterization to prove that multiplication in G

b)

TI

is

(separately) continuous and that inversion is continuous. 11.

Let

n : X+Y

the subset of 2'

be a homomorphism of minimal flows, and let

X

- 2"+Y

for some y c YI.

is a closed invariant subset of the flow

be the homomorphism defined by ;(A)

71:

= y

X ( 2 ,TI. Let

if A c n-'(y).

Show that the following are equivalent. (i)

n

(ii)

Every A

(iii)

n

(iv)

II

be

defined by

2" = [A E 2 I A c n-l(y),

Note that 2n

2"

is an equicontinuous extension E

2"

is an almost periodic point of

is distal. is equicontinuous.

(2",T).

149

Chapter 1 1 Disjointness Disjointness, which was defined by Furstenberg together with a corresponding notion in ergodic theory, is an independence condition between two minimal flows. In one sense, disjointness may be thought of

as analogous to relative primeness in number theory.

(For s w e time it

was thought that disjointness of two flows was equivalent to their having no common factor.) Disjointness is also a useful way of looking at entire classes of minimal flows.

In fact, an interesting problem is to

determine the flows disjoint from a given class of minimal flows. From this point of view, disjointness may be regarded as being analogous to an orthogonal complement. (X,T)

We now proceed to the formal definition. The minimal flows and

(Y,T)

are said to be dis.ioint if the product flow

(XxY,T) is

minimal. We use a perpendicular sign to indicate disjointness, and write o r just

(X,T) I (Y,TI

(If X

X I Y, when the acting group T

and Y are not disjoint, we write

X L Y. 1

As an example, consider two irrational rotations

circle K

R (x) = x + B ma + nB

@ Z

(regarded as the real numbers modulo /3,

where a and

11,

1.

1

Hence the cascades

Proposition.

Y’.

R,(x)

and

R

B

o f the

= x + a,

--

unless m = n = 0. Then, as we showed in the chapter on

and Y’ factors of

X’

Ra

are rationally independent

equicontinuous flows, the product flow on the torus, is minimal.

is understood.

(i)

Let

(K,Ro,) and (K,R ) B

(x,y)+(x+a,y+P), are disjoint.

X, X’, Y. Y’ be minimal flows with X‘

X and Y respectively, and suppose X

I Y.

Then

Chapter I I

150

Disjoint minimal flows have no non-trivial common factors.

(ii)

Proof.

(i)

If X

x

Y is minimal, then

If X I Y and Z

(ii)

is its factor X'

is a common factor, then by (i)

this is impossible unless 2 = 1, Az

so

x

Y'

2 I 2.

But

the trivial flow (since the diagonal

is a closed invariant subset of

Z x 2).

As we will see later in the chapter, the converse to (ii) is false,

although it is valid in a number of cases. It is convenient to introduce a related notion. The flows (Y,T)

are said to be weakly dis.ioint if the product flow

point transitive (has a dense orbit). weakly disjoint.

(X,T)

and

(XxY,T) is

Obviously, disjoint flows are

On the other hand, a (metric) weakly mixing minimal

flow is weakly disjoint from itself, so weak disjointness does not imply disjointness in general.

Moreover, if X and Y are weakly disjoint

minimal flows and one of the flows is distal, then (corollary 16 of the second enveloping semigroup chapter) the product flow X x Y

is both

point transitive and pointwise almost periodic, hence minimal. Summari z ing , we have proved 2. Proposition. (i)

A metric weakly mixing minimal flow is weakly

disjoint from itself.

(ii) Y

If X

and Y are minimal flows with X

distal, then X and

are disjoint if and only if they are weakly disjoint. Other conditions under which disjointness and weak disjointness are

equivalent will be presented later (theorem 9 ) .

It turns out that in many cases, weak disjointness of two miminal flows is equivalent with disjointness of their maximal equicontinuous factors.

In order to prove this and related results, we recall some

developments from the chapter on the equicontinuous structure relation.

Disjointness

Suppose

(X,T) and

151

(Y,T) are minimal flows and that If N

an invariant measure A .

(Y,T) admits

is a closed invariant subset of X x Y,

then, if x E X, N(x) = [y E Yl(x,y) E N1.

The next three lemmas are

lemmas 4, 5, and 6 of the equicontinuous structure relations chapter. X, A(N(x)) = A(N(x'1).

3. Lemma.

If x, x'

4. Lemma.

If DN is defined on X x X by DN(x,x') = A(N(x)AN(x')),

E

then DN is continuous and T 5. Lemma.

((x,x') E and Q c

% %

invariant.

is the equivalence relation on X

If KN

if DN(x,x') = 0), then

%

defined by DN

is closed and T

invariant

(where, as usual, Q denotes the regionally proximal

relation). 6. Lemma. Let

x

E

Proof. Here N

is a subset of the flow = 0.

to show that A(U\N(x))

A(W\N(x))

X, W open in Y, and let N = ({xO)xW)T.

S

A(N(xo)\N(x))

7. Theorem. Let

Now W c N(xo)

5 A(N(xo)AN(x))

(X,T) and

phase spaces and suppose that measures.

Let

Then W c

%(xo).

E

xo

Then (X,T) and

Y

(Xx2 ,TI. It is sufficient (since {xo)xW c N),

so

= 0.

(Y,T) be minimal flows, with metric

(X,T) and

(Y,T) admit invariant

(Y,T) are weakly disjoint if and only if

their maximal equicontinuous factors are disjoint. x E X, there is a dense subset Ax

of Y

In this case if

such that if a E Ax,

(x,a)

is a transitive point of X x Y.

Proof. Here and elsewhere in this chapter, we write X

eq

for the

maximal equicontinuous factor of the minimal flow X. If X and Y

Y

eq'

are weakly disjoint, so are their factors X and eq Since X x Y is pointwise almost periodic, their product is eq eq

Chapter I I

152

minimal, and X I Y eq eqTo prove the converse, it is sufficient to show that if X and Y eq are disjoint, then X and Y are weakly disjoint, (use proposition 2 ) . X I Y, and let U and V eq X and Y respectively. Let x E X, let W So, suppose

and let N = (0xW)T. that

{x‘)

x

W

c

If x‘

E

Q(x),

be non-empty open sets in f

0

be an open set in Y,

it follows from lemmas 5 and 6

N.

By theorem 15 of chapter 1 int n(U) x V is TI:X-+X/Q = X eq’ a non-empty open subset of X x Y. Let w E W. By the assumed eq and Y, (TI(x),w) is a transitive point of disjointness of X eq X x Y, so there is a t E T such that (x(x),w)t E int n(U) x V. eq Hence (x’,w)t E U x V, for some x’ E Q ( x ) . As noted above, Let

(x’,w) E {x’) x W c N, definition of N, (x,w’)s E

and by invariance of N,

there are w’ E W

and

s E T

(x’,w)t E N.

By

such that

u x v.

Now, let A

u,v

= [a E Yl(x,a)t E U x V,

AU,” is open, and since W Y, it follows that A

was

for some t E TI.

Obviously,

chosen as an arbitrary open subset of

is dense in Y.

u,v

If {U } , { V } are countable bases for the open sets of X and i J respectively, then each is open and dense in Y, so Ax AUi,v i , j=1,2,. n , . AUi, v

is non-empty, in fact dense in Y.

(x,a)T is dense in X x Y so

If a

E

Y

Ax,

(x,a) is a transitive point, and the

proof is completed. 8. Corollary.

Suppose

(X,T) and

(Y,T) are minimal flows

admitting invariant measures, with X and Y metric, and suppose is weakly mixing. Proof.

(X,T)

Then

(X,T) and

(Y,T) are weakly disjoint.

X = 1, so eq

Xea I Y eq‘

By theorem 7, X and Y are weakly

Disjointness

153

disjoint. 9. Theorem. Let

(X,T) and

(Y,T) be minimal flows. In the

following cases, weak disjointness of X

and

Y

is equivalent to

disjointness. (i)

(ii)

One of the flows is distal. The flows admit invariant measures, have metric phase spaces,

and one of the flows is point distal.

(iii)

The acting group T

is abelian, and in one of the flows

proximal is an equivalence relation. Proof.

(ii)

This is proposition 2, (ii).

(i)

If X

is point distal and x E X

is a distal point, then

(x,y) is an almost periodic, for every y

E

Y

(theorem 15 of the second

enveloping semigroup chapter).

E

Y

be such that

Now let y

transitive point (theorem 7). Thus X x Y

periodic, s o (iii)

N1 and N2

(x,y) is both transitive and almost

is minimal.

z = (x,y) be a transitive point of X

Let

are minimal subsets of X x Y.

and let

(xu2 ,y) = zu2

E

N2,

yul = y, yu2 = y.

(x,xul)E P(X),

(xu1,xu2)E P(XI2 = P(X). (zu1,zu2) E P(XxY1.

But

Now

Y, and suppose I2

such that

u1 and u be idempotents in I1 2

respectively such that

x

I 1 and

Let

right ideals in the enveloping semigroup E(XxY)

z12 = N2

(x,y) is a

and

(XU,,~) =

(x,xu2) E P(X),

be minimal

zI1 = N1, I2

zu 1

N1'

so

It follows immediately that zu1

E

N1, zu2 E N2,

so

N 1 = N2,

since points

in distinct minimal sets cannot be proximal. Thus X x Y has a unique minimal set, and since T

is abelian, X

x

Y is minimal.

(Note that if

(xo,yo) is an almost periodic point, then so are all points of the form (xot,yOs) (t,sE TI, so the almost periodic points are dense.)

Chapter 11

154

Therefore X and Y are disjoint. 10. Corollary. Suppose

(X,T) and

(Y,T) are minimal flows

admitting invariant measures with metric phase spaces. Suppose X weakly mixing and Y

is

is point distal. Then X I Y.

Proof. Use corollary 8 and theorem 9 (ii). (X,T) is a flow, recall that a quasi-factor of

If

X is a minimal

subset Z of the "hyperspace" (2', TI. is a minimal flow, and X

Proposition. If X

11.

quasi-factor of X, then X

,L

X.

Proof. Consider the subset L L = [(x,A)

E

X

x

L

Xlx E A ] .

subset of X x Z. Since Z A t X,

so

and

yu = y. (i)

of X x Z defined by

is obviously a non-empty closed invariant is non-trivial, there is an

if x E X \ A, ( x , A ) Q L, and

12. Theorem. Let Let

is a non-trivial

(X,x) and

A = O(X,x)

L

*

X

A E

with

X x X.

(Y,y) be minimal ambits with xu = x

and B = g(Y,y).

Then

The product flow (XxY,T) has a unique minimal subset if and only if AB = G. If Xuou = X, then X.1 Y

(ii)

If the group T is abelian or if one of the flows is point

(iii)

distal, then X I Y Proof. X

x

if and only if AB = G.

(i)

i f and only i f

Suppose G = AB,

and let N

be a minimal subset of

Y. It is sufficient to show that N n (x,y)T

that

x ' u = x'

and

(x',y)

E

(xl,yl)E N, and let p E M

N.

G,

and let g = a@,

f

0.

Let

(To see that such an x'

with ylp = y. Then

(xlpu,y) E N, so we can take x' = xlpu. g E

AB = C.

with a E A ,

f3 E

x'

E

X such

exists. let

(x pu,ylpu) = 1

Now, let x' = xg, with

B. Then x'

=

xg = xa@ = xp,

Disjointness

and

(x',y) = (xP,yfo = (x,yl/3 E (x,y)T. Conversely, suppose

has a unique minimal subset (xg,y) E N

periodic, so h

E

155

N, and let

are therefore

g E

G. Then

and g = gh-1 h (ii)

unique minimal subset N. ntn'

periodic, so

where xn

Let E

(xn,y) E N,

lim(xntn,ytn) E N. (iii)

for some

gh-' E A,

so

h E B,

AB.

E

Suppose Xuou = X and AB = G.

X' = lim x

(xg,y) is almost

(xg,y) = (x,y)h,

G. Then xg = xh, xgh-l = x and yh = y,

(XxY,T)

x'

E

Xu, and and

X x Y contains a

By (i)

X. Then x' tn+u.

Then

(xntn,ytn) E N.

Therefore X x {y) c N,

Xuou so

E

(xn,y) is almost

Hence

X x Y c N,

and

It is easy to see that in both cases Xu

Xuou = X, and therefore (iii) follows from If T

independent proof.

(x',y) = so

X

I

Y.

is dense in X, so

(ii).

We also give an

is abelian, then the set of almost periodic

points is dense (as was noted in the proof of theorem 9). Hence, if (XxY,T) has a unique minimal set N, Similarly, if

N = R = X

Y, and X

I

idempotent in M,

is a distal point, for all

t

E

E

T. Hence if v

x tv = x t, and it follows that 0

X is a distal

0

13. Corollary.

If the acting group T

is preserved by proximal extensions. and @ : (Y',T)-+(Y,T)

t

E

T.

is abelian, then disjointness

That is, if

are proximal, then

is an

(xot,y) is an

almost periodic point in X x Y for all y E Y, and all

if

Y.

(X,T) is point distal, we show that the almost

periodic points in X x Y are dense. For, if xo point, then xot

I

ic:

(X',T)+(X,T)

(X,T) I (Y,T) if and only

(X',T) I (Y',T). Proof. By theorem 2, (ii) of the "algebraic theory" chapter, a

minimal flow has the same group as a proximal extension. Now apply the previous theorem.

Chapter 11

156

Now we introduce highly proximal extensions, which, as the term indicates are a subclass of the proximal extensions. As we shall see, disjointness is preserved under these extensions. The concept of high proximality is also useful in the consideration of the relation between disjointness of two flows and the property of having no common factor, as well as in determining the flows disjoint from certain classes of minimal flows. Let

X and Y

be flows with

homomorphism. We say that

n : X+Y

Y minimal and et

be a

is highly proximal if some fiber is shrunk

H

uniformly to a point by a net of group elements. Precisely, highly proximal if there is a y E Y, and x E X and a net such that

n

-1

in T

{tn}

(Convergence is with respect to the Hausdorff

n-l(y)tn+{x}.

topology in 2 ' ) .

is

Equivalently, if x E n-l(y)

and p

M, then

E

(YlOP = {xp).

If a homomorphism is highly proximal, the defining property holds for all fibers. For, if y' as above. Then n-'(y')oq so there is a net

If X

isj)

E

Y, let q c II

-1

(y),

E

M

and

in T such that

with y'q = y, where

71

is

c n-'(~)~p = {xp},

n-'(y'Ioqp

-1

y

(y'ls -+{xp).

is also minimal, it is easy to see that

J

proximal if and only if every non-empty open set U

is highly

X+Y

H :

in X contains a

complete fiber 71-1 (y). Note that highly proximal is a purely relative notion "highly proximal flows. " so

(if X

For if

n : X+ 1, then n-'(lltn

is non-trivial) rr-l(l)tn

is not highly proximal.

H:

X+1

there are no = Xtn = X

X.

cannot approach a singleton in 2

A highly proximal extension is obviously proximal.

proximal minimal flow, then

--

If X

is a

is a proximal homomorphism which

In the final chapter, we will give examples of

proximal extensions of minimal cascades which are not highly proximal.

Disjointness

157

The almost one-to-one extensions are a subclass of the highly proximal extensions. The homomorphism x : X+Y

is almost one-to-one or almost

automorphic if some fiber is a singleton

--

x

E

X, y

A

-1

(y) = {x},

for some

Y. Obviously, an almost one-to-one extension is highly

E

proximal. The “two circle” minimal set is an example of a highly proximal extension of an irrational rotation of the circle for which every fiber consists of two points.

In the case of metric phase spaces,

every highly proximal extension is almost one-to-one (exercise 5 ) . According to corollary 13, disjointness is preserved under proximal extensions if the acting group T

is abelian.

In fact, disjointness is

always preserved under highly proximal extensions. To prove this, we require some notation and a lemma. If

A: 1

XjY

then 2x

is a homomorphism of (not necessarily minimal) flows,

will denote the set of A E 2’

such that

n(A) = Y

(thus

I

2x n

-1

is the collection of closed subsets of X which meet every fiber (y). 1 14.

Lemma. Let

X, X’ and Y be minimal flows and let A : X‘+X

be a highly proximal extension. Let $(x’,y) = (n(x’),y).

I(, =

k x

id: X’ x Y-+X

Y,

x

Then

I

(i) (ii)

= {x’ x Y}.

2’

If X x Y has a dense set of almost periodic points,

X‘ Proof.

x

so

does

Y.

(i)

Let

W

# 0

be open in X’ x Y.

Then W

3

U’

x

V, where

U’ and V are non-empty open sets in X’ and Y respectively. Since A

y E

is highly proximal, there is an x E X

v, (if)

$-l(x,y) c U’ x Let A

and

A’

v

c

with n-l(x) c U’,

so

if

w.

denote the almost periodic points of X x Y

Chapter I I

158

and

X'

A'

2*

E

and

=

X

X, X', Y and Y'

Proof. 14 X'

x

I

Y respectively. Then X

1

Y, then X'

Y has a dense set of almost periodic points. id: X' x Y+X

x

so

and I

Y

Y'.

It is sufficient to show if X

homomorphism n

Y,

be minimal flows with X'

highly proximal extensions of X and

if and only if X'

x

Y, by (i).

x

15. Theorem. Let

Y'

@(A') = A = X

Y respectively. Then @ ( A ' ) = A ,

x 1

x

Y

is proximal

1

Y.

By lemma

Since the

X' x Y

is minimal.

We may call two minimal flows "highly proximally equivalent" if there

is a third minimal flow which is a highly proximal extension of each of

(It is not difficult to show that this is in fact an equivalence

them.

relation.) Thus the theorem just proved shows that the disjointness of two flows depends only on their highly proximal equivalence class. 16. Theorem. Let let

I[:

X+Y

be a homomorphism of minimal flows, and

Y' be the closed invariant subset of the flow 2X ,

-1

Y' = closure [ n (y)Iy of Y'

homomorphism u B c n-'(y),

Y].

Then Y'

Y' -- that is,

A E

onto Y defined by

Y and let A

E

Let

Y'

and if B

there is a net

so

(Since Y'

( y l ~ y ,Y#

if B

E

Y'

with

E

with B c A,

Y'

then B = A.

(The

{yi)

in Y with n-'(yi)-+A.

and let Bi E 'Y

Now

with

projects onto the minimal flow Y via the map

also pro- jects onto Y, and such Bi always exist. )

(a subnet of) B i + B * E Y# .

*

= y.

be an inclusion minimal element of

let Y# be a minimal subset of Y',

n

dB)

follows by a Zorn's lemma argument.)

A

Since A E Y'

Bi c n-'(yi). -1

has a unique minimal set. The

is highly proximal.

Proof. Let yo

existence of

E

defined by

B = A, since

A

Since n-l(yi)+A,

is inclusion minimal.

we have B* c A,

Disjointness

That is, if Y#

159

is a minimal subset of Y',

A

Y#. Since minimal

E

subsets of a flow are disjoint or equal, there is only one minimal set in Y'.

Note that if

y

#

Y, Y = [~-'(y)'plp

E

certainly a minimal subset of

-1

n

0

*

B

(yo)ti-+A. Let c

A

so

*

B

=

Bi c n

-1

(since this set is

be an inclusion minimal element of

as in the preceding discussion. Let

Y'

MI

Y').

yo E Y and let A c n-'(y

Let

E

{ti)

(yo) and let

be a net in T with Then, as above,

Biti+B*.

*

A and Biti+B . This proves that a is highly

proximal.

It follows that the unique minimal subset 'Y proximal extension of 17. Theorem.

of Y

is also a highly

Y.

Let X and

Y be minimal flows with X C Y. Then

*

there is a highly proximal extension X

*

of X such that X

has as a

Pwtor a non-trivial quasi-factor of Y. Proof. Let xu = x

u be an idempotent in M,

and yu = y. Let

phisms defined by

X*

Y

= [r-'(x)

7 :

M-+X

and 6

~ ( p )= xp, 6(p) = yp

0

plp E MI

M+Y

E

X, y

sufficient to show that

-1

a(;y

(XI 0u)

*

6(r-'(x))

then if y' E Y, there is a p E M

(p E MI.

Then, by theorem 16,

c 6(r-'(x)),

X

x

and

Y. To show that so

it is

Y. In fact, if &(r-l(x)) = Y, with

xp = x

and

{XI x Y and it follows that (x,y)T = X x Y.

(x,y)u = (x,y), this says that

Y with

E

be the homomor-

is clearly a quasi-factor of

Y is non-trivial, note that

3

:

x

is a highly proximal extension of X

0 = [ 6 ( 7 - ' ( x ) p)lp E MI

(x,y)T

and let

yp = y',

so

Since

Y is minimal, and so X

I

Y,

contrary to assumption. Several of the following results are concerned with regular minimal flows and the regularizer of a minimal flow. The definition and

Chapter 11

160

properties of these flows are in exercises 5 and 6 of the "algebraic theory" chapter. (X,T) be a distal minimal flow, and let X

18. Lemma. - Let

Then X

quasi-factor of X.

is a factor of the regularizer of

Proof. We first show that X

u

with Aou = A.

(the set of idempotents of MI

J(M)

E

A E X

is distal. Let

be a X.

and let

v

Let

E J

MI,

and let B = Aov.

Since X

A = Av c Aov = B.

Similarly B = Bu c Bau = (Aov)ou = Aovu = Aou = A,

A

and

B. That is, if A

=

Therefore T Now fix A by

A

E

= Aq

X and v

Suppose xp = x, for all x E X. E

M

with pq = qp = w.

c Aoq, and Aop c Aoqp = Aow = A, Hence Aop = A

Let

defined

w E J(M)

Then A = Ap c Aop. so

(by the first part of this

and a similar argument shows that if xpl = xpz

for all x E X, then Aopl = Aop2. to X

so

then Aov = A.

x E X, xu = xpw = xp = x and xq = xpq = xu = x,

proof).

M

E J(M),

X and consider the homomorphism from M to X

pw = p, and let q

Now if

E

xv = x, f o r all x E X, and

is distal.

p ~ A o p (p E M).

with

is distal

It follows that the map p-+Aop

induces a homomorphism of the enveloping semigroup E(X)

T. (Since X is distal, E(X) is onto.) But since X

of to

is minimal and the map from M to E(X)

is distal, E(X)

is isomorphic with the

regularizer of X. This completes the proof. 19. Corollary. If

(X,T) is a distal minimal flow, and T

quasi-factor of X, then 20. Corollary. If

is a

(X,T) is distal.

(X,T) is a regular distal minimal flow, and T

a quasi-factor of X, then X

is

is a factor of X.

Now we consider the relation between disjointness of two flows and the property of the flows having no common factor.

Disjointness 21.

Theorem. Suppose

(X,T) and

161

(Y,T) are minimal flows with

(Y,T) regular and distal. Then X I Y

if and only if they have no

common factor. Proof. As we noted earlier (proposition 1 ) d sjoint flows have no common factor. Suppose that

X and Y are not disjoint. By theorem

*

17, there is a highly proximal extension X quasi-factor Y of Y such that

of X and a non-trivial

* Y is a factor of X . Since Y is

distal, it follows from lemma 4 of chapter 10 that Y X. But since Y Thus

is regular,

is a factor of

Y is a factor of Y, by corollary 20.

Y is a common factor of X and Y, which is a contradiction.

22. Corollary. Suppose the acting group T Then X I Y

(X,T) and

is abe ian.

(Y,T) are minimal flows, where

Suppose

if and only if X and Y

(X,T)

is equicont inuous.

have no common factors.

Proof. An equicontinuous flow with abelian acting group is regular and distal. 23. Theorem. Suppose

abelian, and X and Y X I Y

(X,T) and

metric.

(Y,T) are minimal flows with T

Suppose

(Y,T) is point distal. Then

if and only if X and Y have no common factor.

Proof. Suppose X and Y have no common factor. Without loss of generality X

is not weakly mixing (corollary 10). Then X eq

and

Y eq

I Y are non-trivial and have no common factor. By corollary 22 X eq eq' By theorem 7, X and Y are weakly disjoint, so by theorem 9 ( i i )

x1

Y.

Minimal flows with no common factor need not be disjoint.

In fact, an

example of this phenomenon can occur with finite acting group T.

If T

is a finite group, then all minimal sets are of the form (T/H,T), where

Chapter 11

162

T/H

denotes the space of r i g h t c o s e t s

(Ht,s)+Hts.

is given by

a f a c t o r of g E G,

(T/H,T),

(t

r ( H t ) = Kgt

E

every

g

and

E T,

T-'L~

is a l l of

T.

and

(Tb,T)

i f f o r every

r

T

E

(T/H,T)

T

H c g-lKg.

g, ;r E T.

t

generated by

r

Q

and

and

T,

but

gHg-'

and

T

u T.

generate

T

odd permutation Now, let

L

gHg-'

7Ly-l

(T/L,T)

t h a t is, i f

generate

is i n G ,

r H

so

A4 G = S

be the subgroup of

T

For if

G

tLt-l

HL

* T.

is a

T

contains all three

G

is of index

A4, in

2

the S4,

and the

4'

generated by a f o u r c y c l e T.

If

r,

g , t E T,

are a l s o generated by a f o u r and t h r e e c y c l e

have no common f a c t o r .

elements, and s o

for

is t h e subgroup

be t h e subgroup generated by a three c y c l e and

T,

is a f o u r c y c l e , and

it can be checked t h a t

A4 c G. .But

H

t h e symmetric group

T = S 4'

r e s p e c t i v e l y , s o t h e subgroup they generate is a l l of and

and

and subgroups

I t is w e l l known t h a t t h e t h r e e c y c l e s generate

a l t e r n a t i n g group, s o

then

T,

#

W e first note t h a t if

t h r e e cycle, then

and l e t

g-'Hg

He = H r ,

such t h a t

L

E

generated by

An example is provided by

on f o u r letters.

cycles.

Therefore, the

fT/L,T) are d i s j o i n t i f and only

and

t h e r e is an

HL

f o r which

every

f o r some

I t is easy t o see

TI.

Hence i t is s u f f i c i e n t t o f i n d a f i n i t e group

L

is

HL = T.

and only i f

and

(T/K,T)

have no common f a c t o r if and only i f , f o r

t h e subgroup of

Now t h e minimal flows

If

n(H) = Kg,

n : T/H-+T/K, then

and

t h a t t h i s map is w e l l defined i f and only i f

(T/H,T)

T

and t h e a c t i o n of

is f i n i t e i t is equicontinuous.

and i t follows t h a t

flows

TI

E

This a c t i o n is obviously t r a n s i t i v e , hence

T/H

minimal, and s i n c e

{Htlt

Thus

However,

(T/H.T)

and

HL

Hence

(T/H,T)

has at most twelve

(T/L,T)

The example j u s t discussed is due t o A. Knapp.

T.

are not d i s j o i n t .

I f t h e a c t i n g group

T

Disjointness

163

is abelian, the construction of an example is considerably more difficult. In particular the flows involved cannot be equicontinuous or even point distal, nor can proximal be an equivalence relation (theorem 24 and exercise 3). Examples

(for T = Z!)

have been obtained by

S. Glasner

and B. Weiss (Israel J. Math. 43 (19831, 1-81. We conclude this chapter by discussing the flows disjoint from a given class of flows. If Y

9'

is a collection of minimal flows, write

the set of minimal flows disjoint from every X E 9 .

Let

2,

for

denote the

class of distal minimal flows. 24. Theorem. 'D1 is the set of minimal flows with no non-trivial distal factor.

Proof. If X has a distal factor Z

X

6!

D',

then X

*

theorem 17, X ,

J!

D where D

then X J Z

and X

S !

.'D

If

is the universal distal minimal flow. By

a highly proximal extension of X, has as a factor a

quasi-factor 2 of D. By corollary 19, Z

is distal, and by lemma 4

of chapter 10, 2 is a factor of X, that is,

X has a distal factor.

25. Theorem. Let & denote the class of equicontinuous flows. Then

EI

= .'D

Proof.

If we substitute "equicontinuous"for "distal" in the

preceding proof, then we can conclude that no non-trivial equicontinuous factor.

&

is the set of flows with

(It is an easy exercise that

(ZX,T) is equicontinuous whenever X is equicontinuous -- in particular, a quasi-factor of an equicontinuous flow is equicontinuous.) Now, by the Furstenberg structure theorem, a minimal flow has a distal factor if and only if it has an equicontinuous factor. Hence

d

=. ' 9

Recall, from the "equicontinuous structure relation" chapter, that if

T is a group such that every minimal flow (X,T) admits an invariant

Chapter 11

164

measure (in particular, if T

is abelian) then a minimal flow

(X,T)

has no equicontinuous factor (equivalently no distal factor) if and only if it is weakly mixing. Thus if we write W H

for the class of weakly

mixing minimal flows, we obtain 26.

Theorem. In the case of acting groups which admit invariant

measures,

n1

= E'

= WM.

27. Theorem. The minimal flow X

is in

'I

I,

if and only if every

non-trivial quasi-factor of X has a non-trivial distal factor. Proof.

such that

Suppose X

ti?

zl".

Y has no distal factor and such that X and Y are not

disjoint. By theorem 17, X -1

Y*

(where Y* = [ 6

Y

has a quasi-factor X

( ~ ) ~ p lEp MI,

*

has no distal factor. Therefore X Conversely, if X

factor, then X

G

is a proximal extension of Y,

and X

x

, l X

so

has no distal factor.

has a non-trivial quasi-factor X

bl,

a quasi-factor 1, so

which is a factor of

for some homomorphism 6 : M+Y).

Y has no distal factor, and Y

Now

*

By theorem 24, there is a minimal flow Y

with no distal

(recall a flow is never disjoint from

D".

28. Corollary. Suppose all minimal flows with acting group T admit invariant measures, and suppose (WM)'

(X,T) is minimal. Then X is in

if and only if every non-trivial quasi-factor of X has a

non-trivial distal factor. Exercises.

1.

In his seminal paper, "Disjointness in ergodic theory, minimal sets,

and a problem in diophantine approximation" (Mathematical Systems Theory,

-1, (19671, 1-49) Furstenberg defines disjolntness'of (not necessarily

Disjointness

minimal) flows as follows:

(X,T) and

165

(Y,T) are disjoint if whenever

there is a flow (Z,T) and homomorphisms a : Z+X, there is a homomorphism 7 : Z+X

x

f3: Z+Y,

Y such that a = nX;y, f3

(where nX, ny are the projections of

X

x

Y onto X

then = nyi

and Y

respectively.1 a) Show that X

is disjoint from Y

(as defined above) if and only

if the only closed invariant subset

of X

x

Y

such that

n (I-) = X,

71

If X and

Y are disjoint, then at least one of X and Y is

X

b)

(I-) = Y

r

minimal.

Y

is

X x Y

itself.

If they are both minimal then the two notions of

disjointness coincide.

I: = {(Xi,T)li

2. Let

to define Z

E

I)

be a family of minimal flows.

It is natural

to be "multiply disjoint" if the product flow ll (Xi.T) is i

minimal. Let

u E J(M),

(i E I).

nGiai # i

let xi E Xi with x i u = x i

and let

Gi = 3(Xi,xi)

Then ll (X ,T) has a unique minimal subset if and only if i i 0

whenever a E G. i

(Hence, if T is abelian, this condition

is equivalent with multiple disjointness.) 3. Let

(X,T) and

and suppose T

(Y,T) be minimal flows with metric phase spaces,

is abelian. Then, if X and

Y have no common factor,

they are weakly disjoint and, if in one of the flows proximal is an equivalence relation, they are disjoint

4. For any class of minimal flows 9, "'3 5. Let

(X,T) and

and let n : X -+Y almost one-to-one.

= 9'

(Y,T) be minimal flows with metric phase spaces, be a highly proximal homomorphism. Then

H

is

(The proof depends on the following two lemmas, the

Chapter 11

166

first of which is purely topological.) a) Let

continuous and onto. Then n for y b)

Let

TI:X+Y

X and Y be compact metric spaces and let

E

TI:

Yo, a dense (X,TI+(Y,T)

is open at all points of x

Gg subset of

-1

(y)

Y.

be a homomorphism of minimal flows. Let

y

E

Y such that

p

E

M such that yp = y. Then n-1 (y)op

-1

is open at all points of

TI

be

T[

-1

= TI

(y), and let

(y).

6. Show that "highly proximally equivalent" is an equivalence relation.

7. Let X, Y and 2 be minimal flows, and let # : X+Z, be homomorph sms. We say that

@

and @

are disjoint

# : Y+Z

(# I

$1

if

R = [(x,y) @(x) = Jl(y)l is a minimal subset of X x Y. (We also say #* that X and Y are relatively disjoint over 2 . ) Obviously if 2 = 1, this reduces to disjointness of the flows X and a)

Suppose #

:

$ : Y+Z,

X+Z,

@' :

X--2' , JI'

homomorphisms of minimal flows and 4(xo) = $(YO)'

#'(xo)

= *'(YO)'

xo

and

f

Y. :

Y+Z'

X,

yo

# I JI,

E

@'

are

Y are such that I

JI'.

Then 2

and 2' are isomorphic. b)

p : Z+Y,

Suppose a : W+Z,

minimal flows and p I $.

8. Let

Then

(poor)

are homomorphisms of

Suppose either ( i ) cz

is open, or (ii)

proximal and $ distal.

I): X+Y

$

is highly

is proximal and a is

I $.

X,Y, and Z be minimal flows with metric phase spaces which

admit invariant measures, and let

II :

X+Y

be a distal extension.

Suppose X and Y have the same maximal equicontinuous factor. Then

X iZ if and only if Y 9. Let TI:

I 2.

(X,T), (Y,S) and

(X,T)-+(Y,S)

(2,U) be minimal cascades and suppose

is a group extension Y = X/H,

and

(Y,S)I (Z,U).

Disjoinrness

Suppose, for every

h E H,

the cascade

167

(X,hT) is minimal.

(X,T) I (2,U). (In particulnr, the conclusion holds if totally minimal and every h E H

is of finite order.)

Then

(X,T) is

This Page Intentionally Left Blank

169

Chapter 12 Invariant Measures on Flows An invariant measure for a flow (X,T) is a regular Borel probability measure p

on X such that

X and all t

E

T.

p(At) = p(A)

If such a p

for all Borel subsets A

exists, we say that

of

(X,T) admits an

invariant measure. The existence of an invariant measure is a kind of "incompressibility"condition on the flow, and frequently has significant dynamical consequences. In the present work, we have seen this in the chapter on the equicontinuous structure relation and disjointness. Also, in the "Furstenberg structure theorem" chapter, we showed that a distal minimal flow always admits an invariant measure. focus on the question:

if T

In this chapter we will

is a topological group, when does there

exist an invariant measure f o r all flows with acting group T? Our approach to invariant measures is via strongly proximal flows. (Our main result

- theorem 7 - is that all flows with acting group T

admit invariant measures if and only if T has no strongly proximal minimal actions.) This is not the shortest route, but we pursue it since

it provides a "topological dynamics" approach to the problem. We follow the development in Clasner, Proximal Flows. For the proof of our main result, we require some "Choquet theory," which we develop in some detail. At the end of the chapter, we sketch an alternate proof of the existence of an invariant measure in case the acting group is abelian. Let

X be a compact Hausdorff space, and let &(XI

regular Borel probability measures on X. L

I'

is defined by L ( f ) = fdp, then L

of norm one on C(X).

denote the set of

If p E A(X), f E C(X),

and

is a positive linear functional

Conversely, the Riesz representation theorem tells

us that all such linear functionals arise in this manner from a

Chapter 12

170

u n i t sphere i n

and only i f E

T.

( t h e dual space of

p E M(X)

Clearly,

t

C(X)

*

Thus we may regard

X.

p r o b a b i l i t y measure on

M(X)

and write

C(X))

p(f) =

I

(X,T)

if

is a n invariant measure f o r t h e flow

is a f i x e d point f o r the flow

p

as a s u b s e t of the

(I(X).T)

:

pt = p

fdp.

for all

In f a c t , another condition equivalent t o t h e e x i s t e n c e of an

T

invariant measure is t h a t

*

With the weak Hausdorff space.

C ( X ) , A(X)

topology i n h e r i t e d from

*

(The weak

pointwise convergence - if only i f

has t h e " f i x e d point property" (theorem 7 ) .

Ln(f)-+L(f)

*

topology on

{L,)

for a l l

f E C(X).

is t h e topology of

C(X)

Ln+L

is a n e t , then

is a compact

*

weak

Compactness of

#(XI

i f and

in this

topology is a consequence of Alaoglu's theorem, see c h a p t e r 8 . )

is a flow,

(X,T)

If

defined by

ft(x) = f(xt).

is defined by a flow.

f E C(X)

If

ax

x,

(This embedding is a l s o topological:

if

if and only if

then

t

and

E

T,

f t E C(X) then

ax

(M(X),T)

for

f E C(X).

is a n e t i n

{xn)

*

X,

then

topology.)

W e now review some elementary notions concerning convex sets. subset

of a ( r e a l ) vector space

Q

for all

ax+gy E Q in

a,B 2 0

an extreme point if whenever

If

z = x = y. E

h u l l of z(X)

is convex if

a+@ = 1.

W e write

z = ax+py,

for

x

Q.) The point

A

implies and

y

are

z E Q

is

x , y E Q, a,@ > 0, a+P = 1,

ex(Q) f o r t h e set of extreme p o i n t s of

is a topological vector space, and

X

x,y E Q

(That is, i f

t h e l i n e segment j o i n i n g them lies i n

Q,

then

with

E

is

(M(X),T) by i d e n t i f y i n g

ax(f) = f(x),

i n t h e weak

+6x n

is

pt E M(X)

I t is e a s i l y v e r i f i e d t h a t

at

xn + x

E T,

as a subflow of

(X,T)

with t h e "point mass"

x E X

t

p E #(X)

(pt)(f) = p(ft).

W e may regard

and

X c E,

Q.

t h e n t h e c l o s e d convex

is the smallest c l o s e d convex set c o n t a i n i n g

X.

W e write

f o r t h i s set.

If the group

T

acts on t h e compact convex set

Q

and if each

t

f

T

Invariant Measures on Flows

171

acts as an affine transformation ((ax+py)t = a(xt)+p(yt),

a,p

2

0, a+@ = 1)

then we say that

for x,y E Q,

(Q,T) is an affine flow. I t is

easy to verify that the space of measures M(X)

is convex and the flow

(M(XI,T) discussed above is an affine flow. An affine flow (Q,T) is irreducible if it contains no proper non-empty closed convex invariant subset. A direct application of Zorn's lemma shows that an affine flow always contains an irreducib e affine subflow. Now we introduce the notion of a strongly proximal flow. Recall that a flow is proximal if every pair of points is proximal. The flow is said to be strongly proximal if

(I(X),T)

(X.T)

is a proximal flow. If

(X,T) is strongly proximal, it is proximal (since, as we observed above, An

it can be regarded as a subflow of the proximal flow (M(X),T)).

example of a proximal minimal flow which is not strongly proximal will be given later. Strongly proximal minimal flows can be characterized without any reference to measures, as the following theorem shows. Theorem. Let

(X.T) be a minimal flow, and let xo

is strongly proximal if and only if for every hood V

E

> 0

X. Then

E

(X,T)

and every neighbor-

of xo, there is a finite subset F of T such that if is any finite subset of

. . ,'n

{XI,.

are in V

with at most

X, then for some t

[nel exceptions.

(Here [

1

E

xit

F all

denotes the

greatest integer function.)

For the proof, see Clasner, Proximal Flows, VII.2. 1.

Lemma. The flow (X,T) is strongly proximal if and only if

O m

n

x*

0

for every

/J

E

&(XI.

Proof. Suppose (X,T) is strongly proximal. Then, if p

E

M(X)

and

Chapter 12

172

x

E

X, (p,x) E P,

so

there is a net

E

1

E

2

so

p'

uti+v'

Let

and we have

But, as is easily proved, x = 6x is an extreme point of

x = -(p'+u').

MX),

Let p , u E M ( X ) and let 0 = L (p+v). 2

X. Then (choosing subnets) pti+p',

1

so we may

X and pti--+x'. Conversely, suppose every orbit

closure in M ( X ) meets X. et.-+x

in T such that

X is an invariant subset of M(X)

(pti,xti)-+AM(X). But suppose xti+x'

{ti)

- ax, and

= u'

and u

p

are proximal.

The next flow results have nothing to do with flows per se, but constitute an exposition of the necessary parts of Choquet theory. We follow the book of R.R. Phelps (Lectures on Chosuet's Theorem, Van Nostrand Mathematical Studies, #71. 2.

Y be a compact subset of the locally convex space E

Theorem. Let

-

Q = co(Y)

and suppose that

If p

is compact.

E

M(Y)

there is a unique

point x = @ ( p ) E Q such that f(x) = Iyfdp, for all

p

is affine, weak

Proof. that

If f

n *Hf #

*

0.

Since Q

E

E*. The map

T(y) = (fl(y), . . . , fn(y)).

Qlf(x) = lyfdp]. We want to show

is compact, it is sufficient to show that

fEE * for any finite set fl,...,fnE E , by

E

continuous, and onto.

E*, let Hf = [x

E

f

n

# 0.

Define T

:

i = l , .. . ,,Hfi

Since T

E+Rn

is linear and continuous, T(Q)

is compact and convex. If p = (p(fl),. . . ,p(fn)) JYfidp), we need to show that p E T(Q).

(where p(fi) =

If this is not the case, there

is a linear functional on Rn which strictly separates p and T(Q). Since linear functionals on Rn element of Rn,

are given by inner products with a fixed

. . ,an) E Rn such that this means there is an a = (al,.

(a.p) > sup[(a,T(x)(x

E

QI. Define

*

g E

E

n by

g =

C aifi i=1

and this

Invariant Measures on Flows

173

I,

gdp > sup[g(x)Ix E 81.

last assertion translates to

impossible, since Y c Q and p(Y) = 1. there i s an x E Q such that

It follows that if

so we have

f(p(p)) =

fdp. Y

Next, we show that if x E Q, there is a p

y -+x. a

f E E*.

It is immediate that the map p Q and the

*

separating property of linear functionals) that it is weak

Since Q = G(Y),

A(Y),

is unique, and we write

is affine, and easy to see (using the compactness of

p ( p ) = x.

p E

f(x) = p(f) = Syfdp, for all

Such linear functionals separate points, this x x = p(p),

But this is

there is a net

E

continuous.

N(Y) such that

{ya)

a a Then y = Chiyi, with A; > 0, C A Y = 1, a i i

in co(Y) and

such that

a

yi E Y.

Let

a

(where 6. denotes the point mass at yi), and let (a i,a subnet of) pa converge weak * to p E W(Y). Now wa(f) = f(ya), for

pa = C A Y 6

f

i,a

*

E

E , and pa(f)+p(f),

p ( p ) = x.

This shows that fl

We say that

p

f(ya)-+f(x).

p

p;

ax

represents x.

f(x) = p[f), and

is onto, and the proof is completed.

represents x = p ( p )

is the barycenter

of

so

m.

and that x

is the barycenter

If x E Q, obviously the point mass

Our next result shows that the extreme points of Q

are characterized by the uniqueness of the point masses as representing measures. 3.

Theorem. Let

space E.

ax

Then x

Q be a compact convex subset of the locally convex E

Q

is an extreme point if and only if the point mass

is the only probability measure on Q which represents x.

Proof.

If x e ex(Q), x = ay+flz, with a,@ > 0, a+fl = 1, y

z.

Then

+ = p represents x and p # ax. Suppose x E ex(Q) Y represents x. To show that ~1 = 6x, it is sufficient to show

obviously a8 and p

*

(That

Chapter 12

174

that p(D)

=

p(D) > 0, for some such

p(U n Q) > 0. Then K = U n Q 0 < r = p(K) < 1.

f(z) 1 1 for z

K. Then 0

contradiction.) Define pl,p2 1

p ( B ) = -p(B

p = rp

1+(1-r)p2,

x d ex(Q1.

let

so

*

f

such that

E

1

E

f(x) = 0 and

1, a

1

pl(B) = F p ( B n K) and

M(Q) by

K and so x1

x = @ ( p ) = r@(pl)+(1-r)/3(p2)

*

x.

xi = fS'(pi)

Let

Clearly

= rxl+(l-r)x

2'

and

This is a contradiction, and the proof is completed.

4. Corollary. Suppose Q

is a compact convex subset of the locally

convex space E and Z is a subset of ex Q c

E

E

= f(x) = JQfdp = lKfdp 1

(i = 1,2). Since p (K) = 1, x 1

D

x d U n Q and

of y such that

for B a Bore1 set in Q.

n (Q\K)),

1-r

E

is compact and convex and

(If p(K) = 1, E

Suppose

D. Since D is compact, there is some y

and some closed convex neighborhood U

2

D of Q with x d D.

0 for every compact subset

Q such that G i Z ) = Q.

Then

Z.

Proof. - Let

Y = 2 and x

which represents x. concentrated on Y.

E

ex Q.

By theorem 2 there is a p E M ( Y )

We may regard I./ By theorem 3 ,

as a measure on Q which is

p = 6

X'

Therefore x

E

Y = 2.

Now we return to dynamical concerns. 5. Theorem. Let

(Q,T) be an irreducible affine flow. Then

strongly proximal. If X = Q

(so

m, then

X

(Q,T) is

is the unique minimal set in

(X,T) is strongly proximal and minimal).

Proof. We first show that invariant set.

X

is a minimal set. X

--

If x E Q, co(O(x))

by irreducibility

( 0 0 ) = Q.

m)X = ex(Q) c cl(x).

is obviously an

is closed, convex, and invariant, Hence by corollary 4 (applied to

This shows that

X

is a minimal set, and indeed

is the unique minimal set in Q. Now let p

E

N(Q)

and

so

let 6 : N(Q)+Q

be the barycenter map.

Invariant Meusures on Flows

--

Then /3(co(O(p))

175

is a closed convex invariant subset of Q

so by

--

--

irreducibility P(co(O(p1) = Q. Since /3(co(O(p)) = G(P(O(C1)),

=(/3(m) = Q. Again by corollary 4 , -

X = ex(Q) c P ( O ( p ) ) .

< E 00. so

r;

= 6

X

x

Now let

E

ex Q c

ex(Qf c X.

we have

/3(o(c1)),so Then x = /3(<)

for some

But

6x is the only representing measure for x, (theorem 3) and x = 6 E O ( p ) . By lemma 1, (Q,T) is strongly X

proximal. 6.

Theorem. Suppose (X,T) is either a minimal strongly proximal flow,

or an irreducible affine flow. Then, if

measure, X

is trivial.

Proof. Suppose X

is minimal and strongly proximal with

invariant measure, so (lemma 11,

so

pt = p

pti+x

irreducible affine, then X

t

E T.

for all

for some net

But then xt = pt = p = x, (t

if p

(X,T) admits an invariant

E

TI

t

{t}

T. Let x

E

{XI.

so

E

p =

x.

Similarly, if X

is strongly proximal (theorem 5 ) .

is an invariant measure then p = x

an

o(cl) n x

E

and pti = p ,

and X =

p

X and xt

= x,

is

As above, for all

By irreducibility of X, X = {x).

Now we are ready for the theorem which provides a dynamical characterization of those groups T such that all actions of T

admit invariant

measures. 7. Theorem. Let T be a topological group. Then the following are

equi Val ent : (i)

Every flow with acting group T admits an invariant measure.

(ii)

Every affine flow (Q,T) has a fixed point.

(In this case, we

say T has the fixed point property.) (iii) There are no non-trivial minimal strongly proximal actions of T.

-

Proof.

(i)

*

(ii).

Let

(Q,T) be an affine flow, and let Qo be an

Chapter 12

176

irreducible affine subflow. By (i),

(Q0 ,TI has an invariant measure.

By theorem 5, (QO,T) is strongly proximal and by theorem 6, Qo consists of a single point, Q, = {x,)

and xo

is a fixed point for T

in Q. (ii)

*

(iii).

Suppose

(X,TI is minimal and strongly proximal. Then

(M(X),T) is an affine flow which by hypothesis has a fixed point Thus p

is an invariant measure and by theorem 6, X

(iii) + (i).

subflow of

Let

(M(X),T).

By assumption, Q measure for

(X,T) be a flow, and let Then

M.

is trivial.

(Q,T) be an irreducible

(Q,T) is strongly proximal by theorem 5.

is trivial, Q = { p ) ,

and so

is an invariant

p

(X,T).

Note that theorem 7 applies to abelian groups since, as we have noted several times in earlier chapters, abelian groups have no non-trivial proximal minimal actions, and a strongly proximal flow is proximal. More generally we have 8.

Theorem. If the group T

is solvable, every flow

(X,T) admits an

invariant measure. Pro.of. By definition, there is a sequence of subgroups {e) = T c T c 0 1 groups Tj+l/Tj Let

...

c Tn = T with T

J

abelian. We show that

normal in T

J+l

and the factor

T has the fixed point property.

(Q,T) be an affine flow, and consider the affine flow

(Q,T1).

Since T1 is abelian, then by the remark preceding this proof and theorem 7, (Q,T) has a non-empty fixed point set Q1; clearly Q,

is

compact and convex. Since T 2 is normal in T1, Q, is invariant under the action of T, (i.e., if x1 E Q1, t, E T, then xlt2 E Ql), and the group S

2

= T /T

2 1

acts on Q,

abelian so the flow (Ql,S,)

by

x (t T 1 = xlt2. But S2 is 1 2 1

has a non-empty fixed point set Q, c Q,

Invariant Measures on Flows

and it is immediate that

for x2

x2tZ = x 2 ,

177

Q,

E

and

t2

E

T2. The

proof is now completed by an easy induction. Let

T be a topological group and let B

=

B(T) be the bounded right

uniformly continuous functions on T. An invariant mean on B on 73

continuous positive linear functional m m(f

t)

t (where f ( s ) = f(st),

= m(f)

is a

m(1) = 1 and

such that

for s,t E TI. The group T

is

said to be amenable if there exists an invariant mean on B(T). Using the correspondence between B the maximal ideal space B of

73

and the continuous functions on

(as developed in the chapter on

universal minimal flows and ambits), it is easy to show that a group T

is amenable if and only if every flow with acting group T admits an invariant measure. = <(f)

:(<)

C(B).

for

Recall that if f

<E

B and the map

E

11,

?

by onto

73

It follows that there is a one-to-one correspondence between

(Note that

given by the relation

is invariant if and only if m

Moreover p

p ( ? ) = m(f).

Now

is defined

is a bijection of

f-?

probability measures on B and means on 73

hence

C(B)

E

&(<) = ?(
t

=
t

= f (51,

so

is invariant.

?t = ft and

t

t

p(k) = p(f 1 = m(f 1. (B,T,pe) is the universal ambit

f E C(B)),

s o the above discussion shows that

only if there is an invariant measure for case if and only if every flow (For, let

xo

E

X, and let Xo

B+X0

flow. Let

II :

measure

on Xo

u

(where pe(f) = f(e),

for

T is amenable if and

(B,T), which in turn is the

(X,T) admits an invariant measure.

-

= xoT,

so

(X0 ,T) is a point transitive

be a homomorphism, and define an invariant by

u ( g ) = p(gon)

for

g E C(Xo); u

may be regarded

as an invariant measure on X which is concentrated on X,.) Hence we have proved 9.

Theorem. Let T bt a topological group. Then T satisfies the

Chapter I 2

178

condi ions of theorem 7 i f and only if T

is amenable.

An intrinsic characterization of discrete amenable groups has been

T is amenable if and only if for any

given by F$lner, who proved: E

> 0 and any finite subset

I k M AI < lAI

such that

c,

K of T there is a finite set A of T k

for all

I I denotes cardinal

K, where

E

number.

For a proof, see Jean-Paul Pier Amenable Locally Compact Groups, Wiley, 1984, pp. 62 f f .

In fact, a more general result is proved:

if

T is a locally compact group, then the above characterization of amenability holds with set and A

I I

denoting left Haar measure, K a compact

a measurable set of finite measure.

We now present two examples. We first make note of the following obvious fact. If S1 and

S2

(same) space X such that

(X,S1) is minimal and

and if T

are groups of homeomorphisms of the

is the group generated by

and

S1

is minimal and proximal. In particular, let

(realized as the real numbers modulo 1) homeomorphisms of X defined by q(0) = 0

2

.

Then the cascades

minimal and proximal, so if generated by

cp

X be the unit circle and I) be the

(with f3

irrational) and

(X,I)) are, respectively,

T is the group of homeomorphisms of X

and I) then

(X,T) is strongly proximal.

and

(X,T) is minimal and proximal. We show that if p

E

In fact,

M(X1, JInp+S,,

point measure at 0. If f E C(X), (JInp)(f) = p(fJIn) and f$(0) f(O

zn)-+f(O)

(+"p)(f)-+f(O)

X, TI

then the flow

S2,

and let cp

( ~ ( 0 =) 0+p

(X,cp)

(X,Sz) is prox mal,

the =

so by the Lebesgue dominated convergence theorem

= tiO(f).

Thus $p+tiO

and by lemma 1,

(X,T) is

strongly proximal. An example of a proximal flow which is not strongly proximal is 3L

obtained as follows. Let X = {0,1) ,

the space of bisequences of 0's

Invariant Measures on Flows

and

and let

1's

*

xi = x

where

e

e(x)

if

i

be the shift homeomorphism of X, so if x E X,

(T

(ox)i = xi-1' Let

i

be the homeomorphism of X defined by

*

*

x1 = xi

1 and

changes the first coordinate of and p2

homeomorphisms p1

p2(x) =

f

if xo = 1

e(x)

if

generated by

179

.

(where 1' = 0, 0' = 1).

x and nothing else.

X by p,(x)

of

e(x) = x

Now define

if xo =o

=

Thus

and

T be the group of homeomorphisms of X

Let

xo = 0 and p2.

u,pl,

(X,T) is minimal and

We show that

proximal, but not strongly proximal. To show minimality, note that p2pl = 8 x,y

E

X, we may, by applications of

block of x

8

(so

TI.

8 E

and powers of

to the corresponding block of

y, so

y

Hence, if (T,

change any

is in the orbit

closure of x. To show (X,T) is proximal, we show if x,y positive integer, there is a t on the left) tx and

f

yo and x1

#

n

is a

in common. The

I t is clearly the case for n = 1.

yl, then either

( ~ ~ y ) ~ Suppose, .) inductively that n

and

T such that (writing the action of T

ty have a block of length n

proof is by induction OR n. xo

E

X

E

(p1xl1 = tx and

in common. Call these x and y

or

(If

(p2x)1 =

have a block of length

ty

(instead of

tx and

ty).

We may

suppose (by applying an appropriate power of the shift r) that ' 3 2 '

n+l - ~ ~ y ~ . . . yIf ~ +x1 ~ .= y1 then indeed x

-

* '

block of length n+l

in common. So suppose x1

largest integer less than 1 such that i < I

then obviously x

above, for one of Suppose

and

f

f:

y1 and let

i

be the

yi. (If xi = yi for all

y are proximal.)

t = p1 or p2

i < 0. Then xi

xi

#

and y have a

If 1 = 0, then as

(tx) = (tyIj for j = 1,2,. . . , n+l.

J

yi, xi+l = y i + l .

Then we apply a power of

(r

Chapier 12

180

to shift these to the zero'th and first positions, apply p1

i -i shift back again. That is, let s = cr ta z = sx, w = sy,

zi + l * wi+l'

we have

t = p1

where

or

and

p2

or p2;

if

Repeat the process until the

transformed elements differ in both the zero'th and first positions, and then, as above, apply one of the p1

or

p2

that the first positions

so

coincide and the transformed elements have a block of common length n+l. This completes the induction. Thus the flow

(X,T) is minimal and proximal. However, it is not

strongly proximal, since r . p l p

all preserve the product measure

and p 2

obtained from the measure on the two element set

measure

1

to each point.

(0.1) which assigns

(This is Haar measure on X

regarded as a

topological group.) Therefore, 1.1 is an invariant measure for the flow

(X,T I . Exercise.

If the acting group T

is abelian, a more direct proof of the

existence of an invariant measure for the flow Define a functional q on C(X)

(X,T)

is available.

1 by q(f) = inf --/lftl+ . . . +ft,R,

the infimum is taken over all finite subsets {t l,...,tn} q(f) 5 llfll

and q((Yf) = cxq(f)

then q(f+g)

5

(Let

q(f)+q(g)

for E:

2

(Y

. . ,n.j=1,.. . ,m

> 0, let

l!.(c)l I IcI = q(c).

T, q(f-ft)

5

is abelian.) Now let S

defined by

L(c) = c.

be the L

Then

By the Hahn-Banach theorem, there is a linear

functional L on C(X) E

it is this

consisting of the constant functions, and let

be the linear functional on S

t

and use the set

to show q(f+g) 5 q(f)+q(g)+2&;

step which uses the assumption that T subspace of C ( X )

T. Then

0. Moreover, if f,g E C(X),

c T such that {tl,. . . ,tnL is1,.. . ,'m' 1 -Ilftl+.. . +ftnll < q(f I+&, illgsl+.. . +gsmll < q(g)+&, n

{tiS,j'i=l,.

of

where

such that L(f) 2

S

0 (show q(f-ft) I --Ilfll

q(f)

and L(c) = c.

If

for every positive integer

Invariant Measures on Flows

n).

Then

and it follows that

L(f) 5 L(ft),

is a positive functional. constant

c

such that

(If g 2 0

IL(c+g)I

and

5 IIc+gll

invariant positive linear functional on defines an invariant measure for

(X,T).

181

L(f) = L(ft).

L(g) = p < 0

fails.) Hence C(X)

with

Finally, L

show there is a

L

is an

L(1) = 1,

so

L

This Page Intentionally Left Blank

183

Chapter 13 Kakutani-Bebutov Theorems This chapter has a somewhat different emphasis than the rest of the book.

Thus it is not concerned with the structure of minimal flows and

their homomorphisms, but rather with a general representation theorem for flows. Recall that in chapter 1 we briefly considered flows on spaces of continuous functions on a group. Thus, if f the function defined by f (t) = f(tt). t

:

T+R

and

t E

T, f

t

is

If the group T is u-compact

and B (TI denotes the bounded continuous functions on T with the 0 topology of uniform convergence on compact sets, then defines a flow on BO(T), If

(f,t)Hft

the Bebutov system.

(X,T) is a flow, we say that

(X,T) is embeddable in the Bebutov

flow if there is an isomorphism of

(X,T) into (BO(T),T). That is,

there is a homeomorphism u of X

into BO(T) such that if x E X,

t

E

T and fx = u(x),

then d x t ) = (fxIt.

There are two obvious necessary conditions on a flow be embeddable in the Bebutov system

(X,T) for it to

First, the phase space

(BO(T),T).

X must be metrizable (since B0 (TI is metrizable). Secondly, if F = [x E Xlxt = x

for all

t

E

TI

(the fixed point set of the flow

(X,T)) then F must be homeomorphic to a (possibly empty) subset of For if u

:

X+B 0 (TI is an isomorphism and x

equivariance of the map function. Then F

u

E

F, then the assumed

implies that f = d x ) X

is homeomorphic with u ( F )

R.

is a constant

which may be regarded as

a closed subset of R.

It is a remarkable fact that in many cases (including when the acting

Chapter 13

184

group T = R )

these two necessary conditions are also sufficient. This

is the content of the Kakutani-Bebutov theorems, to which this chapter is

devoted. (X,T) be a flow where

We first reformulate the problem. Let compact metric.

Let

the set of f E C(X) C (X) = C(X1.1

P

with x

#

by

is

: F+R

p

be a homeomorphism and let C (XI be P such that f(x) = p(x), for x E F. (If F = 0,

Now suppose there is an f

y, then f(xt)

into BO(T)

X

f

C (XI such that if x,y E X P for some t E T. Then we can map X

f(yt)

x H f X where fx

:

E

T+R

is defined by

fx(t) = f(xt).

It is easily checked that this map is continuous and equivariant. If x

*

y

and t E T such that

f (t) Y is an isomorphism and we have an

f(xt)

f

f(yt)

f and so the map x H f X Y embedding of X in BO(T). The function f fx

f

then fx(t)

f

so

is called a dynamical

embedding function. Conversely, if

Q

:

X+BO(T)

is an embedding of the flow (X,T) in

the Bebutov system, it is immediately verified that if f f(x) = o(x)(O),

then f

is defined by

is a dynamical embedding function. Therefore a

flow is embeddable in the Bebutov system if and only if there is a dynamical embedding function. Theorems which assert the embeddability of flows in the Bebutov system are called Kakutani-Bebutov theorems. (The reason for this will be explained at the end of the chapter.) Our main result is a KakutaniBebutov theorem for actions of R.

1.

Theorem. Let

Suppose F(cp), subset of R.

(X,c

t

be a real flow on the compact metric space X.

the fixed point set of the flow, is homeomorphic to a Then the flow is embeddable in the Bebutov system on R.

(Equivalently, there is a dynamical embedding function for the flow.)

We first require a number of technical definitions and lemmas on C1

Kakutani-Bebutov Theorems

185

functions and local sections. If f

C(X),

E

and x If A

hand side exists. f

C(X)

E

such that

If K c R

2. Lemma. Suppose A E

C1(X),

It=O

d -f(cp dt t (XI) 1 = C (A)

if the right

be the set of

exists and is continuous on A.

af(x)

acp

A c X,

and

there is an f

= X, let %x) acp 1 c X, let C (A,cp) E

cpK(A)

write

and B

for the set

[cpt(a)lt

E

K,a E A ] .

are disjoint closed subsets of X.

with 0

5

f(x)

5

1 for

x E X

n cp[o,61

(B) =

Then

such that

flA = 0 and flB = 1. 6 > 0 such that

Proof. Let h

h = 1 on cp f = S h 6

E

(B). Now set S6 h(x) =

[0,6l

and let

f

and

6

Joh(cpt(xl)dt.

Clearly

1

C (X) and satisfies the other required conditions.

Note also that 6+0,

0

such that 0 5 h(x) 5 1 for x E X, h = 0 on cp [ 0 , 6 1 ( A )

C(X)

E

cplo,61(A)

and that

S6h

S6h(x) = h(x)

3. Lemma. Let f there is a g

E

approaches h uniformly on compact sets as

E

1

for x

Proof. Let

K1

E

F(p).

E

C(X), K a compact subset of

C (K,cp)

g(x) = f(x)

if x

such that

Ig(x)-f(x)l

<

X and

E

for x

E

> 0. Then X and

E

F(cp1.

and Kz

be compact subsets of X

such that

0 0 K c K1 c K1 c Kz (where KY denotes the interior of Ki). Choose 6 > 0 so that S f is an E approximation of f on Kz and let 6

a,p E C(X) so that and a(x)+P(x) = 1

a = 1

on K1, /3 = 1 on

for x E X.

the conclusion of the lemma. g

E

(KZIC, 0 5 a(x), p(x) 5 1

Then if g = aS6f + pf.

(Note that

g = S6f

g

on K1,

satisfies so

1 C (K,cp).)

Let

(X,cpt) be a flow and let x0

subset S of X

E

X with xo

is called a local section at

xo

!6

F(cp).

A closed

if there is an

E

> 0

Chapter 13

186 ff

ff

such t h a t t h e map

cp

one-to-one

SX[--E,E] onto a neighborhood

map of

U

neighborhood

{x,)

Let

) t S ( x n )1 5

( x )+cp n

5.

to

so

Let

to

with

(x) = y

E

U.

f

E

1

C

t

U

)

of

acp

xo

> 0.

Let

x

E

W.

E

V',

f(cpt(x)) > f ( x ) + a t f o r

-E

5

x

0

let

such t h a t

S = f-'(f(xd))

x

for

If(x)-f(x n U.

6.

t E [-&,&I, s o

Corollary.

flow

pt

at

If

xo.

xo

E

> 0

E

Let

f o r all

x

E

U.

x 0

E

R.

Let

cpt(V') 0 < t 5

be

V'

c W

for and

E,

be a compact neighborhood

V" E

V".

Let

U = V' n V"

and

t h e i n e q u a l i t i e s above

a f ( x ) > 0 on U acp S

t

be such t h a t

) I < a& f o r x

Then if

t o g e t h e r with t h e f a c t t h a t a unique

(x))ds,

t < 0.

0

is

By t h e fundamental theorem of

and

Then, if

Then

0.

be a neighborhood of

W

xo

s

*

f - l ( f ( x 1) n U 0

such t h a t

a compact neighborhood of

af -(cp

af(x 1 acp 0

with

I,a.

of

By uniqueness

xo.

at

0 f o r all

af(x) > a > 0

f(cpt(x)) < f ( x ) - a t

is compact we

E [-E,E].

0

xo E X

and l e t

(X,(p)

af(x

W e may suppose

[-&,&I.

there

Since S

c a l c u l u s , f(cpt(x)) = f ( x ) +

E

U

E

S

and s i n c e

and

a(P

t

This

By j o i n t c o n t i n u i t y of t h e flow

a l o c a l s e c t i o n f o r t h e flow cp t

such t h a t

x

xo.

cp-(x,t ( x ) ) =

xn+x

tS(xn)+tO

t h e r e is a compact neighborhood

Proof.

of

tS is continuous.

and so

Theorem.

(p

S,E

is a

is continuous.

U

( x ) - + y E S. n

(x),

= U

U

@,(XI

Note t h a t if

S.

such t h a t

U+[-E,E]

:

be a sequence

'tS(x,,)

to = ts(x),

[-&,&I

E

we may suppose

E,

may suppose 'tS(xn)

tS

The map

Lemma.

Proof.

ts(x)

cp ( x , t ) =

defined by

is c a l l e d a flow box f o r

is a unique number

4.

Sx[-&,el+X

:

imply t h a t

cpt(x)

E

S

for

is a l o c a l s e c t i o n .

X \ F(cp),

then t h e r e is a l o c a l s e c t i o n f o r t h e

Kakutani-Bebutov Theorems

Proof. x

is a metric for X, set f(x) = ~p(xo,cps(xl)ds, for

If p

X and

E

af(x ) = a(P 0

187

such that cpT (xO 1

T E R

*

(xO ) )

p ( x ,cp O t

xo

U

n

X

E

xo. Then f

1

C (X,cp) and

E

0. By theorem 5, there is a neighborhood U

xo such that f-'(f(x0l) 7. Lemma. Let

*

\

is a local section for the flow at

F(cp)

and let U

of

x0 '

be open with

Let S c U be a compact local section for the flow 1 = v E C (X,cp) such at x and let M > 0. Then there is a v 0' S,M, U that Iv(x)l S 1 for x E X, ?(XI > M for x E S and v(x) = 0 for

x0

E

U c X

\

F(cp).

acp

XEX\U. Proof. Let

> 0 be such that the set V = cp

E

( S ) is a flow box

[-&,El

for which V c U and such that c p [ - & , E l (X-U) n V = 0. Let h : R+[R dh be a C1 function satisfying h ( O ) = 0, > M, and Ih(t)l S 1 for t

E

Let

W.

v1

E

C(X)

v 1 (XI = h(-ts(x)) Then if x

Let

E

E

S. Let v

x

E

X

\

for x

S, v

0 < f3 <

x

such that

=

E

(cp

1 t

E

Iv,(x)l

1 for x

S

V and v,(x)

(x)) = h(t),

= 0

E

X

for x

for small

t,

Then

13 1'

U. Also, if x

E

S,

Iv(x)l

5

E cp

[ - & , & I (X \ U ) .

avl

-(XI

so

such that . acp(cpt( avl x)) > M for t

Sv

and satisfying

acp

E

=

[-@,PI and

1 on X and v(x) = 0 for

then

Now we can proceed with the proof of the Kakutani-Bebutov theorem f o r

T = R.

As we mentioned earlier, it is sufficient to find a dynamical

embedding function for the flow (X,T). Write F for F((p) and let A = XxX \ (FxF u A).

D(K)

be the set of f

E

C (XI such that if Cx,y) E K

P

If K c A

let

there is a

Chapter 13

188

t E R

f o r which

f(qt(x))

show t h a t every

D(K)

(x,y)

E

A

is open and dense i n

*

f(qt(yl).

The s t r a t e g y of t h e proof is t o

K

has a compact neighborhood

C (XI. P

such t h a t

(C (XI is given t h e topology induced P with t h i s topology C (XI is a P

by t h e sup norm llfll = sup I f ( x ) l ; XEX

complete metric space. 1 Once t h i s is accomplished, we a p p l y t h e Lindelof covering theorem of

*

t o o b t a i n a countable f a m i l y

of compact s u b s e t s

D ( K 1 open and dense i n C ( X I and A = u K n . Then i f n P (by t h e Baire category theorem, t h i s set is non-empty, i n f a c t ,

with each

A

f E nKn

C ( X I ) and ( x , y ) E A, ( x . y ) E Kn P f(pt(x)) f ( q t ( y ) l for some t . I f x

dense i n

*

so

{K } n

f ( x ) = p(x)

p(y) = f ( y ) .

f

Hence

f o r some and

n

f E D(K,)

and

are both i n F

y

then

is a dynamical embedding f u n c t i o n .

f

Thus theorem 1 is a consequence of t h e following theorem. 8.

Theorem.

of

(XO.YO)

Let in

A

such t h a t

W e first show t h a t

Proof.

Then t h e r e is a compact neighborhood

(xo,yo) E A.

compact subset of

Let

A.

D(K)

D(K)

C (XI P and d e f i n e

f E D(K)

p f ( x , y ) = sup I f ( q t ( x ) ) - f ( q t ( y ) ) I

C (XI.

is a n open dense s u b s e t of

is open i n

for

x , y E X.

whenever

K

P

K

is a

I t f o l l o w s from t h e

tER

D(K)

d e f i n i t i o n of is a

6

that

> 0 such t h a t

pf(x,y) > 0

pf(x,y) > 6

for

for all

were not t h e c a s e , t h e r e would be a sequence pf(xn,yn)+O.

By compactness of

(xn,yn)+(x',y')

E K

D(K)

*

In f a c t , there

For, i f t h i s

( x , y ) E K. (xn,yn)

E

K

f o r which

( a subsequence o f )

and i t is e a s y t o see t h a t

c o n t r a d i c t s t h e o b s e r v a t i o n above. IIf-gli < 614

K

( x , y ) E K.

p (x',y') f

= 0

which

g E C (XI such t h a t P g E D(K). T h i s proves t h a t

Now, if

t h e n it f o l l o w s e a s i l y t h a t

is open.

I n a second countable space, every open cover has a countable subcover. For a proof, see J . L . Kelley, General T o ~ o l o n u .

KakutanCBebutov Theorems

is much more difficult to

As is usually the case in such proofs, it

show that D(K)

189

is dense in C (XI for some neighborhood K of

P

(xo,yo). The proof makes extensive use of the lemmas in the first part

of this chapter. Without loss of generality, suppose xo 6 F. Let neighborhood of

x

compact subset of U

xo.

the flow at

W c X

such that

Let

0

(S) c

'P[-S,Sl

U

and such that

K = cp

Now, let

E S

be a compact

and let S

Q

(W) c X

cp[[-6,61

be a

is a local section for

and S

W be a compact neighborhood of y0

Pick 6 > 0 such that

U.

\

{yo) v F c X \ U,

such that

0

U

U

\

with

and

is a flow box for S.

(S)

[-S,Sl

(S)xW. We will show that [-6,Sl

is dense in

D(K)

C(X). Let

f E C(X)

there is a g

and let 1

C (K* , c p )

E

for x E F

g(x) = f(x)

*

E

> 0. Let K = cp

such that Let

fE E D(K).

Let

ag

xeK* a(P

fE = g+Fv so 2

(x,y) E K and let

IIf-fcll <

(S

to = t,(x)

cp

so

to

af& acp

ag

av

f

E

af&

to

(x) E S

and

(y))l =

af&

(y))l 5 N. Therefore -(q (y)) # -(pt to a0 to av 0(XI). Since 1 * C (K ,cp) there is a tl E R with fElcpt (XI) fE(cpt (y)). Since 1

E

v = 0 on X

\

U and F c X \ U then if x

Therefore fE E D(KI,

E

*

1

F, fE(xf = g(x) = f(x).

and the proof is completed.

The proof just given for T = R

can easily be generalized to yield a

proof of the Kakutani-Bebutov theorem for T Let

By lemma 3,

We claim that

E.

On the other hand, since v = 0 on X \ U, i---(cp

I-(p

u W).

< &/2 for x E X and 6N M = r , and v = v S,M, u

Ig(x)-f(x)l

N = sup l-(x)i,

cp'

as in lemma 7. Let

[-6,61

locally compact connected.

h ( T ) denote the collection of one parameter subgroups in T

is, continuous homomorphisms from the additive group R

(that

to T) and let

Chapter 13

190

P(T)

denote the set of

t

E

t = p(s)

T for which

for some cp

E

h(T)

m

and

If T

s E R.

is locally compact connected, then

u P(TIn n=1

is

dense in T. (X,T)

Now, if

is a flow and x t F

(the fixed point set of the

flow) then it follows from the above discussion that cp E h ( T )

and

s E R.

xcp(s)

Therefore, the proof for T = R

f

x

for some

can be adapted

t o the more general case. (For example, to prove theorem 8 , if (xo,yo)E XxX \ (FxF u A)

xocp(s)

#

xo

with xo

CC

F, let

cp E

h(T)

with

and proceed word for word as in the proof given above using

the real flow defined by

cp.)

Now we turn to the case of discrete groups where, as we shall see, the situation is quite different. We consider only Z a homeomorphism of the compact metric space X. (X,T)

cascade

actions. Let T

be

In order to embed the

in the Bebutov system on the integers, it is necessary

and sufficient to find a dynamical embedding function - that is, an f

C(X) such that if x,x'

E

f(TJ(x))

#

f(TJ (x'))

X with

E

x

f

x',

then

for some integer j.

The obstruction to the existence of a dynamical embedding function is not only the fixed point set, as in the continuous case, but also periodic points.

This is shown by the following trivial but instruct ve

examp1e . Let Y Let by

be a compact metric space, and let q be a positive integer.

X = Yx{O,l, ...,q- 1)

and let T be the homeomorphism of X

T(y,j) = (y,j+l(mod 9 ) ) .

Now suppose f

function for the cascade ( X . T ) . Fly) = (f(y,O),f(y,l),

is a dynamical embedding

Define F : Y+Rq

..., f(y,q-l)l.

defined

by

Then F is a homeomorphism, so

Y

must be homeomorphic to a subset of Rq. Thus, (by choosing a compact metric space which is not embeddable in

Kizkutani-Bebutov Theorems

191

some Rq) it is easy to construct cascades without fixed points for which there do not exist dynamical embedding functions. We consider cascades (X,T) which have no periodic points ( s o if

x

E

X and m

*

*

0, Tm(x)

XI. Moreover, we restrict ourselves to

cascades with phase spaces of finite dimension. cal space X has covering dimension 5 n open refinement f3 of 6 .

dimension n

if every open cover a has an

such that each point is

In this case we write dim X if dim X

S

5

(Recall that a topologi-

n.

in at most n+l elements

We say that

X has covering

n but it is not the case that

dim X

S

n-1.)

We will prove the following Kakutani-Bebutov theorem for cascades. 9. Theorem. Let

(X,T) be a cascade with no periodic PO nts, where X

is a compact metric space of finite dimension. Then

(X,T

is

embeddable in the Bebutov system on the integers.

For the proof, we will make use of the following well known embedding theorem. A proof is in Dimension Theory by W. Hurewicz and H. Wallman (Princeton University Press), theorem V.2. 10. Theorem. Let

X be a compact metric space with dim X

S

n. Then

X is homeomorphic to a subset of R2n+l We will also require an elementary result from differential topology.

11.

Lemma. Suppose A c Rn, i n > n, and f has Lebesgue measure 0 in Rm.

Then f(A)

:

A+Rm

is a smooth map.

For a proof, see

Differential Topology, by M. W. Hirsch, (Springer-Verlagl,Chapter 3, Proposition 1.2. 12. Corollary. Let A and let f

:

A+#

and B

and g

:

be closed subsets of Rn.

B+Rm

be smooth maps.

there is a vector v E Rm with

llvll <

Proof. -

h(a,b) = f(a)-g(b)

Define h

: AxB+Rrn

by

E

such that

Let

Let E

m 2 2n+l

> 0. Then

f(A) n (g(B)+v) =

(a E A, b E B)

0.

Chapter 13

192

Then h is smooth, and it follows from lemma 11 that there is a vector

llvll <

v with f(A)

A

such that v

E

(g(B)+v) =

is not in the range of h. Then

0.

A and B be compact metric spaces of dimension

13. Lemma. Let

and let m 2 4n+3. Then if f there are continuous maps fc

A+Rm

: :

A+Rm

5

n

and g

:

B+Rm

are continuous

and gE

:

B+#

which are

E:

approximations to f and g respectively such that fc(A) n gc(B) =

0.

Proof. We may regard A and B as closed subsets of R2n+1 (theorem R2n+1 to 101, and then extend f and g to continuous functions from Rm. Let fl and g1 be smooth ~ / 2 approximations of f and g respectively, and then, using corollary 12, choose FE and approximations of fl and g1 respectively such that Fc(A)

c/2

Gc, A

GE(B) =

0.

(Actually, it is only necessary to change one function.) Finally, let fc and ge be the restriction of Fc to A and GE to B, respectively. (X,T) be a cascade and let x,y E X be non-periodic

14. Lemma. Let

points with x

f

y. Let m

integers i l , . . . , im

be a positive integer. Then there are

such that the set

i i {T ' ( X I ,... ,Tirn(x),Tii(y),. , . ,T "(y))

consists of 2m distinct points.

It is only necessary to consider the case y = Tn(x)

Proof.

n > 0. If n

2

m,

choose O , l , .

. . ,m-1.

where

If n < m, choose

Q,m,Zm,.. . , ( m - l l m . If K c XxX \ A and I D (K) = [f

I

E

is a subset of Z,

CCXllif (x,y) f K, f(Tm(xl

f

let

f(Tm(y))

for some m

E

11,

and let D(K) = DZ(K). The following theorem corresponds to theorem 8 in the continuous case. 15. Theorem. Let

dim X

5

n. Let

(X,T) be a cascade without periodic points with

( x o , y o ) E XxX \

A.

Then there are compact neighborhoods

Kakutani-Bebutov Theorems

U

and

193

I of Z

V of xo and yo respectively, and a finite subset

such that

DI (UxV) is open and dense in C(X).

Proof. Openness follows exactly as in the proof of theorem 8 . density, let m t 4n+3 and let

I = {il,.. .,im}

To prove

such that the points

Til(xo),Ti2(xO),. . . ,Tim(xo),Ti'(yo),Ti2(yo), . . . ,Tim(yo), are distinct. U

Let

and V be compact neighborhoods of

xo and

yo

respectively

i such that the sets Til(U),. . . ,T "(U), Til(V),. . . ,Ti"(V) are pairwise disjoint.

If h

C(X),

E

define hI E C(X,Rm) by

i hI(x) = (Til(x),., . ,T "(x)).

hI,V : V+Rm

which are the restrictions of hI

respectively. Let be

E

Consider the functions hI,U : U+Rm

> 0, and let

E

f = (fl....,fm) and

g = (g

gm)

and h respectively such that 1,V hI,U (lemma 13). Now define hE on

0

u

hE(x) = f (T-iJ(x)) for x E TiJ(U) and

Tij(U) u TiJ(V) by j=l,.. . , m

J

is an

domain, he

Now if

approximation to h.

E

approximat on of h

so

Then for x

Extend it to an

in its

E

(still called he) on all of X.

(x,y) E UxV

gj y)

J

y E Tij(V) (j = 1, . . .,m).

for

hc(y) = g.(T-iJ(y))

J (XI *

and V

approximations of

f(U) n g(V1 =

f

to U

and

there is an integer j, 1 I J I m

hE(TiJ(x)

*

such that

hE(TiJ(y)l.

Now the proof of theorem 9 follows word for word as in the continuous case.

(See the discussion preceding theorem 8 , and note that

DI(K) c D(K)

for every

I c

Z

and K c XxX \ A.)

It would be

interesting to determine whether the assumption of finite dimensionality of the phase space is actually necessary.

(This is not known even f o r

minimal cascades.1 The embeddability of a (real) flow (X,cpt) in BO(R)

was first

proved by Bebutov under the assumption that the fixed point set F

cp

has

Chapter 13

194

at most two points. Bebutov’s proof is reproduced in V.V. Nemytskii, ToDolonical Droblems in the theory of dsmamical sustems, Amer. Math. SOC. Translation No. 103 (1954). Theorem 1 in its full generality

was

proved

by S . Kakutani (Journal of Differential Equations 4 (19681, 194-201). Our presentation is based on the 1974 University of Maryland Ph.D thesis of Alan Jaworski; theorem 9 is also due to Jaworski. Exercise. Let metric space and

(X,T) be a distal minimal flow, with X a compact T an arbitrary group.

embedding function.

Then

(X,T) admits a dynamical

195

Chapter 14 General Structure Theorems In this final chapter, we study deeper properties of minimal flows and their homomorphisms than heretofore. For an arbitrary pointed minimal flow, we define a family of flows, and extensions, the “PI tower” of the flow. This leads to consideration of the PI flows

-- these are flows

which can be obtained, modulo a proximal extension, from the trivial flow by a succession of equicontinuous and proximal extensions. Moreover, we obtain a general structure theorem (theorem 30):

any minimal flow (again

modulo a proximal extension) is a weakly mixing extension of a PI flow. In the process of obtaining these results, we introduce an important class of extensions, the RIC extensions, and also obtain more information on the

7-topology.

Many of the flows we have studied in this monograph a r e PI flows.

In

particular, this is the case for the point distal flows, and we prove a structure theorem (due to Veech) for these flows as well. We develop methods for constructing examples of point distal flows, as well as PI flows which are not point distal. Finally, we give an example of a weakly mixing flow. In the development of the PI tower and PI flows, we follow the outline in chapter 10 in the monograph of Glasner, although some of our proofs are different. Also Glasner communicated to us the proofs of theorems 27’ and 28. We begin by developing a characterization of equicontinuous extensions which, in a sense, generalizes the fact that equicontinuous flows are homogeneous spaces of compact groups (theorem 6 of the first enveloping

Chapter 14

196

semigroup chapter). A:X j Y

First suppose that X and Y are minimal flows and If y E Y with yu = y and -1 Mln-l(y)p c A (y)], recall (theorem 17 of

is a distal homomorphism.

E = [p E Mlyp

= y] = [p E Y the "Furstenberg structure theorem" chapter) that, if we identify p 1 -1 and p2 whenever xpl = xp2 for all x E A (y), then E is a group Y for p E Ey. with identity u and n-l(y)p = n-'(y) NOW,

let z

let A = n-'(y),

n.

and let 2 = and hence 2

Since

"'

such that z

is distal, z

A

=

X

x, for x

M.

E

It is clear that

is distal, for if "nzp) = i(zp')

Moreover

proximal, then xp and xp' n(xp) = n(xp') Now, if q

and A E

To see that H

4

,.,

of

A

2

to

is well defined.

with zp and zp'

xp = xp'

x E A,

and since zp = zp'.

so

E

define a map H of 2 by H (zp) = zqp (p E M). Y' 9 4 is well defined, suppose zp = zp' (p,p' E MI. Then,

from the definition of E A

G

are proximal f o r all

is distal, we have

E A,

is an almost periodic point,

is a minimal flow. Define the homomorphism

Y by ;(zp) = yp for p

implies xq

x

z , xp = xp'

we have xqp = xqp',

for all x E A so

and since x E A

zqp = zqp',

and H

4

is well

defined. Moreover, it is easily verified that H is continuous, that 4 H H = H (q,q' E E ) and that H is an automorphism of the flow 4 q' qq' Y q (2.T). If r E M with ;(zr) = y then yr = y and r E E Hence, if Y' 1 we write K for the group of maps [H lq E Eyl. we have A (y) = Kz. 9 Moreover, if p E M then by theorem 17 of the Furstenberg structure

--

--

--1 1 theorem chapter R (yp) = R (y)p = (Kz)p = K(zp). Thus, if x

is a distal extension, K

is a group of automorphisms of

the minimal flow 2 and the quotient flow Z/K

is isomorphic with Y.

We note also a fact which will be used later in the chapter:

x

E

H

-1 (y), A = S ( X , x ) and F = I(Y,y),

then S ( 2 , z ) =

if

n P-lAp. BEF

if

a

E C,

a

E

B(Z,z)

if and only if x'a = x'

for all x'

E tr-'(y).

For,

General Structure Theorems

Now x'

E

-1

n

(y) if and only if x' = xp, for @ E F,

p

E

F o r xi3ap-l

(theorem 7 of

a E 8(Z,z) if and only if xpa = xp,

the "algebraic theory" chapter) so for

191

= x.

It also follows from this representation that 8(Z,z) is normal in F. Now, suppose that

is an equicontinuous extension.

R

the action of K on Z

is jointly continuous. For suppose qn, q

with q -+q, and let pn. p E M n

x

E

action of T uous so

But n(xqn) = n(xq)

A.

if and only if xq p +xqp n n = y,

(and therefore of M I

xqnpn+xqp.

for

since

qn' EY and the -1 on the fiber R (y) is equicontin-

K

Almost the same argument shows that

topology inherited from E I

Ey

E

Then H (zp 1 = zq p n n' 4n

with pn+p.

and H (zp) = zqp. Now zq p +zqp q n n every

In this case,

(with the

is a compact group.

Y

Thus we have proved most of the following theorem. 1. R

Theorem. Let

R :

X+Y

be a homomorphism of minimal flows. Then

is an equicontinuous extension if and only if there is a flow N

homomorphisms

M

11:

Z+Y

and $ : Z+X

with

R$

=

R

and

.,R

Z

and

a compact

group extension. Proof.

If

shows that

R

is an equicontinuous extension, the preceding discussion

2,;

for any x E A =

and $ exist as in the theorem. R

-1

(Define +(zp) = xp,

(y).) On the other hand, if the conditions of the N

theorem are satisfied, then equicontinuous extension, so

R:

Z+Y

R :

X+Y

is a group extension, hence an is an equicontinuous extension.

(This follows from the openness of the distal homomorphism

#.I

extension is frequently taken as the definition of an equicontinuous extension.

Chapter 14

198

Before proceeding, we need to prove several facts about the t-topology of C = Mu. a.+p

If {ui}

is a net in C, and if we write

this refers to convergence in M

inherited from BT).

If we wish to indicate convergence in G relative

to the r-topology, we will write 2. Lemma. Let

{a 1 i

topology of M)

a +p i

that there

1

s

t

a.+a. 1

be a net in C. Suppose (with respect to the E

t

M.

Then a +pu. i

be a net in T with

{t.}

Proof. Let

(with respect to the topology

a subnet {t. } Ji

t +u. i

of {ti}

Then it is easy to see

such that

a t i

Ji

+pu

T

Therefore a +pu. i

Of course, since the t-topology is not Hausdorff, the net

{a,)

will in general converge to other elements of C as well. Let

v E J(M)

with v

#

u.

Then Mv = Cv

v and the T-topology is defined on Mv.

It follows from exercise 7 of

the second enveloping semigroup chapter, that as groups, via the map

awav.

is a group with identity

G and Cv are isomorphic

The next lemma shows that this map also

defines a homeomorphism of the t-topologies. 3.

Lemma. The map a w a v

Proof. Let

defines a r-isomorphism of C onto Cv.

A be a r-closed subset of C. Then A = Avu =

AVVU c ( A v o v )C~ (AOVOV)U = ( A o v ) ~= (AOV)UU c ( A O V O U ) ~= (Aou)u = Hence

(Avov)u = A and

(Avov)v = Av.

Therefore Av

A.

is t-closed in

cv . 4.

Lemma. Let

K and L be t-closed subsets of

C.

Then KL

is

r-closed.

Proof. Let a and

{ti)

E

(KLIou n C.

Then u = lim kieiti where

{ki), {-ti}

are nets in K, L, and T respectively with ti+u.

Let

General Structure Theorems

!. t + r E M. i i

k

a

e. t i

i l

E

E

Then r

K$ti

199

Lou, and ru E Lou n G = L. Also

E

and Kliti+Kor

so

a

E Kor,

ar-lu

E

Kou n G = K and

Kru c KL.

Since in any group the product of two subgroups, one of which is normal, is again a subgroup, lemma 4 immediately yields 5. Corollary.

If A

and B are r-closed subgroups of G with

€3

is a 7-closed subgroup of G.

normal in G, then AB

6. Theorem. Left and right multiplication in G

are t-homeomorphisms.

Proof. It is sufficient to show that left and right multiplication are t-continuous. If {a 1 n if {t 1 n pantn-+pa

is a net in G

is a net in T with t + u n and therefore pa &@a. n

and a E G with a s a , then n then a t -+a, n n

so

if B E G,

Hence left multiplication is

continuous. To prove that right multiplication is continuous, it is sufficient to show that Ap

is r-closed whenever p

E

G and A

is

r-closed. To this end, note that Ap c clr(A/3) = (ApouIu c (Aopou)u = (AoBuIu = (Aop)u =

Aop

n

G, 7p-l

E

AP.

(The last equality holds since if

Aou n G = A

Therefore clz(Ap) = Note that we may

AP

and ;r c A/3;

and Ap

is

r

E

(Aoplu =

clearly Ap c (Aof3)u.)

z-closed.

apply theorem 8 in the "Joint continuity

theorems" chapter to conclude that multiplication in G

is jointly

continuous, since that theorem requires that the group is Hausdorff. Indeed, what we will do next is to overcome this deficiency

--

given a

t-closed subgroup of G, we will construct a normal subgroup so that the quotient group is Hausdorff. Let

F be a T-closed subgroup of G. Then F (with the relative

r-topology inherited from GI

is a compact T1 group on which left and

right multiplication are continuous. Let N denote the neighborhood

Chapter 14

200

filter at the identity u of G

NF

(for the t-topology) and let

denote the corresponding neighborhood filter in F, so

NF = [V n FIV 7. Lemma.

E

N1.

If D c F, then cltD = n[DV-'lV

E

NF].

Proof. This follows from the fact that the sets XV for V E NF constitute a neighborhood base at

x for the t-topology of F

(which

in turn follows from theorem 6). If F

t-closed subgroup of G, we define

is a

H(F) = n[clTVIV E NFl. H(F)

is of fundamental importance for the

developments of this chapter. If a E C,

then a

H(F)

E

if and only i f

every t-neighborhood of a meets every t-neighborhood of u. a

H(F)

E

{p } J

if and only if there is a net

ly converges to a and t o

u

Hence

in F which simultaneous-

(with respect to the

r-topology, of

course 1. is a t-closed subgroup of F which is closed under

Theorem. H(F)

8.

topological automorphisms of F

(so

is a semigroup. Let

Proof. We first show that H(F) let

V,W

E

NF. Then yW

n V

x E H(F) c cltU, so

xy

E

also V E NF H(F)

E

E

V for some w E W.

and xy

NF,

so

E

-1 clTVu c cl,VW-l.

by lemma 7 ,

is arbitrary, we have xy E n[clrVIV

is a semigroup. Now if

x,y E H(F),

and

Then

(clTU)yw = clT(Uyw) c cltV. Also

xyw E cl7V

for every W

(cltV)W-'

so yw

# 0,

Uyw c V for some U E NF, so

is normal in F).

H(F)

x

E

H(F),

xy E

E

cltV.

That is Since

IF]= H(F),

then H(F)x

and

is also a semi-

group, and by lemma 6 of the second enveloping semigroup chapter, H(F)x contains an idempotent v. v = u, u E H(F)x group.

and x

Obviously H(F)

But H(F)x c F2 = F and F

is a group so

is invertible. This proves that

is t-closed. Now let

1)

H(F)

is a

be a topological

General Structure Theorems

automorphism of F

(that is,

If V

homeomorphism).

r)(clTU) c clrV and so

is a group automorphism which is also a

7)

NF, there is a U

E

20 1

v(H(F)) c H(F).

E

NF with r)(U) c V. Then

In particular, H(F)

is

invariant under inner automorphisms of F, which is to say that H(F) is normal in F. is a r-closed normal subgroup of G, then H(F)

9. Corollary. If F

is a normal subgroup of C. Proof. If g

E

G, the map p-gpg-l

defines a topological

automorphism of F so gH(F)g -1 c H(F). 10. Theorem. F/H(F)

is a compact Hausdorff topological group.

Moreover, if K is a r-closed subgroup of F, then F/K

is a Hausdorff

space if and only if H(F) c K. Proof. We show that limits of nets in F/H(F) be a net in F/H(F). -1

T

x +b,

J

b E

Then

<J

= H(F)xj

-1 t and x a -+ba

t

then x.a +u J H(F)a and H(F)b = H(F)a.

where xj -1

J

(5

are unique. Let E

If x &a

F.

-1 so ba

J and

J

E

H(F),

This proves that limits in F/H(F)

are

unique (note that we have implicitly made use of the fact that the quotient map from F to F/H(F) Multiplication in F/H(F) theorem

8

is open) so

F/H(F)

is Hausdorff.

is separately continuous, so it follows from

in the “Joint continuity theorems” chapter that F/H(F)

topological group. If K is a r-closed subgroup of F with K then the equivalence relation defining the map F/H(F)++F/K and it fsllows that F/K Hausdorff, and let x t

a +u

J

x

E

t

and a -+x.

J

E

Then there is a net

Then Ka + K

J

and Ka +Kx,

J

{a 1

J

H(F)

3

is closed,

is Hausdorff. Conversely, suppose F/K

H(F).

is a

is

in F with

so Kx = K and

K. This completes the proof. Suppose

R:

X+Y

is a distal extension, xo

E

X with xou = x0’

Chapter 14

202

yo = n(xo), B(X,xo) = A, 9(Y,yo) = F. of

F. Then each 6

F

E

Suppose A

is a normal subgroup

defines an automorphism of X by

H (x q) = xopq (q E MI. To see that H is well defined, suppose P O B xoq = x0r (q,r E MI. Let g = qu, h = ru, so x0g = x0 h and g = ah for some a

E

A.

Then x0/3g = x0 @ah = x0a'ph = xO/3h (where a'

Thus xopqu = x0Pru, n(x,pq)

=

yopq = y@

so =

x0Bq

E

A).

and xoPr are proximal. Since

y0r = n(xOPr),

and n

is distal, xoPq = xa&-.

Hence H is well defined and it is easy to see that it is an automorP phism. Moreover, it is immediate that the map j3-H defines a group

13

homomorphism of F to the group of automorphisms of X and the kernel of this homomorphism is A.

Thus we may regard the maps HP

an action of the quotient group K = F/A

[HP(x)Jp E

F] =

Kx

(for any x

E

n

-1

on X. Note that n

(y))

so

W K

is a homomorphism and K

automorphisms of X with Y

isomorphic with X/K,

for some

(p E

For,

if 6

E

F, a

E A,

-1

(y) =

is isomorphic with Y.

Conversely, if n : X-+Y

is normal in F = 9(Y,yo).

as defining

is a group of then A = S(X,xo)

then xoP = p(xo)

K and xO@a = (p(x,)a = (p(xoa)= (p(x,) = xo@

so

Pap

-1

E A.

It is routine to check that the construction in the previous paragraph defines an isomorphism of F/A

with K.

Note that there is a certain resemblance between the proof just given and the first part of the proof of the existence of a group extension, given an equicontinuous extension. This ,is not accidental; indeed, it is not difficult to show that the minimal flow 2 (the group extension of

Y) constructed in that proof is, in case

A

is normal in F,

isomorphic with X. Suppose now, in addition to the assumptions of the preceding discussion ( n is distal, and A is the same thing, that F/A

is normal in F) that A > H(F), or, what

is Hausdorff. Then F/A

is a topological

General Structure Theorems

203

group (theorem 10).

W e show i n t h i s case t h a t

is a group extension of

K

{pi)

c o n t i n u i t y theorems" chapter, t o show t h a t i f

pi&@,

H

then

(x)+H

pi

we must show t h a t r v = r.

6

(XI, for

xopiq+x0pq.

o i

and

0

0

w

E

s-topology of also

qw = q

with

J(M)

and

Gw)

Since

and t h e r e are

al,a2 E A

x9pqu = x0 ru.

Then

y.p.q+y

piq+rw

a pqr 1

-1

Then

0

0

0

piqr

Now

xopqv = xopq.

-1

t

u+u.

But

Apqr-lu = Au = A -1 xOPqr u = x

u = az, s o

x pq = x pqv = x r v = x r. 0

0

(with respect t o the

is Hausdorff,

such t h a t

with

v E J(M)

y r = yoq.

is d i s t a l

K

F

q E M,

M and l e t so

r

(lemma 3). F/A

E

0

1 1

s

then

piqu&ru

(3iqr-1u&pqr-1u,

Piq+r

and s i n c e

0

is a net i n

Equivalently, if

X.

E

Suppose

y p q = y q

Then

x

n ( x pqv) = y qv = yOrv = yor = yoq If

(with group

For t h i s i t is s u f f i c i e n t , by theorem 8 of t h e " J o i n t

K = F/A).

with

Y

0'

That is,

x p q-+xor =

o i

xopqq, and t h e proof is completed. Conversely, suppose

Y = X/K.

is a compact group extension,

n : X+Y

Then, as we have observed, [H I p E Fl

collection

B

of

K,

H

(8,)

+Ha

F

and t h e

of automorphisms may be i d e n t i f i e d with H(F) c A.

I n f a c t , we show i n t h i s case t h a t t h e r e is a net

is normal i n

A

with

p,&p

for some 6

t

and

F.

E

For l e t

pi+u.

Since

E

K.

H(F).

so

Then, by compactness

pi&u.

t h e r e is a n e t

{ti)

pi in

T

with

ti+u

such t h a t

Biti+u

M.

in

Then

(xoti) =

H

pi x ,9 t . + x O u = xo 0 1 1

and a l s o

T

with

si-+u.

(xoti)+Hg(xOu)

Since

t

f3,-+f.3,

Then, as above

H

pisi+f3 ( x s )+xoB i

6, 0

pi ( xos i ) + H 6 ( x0 ) = xo. Therefore, x0B = x0

H

Suppose now t h a t

= Hg(xo),

so

Hg = HU,

pi

t h e i d e n t i t y automorphfsm. in

H

H(F) c A

(but

A

and

for some net

{siJ

and @ E A.

is not n e c e s s a r i l y normal i n F)

Chapter 14

204

Z

Let

is

(2

be the space constructed at the beginning of the chapter

an extension of X and there is a group K of automorphisms of 7. such

Z/K = Y).

that

n B-IAB. If p E H(F) and B E F. PEF is normal in F, P-'pS E H(F) c A, so p E and we

x,

then, since H(F)

i. Since

H(F) c

have

above to Z

=

Then B ( Z , z ) =

is normal in F, we may apply the discussion

Y and conclude that Z

and

of Y. Therefore by theorem 1, X

is a compact group extension

is an equicontinuous extension of Y.

On the other hand, suppose we are given n: X-+Y, extension. Let theorem 1.

Z

an equicontinuous

be the compact group extension of

Then (if B(X) = A, g(Y) = F, & ( Z ) =

Y constructed in

i),H(F) c i

and

, d

A c A,

so

H(F) c A.

We summarize the preceding discussion in the following theorem. Theorem. Let n : X+Y

11.

be a distal extension of minimal flows,

X with x0u = xo, yo = K(x,),

x0

E

R

is an equicontinuous extension if and only if H(F) c A,

A = B(X,xo)

group extension if and only if H(F) c A which case Y = X/K, Let IT: X+Y and yo = n(xo).

with K

and A

and

F = B(Y,yo). and

is normal in F

Then is a

H

(in

isomorphic to F/A).

be a homomorphism of minimal flows, with The extension n

incontractible) if, for every p E

x0u = xo

E

X

is called RIC (relatively M, II-1 (yap) = xoFop, where

F = G(Y,yO). If Y = 1, the trivial flow and X+l incontractible. Clearly X

x0COU >

is called

is incontractible if and only if X = xoGou.

Note that if the acting group T (X,T) is incontractible

is RIC, then X

is abelian, then every minimal flow

(if t E T, ut E C and xot

E

XG and

= XI.

The next result shows that one can always "interpolate" an

General Structure Theorems

205

equicontinuous extension into a R I C extension. First we require two t-topology lemmas.

12. Lemma. Let a.r.+q

E

1 1

{a.) be a net in C, {ri) a net in

and r +p i

M,

BT and suppose

1

M.

E

then {ai) has a subnet

that a k,qp-'u. ki Proof. Let A. = [ a li J i -1

qp u

E

cl A

qp-lu

E

A ~ O U

n G = cl A t

13. Lemma.

if N

It is suff cient to prove that

Now a r E A or i i j i'

J'

T

jl.

2

Let

2 J,

for

F be a t-closed subgroup of

We have aiti-+q,

in T

with ti+p.

Then if N

q

E

AJop and

J'

where

G and

-1

qp u

is a relative t-neighborhood of

Proof.

so

q

Fop. Then

in F, q E Nap.

{ai) is a net in F, {ti)

is a net -1

By the lemma just proved, a subnet a -+qp u. ki

is as in the statement of the lemma ak E N, a t i ki ki

E

Nt ki

and q E Nop. 14. Theorem. Let yo = x(x0),

TI:

A = G(X,xO).

subgroup of C such that a minimal flow Z

$A

= TI,

and

F = C(Y,yo).

A c B

c F

and

Let

B be a t-closed

FIB Hausdorff. Then there is

and homomorphisms A : X+Z,

# : Z+Y,

such that

is an isomorphism if and only if B = F.

Define a relation

-

and x = xop, x' = xoq, p,q

E

on X by

x-x'

M with qp-lu

whenever n(x) = x(x') E

defined, suppose x = x0r, x' = xos (r,s E MI. -1 -1 (sq-lu)(qp-lu)(rp u) E ABA = B. Obviously tion.

with x0 u = xo,

B(Z,A(xo)) = B, and $ an equicontinuous extension. The

extension $ Proof.

be a R I C extension xo E X

X+Y

B.

To show

-

is well

-1 Then sr u =

-

is an equivalence rela-

We show it is closed. First note that if x-~', xop = x, xoq = x'

Chapter 14

206

then since n

where r

x q = x r,

0

0

E

Now suppose {xn) x'-+x'. n

Let

and

xn = xopn. x; -1

must show that qp u qn E Fop,. -1

qnpn u

so

-1

-1 (y p) = x Fop, 0 0 Fop so we may assume q E Fop.

x0q

is R I C ,

E

= x' E n

{xi)

(n(x)) = n

are nets with xn-x;,

= xoqn,

and let

and suppose p +p,

E

Napn

qp-lu E n[clTNIN

-

-1 we have qnpn u

Nou n G = cl N.

E

7

a r-neighborhood of B

T-invariant.

Nap,.

be the quotient flow Z = X/-, with A$ = n. Let

z g = zo

if and only if xog

0

-1

g = gu u

E

B,

so

C(Z,zo) =

Finally, we show that $

it is sufficient to show $

$(zl) = $ ( z 2 ) . of

2x2.

pv = p, so and

Let

-

yoq = n(xOq). x2

Also

is closed.

(Caution: If

t

E

T,

qtlptl-lu

=

Clearly we have homomorphisms

zo = A(xo).

- xo

Note that i f

i f and only if

is equicontinuous. Since F/B

is Hausdorff,

is distal (theorems 10 and 11).

Then yop =

Suppose

(zl,z,) is an almost periodic point

z1 = zop, z2 = zoq, (p,q E M) and let v

-1 ap = qp up = qv.

is

B.

We will show that

z 1v = z .

Then

x =

so

xt = x0 pt, x't = xoqt, it is not in general the case that -1 -1 -1 -1 need not be in MI. qtt p u, since t p

$ : Z-+Y

N

so

in F1. Since F/B

This proves that the equivalence relation

A similar proof shows that it is

B,

E

Hence

-1 Hausdorff, this intersection equals B and qp u E B,

g E G,

of

NOW let N be a r-neighborhood of B in F. From the

Nop and qp-1u

A : X+2,

We

for every r-neighborhood N

is a t-neighborhood of qnpilu in F, and by above qn E

Let Z

n

B. By the above discussion we can suppose

by lemma 13, qn

in F.

xop-xoq = x'.

x +x,

q +q. n

n

definition of the equivalence relation

q E

so

$(zap) = SO

J(M)

such that

Jl(zoq) = yoq, a = qp-lu E F,

It follows that n(xoqv) = yoqv

-1 qvq u = u E B,

E

= y ap = y p =

0

if x2 = xoq, x v = xoqv

and hence z v = zoqv = A(xoqv) = h(xOq) = z q = z2. 2 0

2

Thus

0

- xoq =

General Structure Theorems

(zl,z2) and

(Z1'Z2 v =

proves that

is dista

$

$

theory chapter that

( z ,z )

1 2

. It

and A c F,

F

B

H(F), F/B

2

B

is an almost periodic point. This

follows from corollary 5 of the "algebraic

is an isomorphism if and only if

We apply theorem 14 to of

207

B = AH(F).

Since H(F)

is a r-closed subgroup of

B = F.

is a normal subgroup

G, and since

is Hausdorff. Therefore, assuming the notation of

theorem 14, we obtain: 15. Theorem. Let

n : X-+Y

be a RIC extension. Then there is a mini-

mal flow Z and homomorphisms $ : Z+Y, G(Z) =

u : X+Z

such that

AH(F) and $ equicontinuous. The extension $

@cr = n,

is an

(in which case no non-trivial

isomorphism if and only if AH(F) = F

equicontinuous extensian can be interpolated into n). Proof. Most of the assertions follow immediately from theorem 14. If

A : Z+Y Ap =

D

5

R

is a non-trivial equicontinuous extension, p : X-+Z

and O ( Z ) = D then A c D $ F

and

with

H(F) c D, so AH(F) c AD =

F.

Note that theorem 15 provides the "largest" equicontinuous extension which can be interpolated into n. equicontinuous, H(F) c A

(In particular, if

and B(Z) = AH(F) = A,

so

R

is already

2 = X

and 3 =

R.

1

Given an arbitrary extension of minimal flows, we develop a method for "lifting" this extension by proximal extensions to obtain a RIC extension. That is, if flows X'

R:

X-+Y

and Y',

is a homomorphism, we will construct minimal which are proximal extensions of

respectively, and a R I C homomorphism

R' :

X'+Y'

X

and

Y

such that the obvi-

ous diagram commutes. For this purpose, we need to develop some

properties of RIC extensions, as well as an alternate characterization.

16. Lemma. Let

F be a r-closed subgroup of G. Then

Chapter 14

208

if a E G, Foa = Fou

fi)

if and only if a E F,

(ii) if p,q E M, then Fop = Foq

-1

Foa = Fa SO

If Foa = Fou, then a = au

(i)

Proof.

c

og:

-1

Foa

E

Fou n G = F.

-1 and since a

oa = F a ,

q E Fop.

E

If a E F,

-1 F, also Foa c Fou,

Foa = Fou. If Fop = Foq, clearly q E Fop.

(ii)

{ai)

in T with

{ti}

in F and

only if a

E

E

Fop, there are nets

aiti+q.

= Fop, so

Foq = Fop.

Foa = Fou

F, so the group of this flow with respect to

0(Fop) = xop

defined by

by (ii) of lemma 16, q

E

if and

FOU,

is a minimal flow, x E X with 0

and B(X,xo) = F, then there is a homomorphism

0

Then

we define the quasi-factor U(F)

By ( i ) of lemma 16,

B(W(F),Fou) = F. Moreover, if X x u = x

and

1

i i

W(F) = [Foplp E MI.

M by

0

ti+p

is a t-closed subgroup of G

If F

If q

and also F0a.t = Fout.+Foup

Foa t +Foq i i

of

if and only if

0 :

( e is well defined, since if Fop Fop and it follows easily that

Since the minimal flows X and U(F1

2I(F)-+X

= Foq,

xoq

=

then

x0p). is a

both have group F, %(F)

is an extension of every

proximal extension of X, and since U(F)

minimal flow with group F, it is the maximal proximal extension of X. 17. Lemma. Let

n : X+Y

be a homomorphism of minimal flows and let

x0 E X with x0 u = xo, yo = n(xo),

n

-1

(i)

If

(ii)

The extension n

(yo) =

F = Y(Y,y0).

and

Then

is RIC, it is open.

TI

is RIC if and only if it is open and

x0 Fou.

(iii) If n

is distal, it is RIC.

Proof. Suppose n = H-1 (yO)op, and

then x0Fop = n

-1

is RIC.

n

Then, if p

is open. -1

(yo)op = n

If

(yap),

E

M. n-1 (yap) = xoFop = x0Fouop

is open and

H

so

n

n-1 (yo) = x Fou, 0

is RIC. The proof of the

General Smtcmre Theorems

converse is immediate. Suppose n "algebraic theory" chapter phisms are open n and xoFou

-1

= H

-1

A

(yo) = n

(yo)ou = n

-1

-1

-1

209

is distal. By theorem 7 in the

(yo) = xoF, and since distal homomor-

(y u) = n

-1

0

(yo).

(y,)ou,

By (ii),

TI

x0 F

SO

= n

-1

(yo)ou,

is RIC.

Note that it follows from (ii) that the property of being RIC does not depend on the choice of basepoint. Recall (exercise 7 of the "Disjointness" chapter) that if H : X+Z, I):

Y j Z are homomorphisms of minimal flows, then X and Y are said

to be dis.joint

Z

O X

(with respect to

and $1

71

if the relation

R = [(x,y)ln(x) = +(y)] is a minimal subset of the product flow XxY. n* (In this case, we also say "H and I) are disjoint" and write TI 1 3 . 1 18. Theorem Let n : X-+Y

(x0u

n(x ) = yo 0

= x 1. 0

SO

x

is RIC if and only if

X and

U(F)

R = R = f(x,Fop)(n(x) = 6fFop) = yopl. Suppose that n ,6

Let

is RIC.

Then n

(where F = C(Y,y,l).

are disjoint over Y Proof.

be a homomorphism of minimal flows with

Then, if

(x,Fop) E R, n(x) = yop and x

n

-1

(yap) =

x0(Fop),

Then (lemma 16) Foq = Fop, and

f o r some q E Fop.

= x q 0

E

(x,Fop) = (xoq,Foq)E (x0,Fou)T. This shows that the relation R minimal set, and X and suppose R €)(Fop),

V(F)

so

there is a q

(xoq,(Fou)oq) = (xoq,Foq) and

x = x0 q

E

are disjoint over Y.

is a minimal set, and let x

(x,Fop) E R

xo(Fop).

Thus

-1

H

71

E

E

-1 T[

(yap).

M such that

Fop = Foq. Then q

(yap) = x0 (Fop)

and

E

TI

is a

Conversely, Then n(x) = yop = (x,Fop) = Fop and is RIC.

Now we are ready to construct the "lift" of an arbitrary extension to

a R I C extension. Let

Y' = [xoFoplp E F

= G(Y,yOI).

MI

Let

Y'

be the quasi-factor of

(as usual X'

x0

E

X defined by

X with x0u = xo, yo = n(xo)

be the minimal flow

and

Chapter 14

210

(x0 ' x0 Fou)T XxY'. yb

=

X' = X v Y',

We write 0

(x,xoFoq) E X',

r E M, so

x

we have

= x r E x For = x Foq. 0 0 0

there are nets

(x x Fouls s

n n

0' 0

if and only if x

(x.xoFoq) E X'

Proof. If

{an} =

(xoansn,xoFousn)+

in F, { s

MI.

To see that 0

0

e

n'

Let

as the "supremum" of X

:

E

xoFoq.

(x,xoFoq) = (xo,xOFou)r for some Conversely, if

x

in T with sn+q,

Y'+Y

E

E

xoFoq

x a s +x. O n n

F = B(Y,yo),

E

xOFop c n

then Foa = Fou,

U(F)

are disjoint over Y'

Let

E

is RIC.

so

J J

and

0

= lim xoajtj,

where

aJ

E

F and

and

t .-+q. J

E

( ( x r,xoFor),For) 0

X'

M. Then x p = x r. Also a.t E Ft so r E Foq, and 0 0 J .I j Foq. Then < = ((xop,xoFop),Foq) = ((xop,xoFoq),Foq) =

a t -+r

For =

0

18) to show that

We show

(with respect to the homomorphisms n'

x Fop = x Foq and x p 0

so

be the projection maps

It is sufficient (theorem

n'

(yap)

and 9(Y'yb) = F.

and 8' : X'+X

that

-1

4 and yop = yoq. Moreover,

0

0

Then

8(ybp) = B(xoFop) = yop

by

n'(x p,ybp) = ybp and B'(xop,ybp) = xop. Then en' = ne'.

x p

then

X' .

is well defined note that

= x Fou = y' 0 0'

X'+Y'

8 :

(x,xoFoq)

(yap) n n-1 (yoq) *

is proximal, for if a

x0Foa

}

n

(x,xoFoq), so

if x Fop = x Foq, n-1

yba =

where

x a s x Fouoa s 1 = (x a s x F o a s 1 = (Onn' o n n Onn'O n n

We define a homomorphism

8

(X',x&) = (X,X,) v (Y',yb),

note that its construction depends on the "basepoints" xo and

19. Lemma.

E

or

is a minimal subflow of

Thus X'

xb = (xo,yb). (We regard X'

x Fou and

and Y';

(p

MI.

= [(xop.xoFop)Ip E

E

((xo,y~),Fou)T, and it follows that €7 is minimal.

General Structure Theorems

21 1

Concerning this construction, we have the following theorem. 20. Theorem. The diagram

X’-Y’ n‘

is commutative, n‘

I

and 0

and 8‘

is RIC.

le

are proximal. 0

Moreover, if X

is RIC

is an isomorphism if and only if

is a metric space, then X’

and Y’

n

are also

metric . Proof.

We have already shown that

(x,y‘), (x,z‘) E X‘ = X v Y‘,

so

n’

s RIC and 0

proximal.

= x = B‘(x,z’).

e’(x,y

Suppose

Then

(x,y’) = (xop,xoFop), (x,z’) = (xoq,xoFoq), and xop = xoq. Then y p = n(xopl = n(xoqf = yoq, and since 8

is proximal

0

(x0Fop,x Foq) 0

E

P(Y‘).

It follows that

(x,y‘) and

(y’,t’f =

( x , z ‘ ) are

proximal. If t3

is an isomorphism, then the RIC homomorphism II‘,

(x,p,y;p)

w y ’0p

may be regarded as the map

is RIC, and suppose B(xoFop) = @(xoFoq).

Then yop = yoq, and x0Fop = n metric, then 2’

which in

that is, the homomorphism n, so

turn may be regarded as xop+yop,

is RIC. Finally, suppose n

(xop,yop)k+y0p,

-1

is also metric,

(y,p) so

= n

-1

(yoq) = xoFoq.

Y’ and X’ = X v Y‘

If X

is

are metric.

This completes the proof. Note that if 0

is an isomorphism, so is

e’,

and

n’

is

essentially the same as n. We are going to use theorems 15 and 20 t o carry out an important construction, the so-called “PI tower“ of a minimal flow. First we develop some notation. Put Go = G, G1 = H(G), G2 = 9(G, every ordinal number a, Ga+, = H ( G a ) . Ga =

nC @
@‘

Clearly each Ga

If a

, and for

is a limit ordinal, put

is a T-closed subgroup of G, and each

n

212

Chapter 14

Ga+ 1

is invariant under topological automorphisms of

an easy induction that Ga

Ga'

it follows by

is a normal subgroup of G. Let

smallest ordinal such that Gu+l = Gu

for all u

Gu = Gu

(so

be the

u

and

2 u)

write Gm for Gu. We require one more technical lemma. 21. Lemma. Let A

normal in G.

and B be vclosed subgroups of G, with B

Then AH(AB) = AH(B).

Proof. Clearly H(B) c H(AB), H(AB1 c AH(B) (a E A, p E B).

(so

AH(AB) c AH(B)).

AH(B)anpn&AH(B)a&

7 = ap E

B, (B n AH(B))pn

(B n AH(B))pn&(B

7

and an/3n+u.

H(AB),

n

in A

(corollary 91,

Similarly, AH(B)Pn&AH(B). =

B

n

AH(B))/3

AH(B)pn

and B

Then

is normal in G

and since H(B)

we have AH(B)p,&AH(B)p. E

Let

We show that

Then there are nets {an) and (8,)

respectively such that anp nL a p

pn

AH(B) c AH(AB).

so

Now, since

and it follows that

and also

(B n AH(B))pn&B

n

But

AH(B).

is a Hausdorff space (since B n AH(B) > H(B)) so B n AH(B (B n AH(B))p = B n AH(B) and /3 E AH(B). Hence 7 = a@ E AH(B),

and

the proof is completed. Now we can construct the PI tower of a (pointed) minimal flow (PI stands for "proximal isometric"). Let let x

0

E

X with x0u = x0'

X

=

(X,T) be a minimal flow and

Recall that, starting with an arbitrary

extension of minimal flows, theorem 20 "lifts" the extension to a RIC extension, via proximal extensions. If a given homomorphism is RIC, theorem 15 "interpolates" an equicontinuous extension. The PI tower is obtained, starting with the homomorphism of X onto the trivial flow, by alternately applying theorems 15 and 20 Let

X0 = X and

K~

the homomorphism of Xo

onto the trivial flow

General Structure Theorems Z

0

xb

= 1. :

By theorem 20, no

X1+Y1.

213

can be lifted to a RIC extension

Thus we have a commutative diagram

The extensions

go

and 96

example, if T

is abelian

X

is precisely when

are proximal.

(In many cases

-- for

-- Y1 will itself be the trivial flow; this

is incontractible.) The construction in the proof

of theorem 20 gives rise to basepoints

x1 and y1 of

X1 and Y1

respectively, but we suppress them notationally for the time being. (This will also be the case for the application of theorem 15 below.) then O(X

If B(Xo) = A,

1

)

= A

and B(Y1) = O ( Z o ) = C.

"6: X1+Y1.

Now consider the RIC homomorphism are homomorphisms x1 : X1+Z1 where

Q1

and

$i1

:

Z1+Y

1

By theorem 15, there such that

"6 = !Iilxl,

is an equicontinuous extension, and O(Z1) = AH(G) = AG1.

Thus we have a diagram

-

y1

*l

where

j,

is the identity map.

a RIC extension,

H'

. Xz+Y2,

1'

Now we apply theorem 20 again to obtain

and then apply theorem 15 to interpolate

an equicontinuous extension JI,: Z +Y2. 2 commutative diagram

Summarizing, we have a

Chapter 14

214

The extensions homomorphisms

(so

0

arid 6;

i

are proximal, ji are the identity

B(X2) = S(X0) = A),

the n;

are RIC and the $2

are equicontinuous. The group of Z 2 , B ( Z 2 ) = AH(B(Y2)) = AH(B(Z1)l =

AH(AG1)

=

AH(G1) = AG

2

(lemma 2 1 ) .

Continue this process, applying theorems 15 and 20 alternately. Inductively, suppose P minimal flows, nP B ( Z ) = AG

P

where

P

B

is a homomorphism from

and Z

X to Z S(X P P’ B

P =

are

A,

is RIC, and 0

L7(YP+l) = AG 1.

P

are proximal ( s o

and 0’

P

S(XP+l)=

diagram

Jp+1 xB+ 1

$B+l

is equicontinuous,

B(ZP+l)= AH(AG

A

Then we apply theorem 15 and lemma 21 to obtain a

P

where

and

By theorem 20, there is a commutative diagram

13’

R’

is an ordinal number and X

B

)

jP+lis the identity map,

= AH(GB) = ACp+l.

and

and

Gerieral Structure Theorems

is a limit ordinal, and for /3 < a,

Suppose a and 2

XP

E

x u = x

such that

Define X a = space

ll X P
and let R

P

whose

P

B'th

(x ) = z a

P' =B

and B ( X 1 = A, % ( Z

= n (x

P

B

P

= AG

8'

be the point in the product

(x 1 = x

coordinate

P

aP

and let

P

-

Xa = x T a

is a minimal flow.) Similarly, let Za =

Xa

P
be the homomorphism induced by the

n a : Xa-+Za

a

P 1, B

=p: P

minimal flows X

have been defined with

(That is, let xa

XP. P
Since x u = x a' a

Then

X 4 2

and a homomorphism

8'

xP

215

where

a'

for P < a .

( z ~ = ) z P~,

the next to last equality, let

y E

n AGB'

(6 < a)

nP,

Also, B ( X a ) =

Then 7 = a g P 8'

P
where

E A, gP E GP. Ue may regard { a P } and { g B } as nets, and suppose P {gi) is a subnet of {g } such that gi+q E M. Then if 6 = qu, P t gi--+6. Since the G are a totally ordered family, 6 E G Now B' P P
a

n

A, and

7 f

and 7 = yu

7 E Aoq,

7 E a i g i E Agi, so

n

A6 c A

G P
E

-1 Aoqu = A o 6 , 76

nG

It is clear that A

P
c

P

n

AG . P
E

Aou n G =

I

This completes the inductive step in the construction. Note that the flows Y

are not defined for limit ordinals.

P

From general considerations, there is a smallest (necessarily successor) ordinal

n

V

: X -2

and

V

r

)

r)

9

:Z

v

+Y r

)

is an isomorphism. Then

It follows that the homomorphisms

is RIC.

0' : Xs+l-+Xv

such that J,

O r ) : Yr)+l+ZT)

are isomorphisms. (Recall, in theorem 20, if the

r)

given homomorphism 8'

H

is already RIC, the proximal extensions

r)

all the horizontal arrows

O i , gg,

are isomorphisms, so that all the vertical arrows n6 identified with n'

q- 1

(and n 1. r)

Also

AG

v

and

is essentially n . )

are isomorphisms, so the RIC homomorphism n'

Then, for 6 2

8

G6

in the PI tower and ni

may be

= D ( Z 1 = B(Yv) = AGr)-l, r)

Chapter 14

216

and so Put

AGa = AG

for all 6

7)

X

=

Xm, Y = Ym and

B(Xm) =

A,

II

7)

r)

(In particular AGm = AG

2 7).

=

and B(Y,)

= AG

Then n m : Xm+Ym

TI

r)

m'

.) r)

is RIC,

It follows from theorem 15 that no

= AG,.

r)

Ym

non-trivial equicontinuous extensions of

can be interpolated into

TI.

m

together with the various

The collection of minimal flows Xa, Y a , Za

connecting homomorphisms constitute the PI tower of the minimal ambit (X,xo).

If the minimal flow X

is incontractible, note that

is trivial, Y1 = 1

Y 1 = [xoGoplp E MI

an equicontinuous extension of Y1 = 1, flow.

(X1

(so

Z1

so

=

X0

=

X) and Z1

is

is an equicontinuous

If Z1 is itself trivial, then (since X = Xo), 1

I I ~=

n o : Xo+l

and an easy induction shows that Xm = X and Ym = 1. By examining the definitions, it is easy to show by induction that is the orbit closure of the point in

ll '2

whose

Y B

a'th coordinate is

a
xoGaou

( a < 6)

(since B ( Z a ) = AC,,

similarly for limit ordinals). (/3 <

r))

(In general, i f

= (x AG ou)T = (x G ou)T, and O a Oa

Now the family of closed subsets {xoCou}

is a decreasing family, so if

countably many. {Ka'a
Ya+l

X

is metric there are at most

X is a compact metric space and

is a family of closed subsets with Ka+l $ Ka,

then by Urysohn's

lemma there is a continuous real valued function f on X such that a fa = 0 on Ka+l and fa (x) = 1 for some x

r)

Ka'

Hence if /3 < a <

(1 11 denotes the supremum norm on C(X).

IIfp-fall t 1, when is separable,

E

0,

Since C(X)

must be countable o r finite.) This discussion

immediately implies

22. Theorem. If X

is metric, then

rj

is countable, and all the flows

which appear in the PI tower are metrizable. A minimal flow

X

is said to be strictly PI if it can be obtained

General Structure Theorems

217

from the trivial flow by a (transfinite) succession of proximal and equicontinuous extensions. Precisely, there is an ordinal number v ,

a

collection of minimal ambits {(Wa,~a))asv and homomorphisms

rra: Wa+l-+Wa

with

H

a

such that Wv = X, W

( w ~ + ~= )w

0

a’

= 1, K

a

is

either proximal or equicontinuous, and for a limit ordinal a, Wa = v Wrj. (Thus W, is the inverse limit of the pointed flows Wp.) B
PI tower for X

is strictly PI.

The minimal flow X

is said to be PI if a proximal extension X‘

of

X is strictly PI. At first sight, the definition of a PI flow may appear to be unnatural, in contrast to the straightforward definition of a strictly PI flow. The reason f o r considering this somewhat larger class of flows will become clear below. 23. Theorem. Let

X be a minimal flow. Then the following are

equivalent.

(i)

X is PI.

(ii)

B(X)

(iii) X,

Proof.

3 G,.

and Ym are isomorphic.

+ (ii). Since the group of

(i)

a minimal flow is invariant under

proximal extensions, we may suppose that X if W

and W‘

are minimal flows with B(W’)

is strictly PI. Note that 3

G,

and c p : W+W‘

homomorphism which is either proximal or equicontinuous, then B(W) (if cp

is proximal, B(W) = B ( W 0

then B(W)

2

with B(W 1

H(B(W’)) > H(G,)

3

= G,).

G,

and if cp

If {W 1

P then B(W’)

is a 3

G,

is equicontinuous,

is a collection of flows

3 G, and W’ = VW = n B ( W ) 3 Gm. Thus if P 8’ P i s the collection of minimal flows which defines the PI flow X, { Wala
Chapter 14

218

then each D(Wa) (ii) +

iii).

S(Xm)

A.

x,

9(X)

3

3 Gm,

Let

so

A = B(X). Since

Now 9(Ym) = AG r)

cm.

Hence nm: Xm+Ym

it fo 1 1ows easily that (iii)

O(X) = 9 ( W v )

+ (i).

K~

3

Gm.

is a proximal extension of

Xm

= AG = r)+1

...

= AG, = A,

is proximal. But

nm

since A =

is also RIC, and

is an isomorphism.

Ym is strictly PI by construction, and Xm

proximal extension of X. is strictly PI, so X

is a

Since Xm = Ym, a proximal extension of X

is PI.

24. Corollary. A factor of a PI flow is PI.

Proof.

Use condition (ii) in theorem 23.

Apparently the preceding corollary does not follow directly from the definition of a PI flow. Moreover, it can be shown (see exercise 1

that

a factor of a strictly PI flow need not be strictly PI. This fact, (as well as the algebraic characterization given in theorem 23) explains why

it is preferable to allow for proximal extensions rather than to work only with strictly PI flows.

25. Corollary. If X

is a non-trivial PI flow which is incontractible,

then X has a non-trivial equicontinuous factor. Proof. We have observed that the f l ~ w Z1 is a factor of

Ym = Xm.

is in fact a factor of Z1 = 1,

is equicontinuous, and

Since Xm is a proximal extension of X, Z1

X. Moreover, Z1 is non-trivial (for if

then as we have noted Ym = 1,

so

Xm = Ym = 1

and X = 1).

Now we wish to investigate the nature of the homomorphism K

m

:

Xm-+Ym

extension.

in the PI tower. We will show that it is a weaklv mixing (An extension IT:X+Y

is called weakly mixing if the

relation R(n) = [(x,x')ldx) = n(x')l

is topologically transitive

every invariant open set IS dense. Obviously X + 1

--

is a weakly mixing

General Structure Theorems

219

is a weakly mixing flow. 1

extension if and only if X

To prove this, we need a lemma which is of some independent interest. Recall that, for each q E M

the map p H q p

tinuous, In particular, if F

"left action

(For this purpose

"

E J(M) n

Proof.

F

such that

F

Note that

F

fi

E

J(M).

Now a.+p

-1

= a

p E K, and since K

If F and w E J(M)

Then there is a

F

Therefore,

T

{a,) in F, so

a.--+pu 1

E

G

= a

is r-closed. Since FK c K,

is minimal

K = fi.

are as in the preceding lemma, it will be

convenient t o regard Fw as acting on a E G, w E J(M), p E M I .

for this

Let p E K, so p = aw where a

for some net

1

(F,M)

is regarded as a discrete group.)

is an F-invariant subset of M.

(lemma 3 ) and a E F since F

w

we obtain a flow by

is minimal under the left action of F.

contains an F minimal subset K. and w

to itself is con-

We write

be a t-closed subgroup of G.

F

26. Lemma. Let

w

is a subgroup of G,

F act on the left on M, (a,p)-+ap.

having

from M

Thus we write

fi (recall awp = ap, for (Fw,G) for the minimal flow

under the left action of Fw. Let let

v

A :

E

X+Y

J(M)

be a homomorphism of minimal flows, let

with yv = y. We write Svv(Y,yf= [avla E G, yav = yl.

Recall (lemma 3 )

Ov(Y,y) is a r-closed subgroup of the group Cv.

Since yp = y for p E Sv(Y,y),

w

E J(M)

y E Y and

it follows from lemma 26 that there is a

such that yw = y and the flow ( S w ( Y , y ) ; m )

27. Theorem. Let

A :

X+Y

is minimal.

be a RIC extension of minimal flows.

Suppose it is not possible to interpolate a non-trivial equicontinuous extension into $I

A.

(That is, if u: X+Z,

equicontinuous, then $I

is weakly mixing.

$I: Z - + Y with

$Iu =

A

and

is an isomorphism.) Then the extension

In fact, if y

E

Y, there is a w

E

JfM)

with

I[

220

Chapter 14 -1

yw = y such that if x neighborhood of x

E II

(y)w, then, whenever U

in n-'(y)w,

= R(n)

Ux{x')T

is a relative

for every x' E

-1

II

(y).

Proof. First note that the conclusion of the theorem does indeed imply that

For let V1 and V2 be open sets in X

is weakly mixing.

K

W = (V1xV2) n R(n) is non-empty. Let x,y and w be as in

such that

the statement of the theorem and let

V2z n n-'(y)

#

4.

Then if x' E V2t n

R(n)

by hypothesis, and also

((V,r

A

=

n

-1

II

-1

(y), (Vlt n n

c (VlxVz)~n R(n))T

(y)w)x{x'))T

R(n) and

R

-1

E

V 1T and

(y)w)x{x'))T

= ((V1xV2)t n R(n))T,

is weakly mixing. By lemma 26,

such that yw = y and the left action

x

T such that

T E

=

so

there is a w E J(M)

(Bw(Y,y),PW(Y,Y))

is minimal.

From now on, we write u for w and F = B(Y,y) = [ a E C = Mulyu = yl. Let

x

E

6

and let of

F.

= [p E Tlxp

Since

be a relative neighborhood of x E n

let U

n-'(y)u,

E

Ul.

Since u

E

c, c

-1 (y)u

is a non-empty open subset

(F,F) is minimal, there are fl,f2,...,fnE F such that

n

u fiG = F

and since

(Fou)u = F, i t follows easily that

I=1

n

F =

u

(fi60u)u. Since F

is t-closed, at least one of the sets

i=1

(fifiou)u has non-empty 7-interior. Since this set is homeomorphic with

(~ou)u, it follows that int,f(&u)u)

(6.~1~is

f

a t-closed neighborhood of

0.

Now some translate of

u, so there is an u

E

F

such

that aH(F) c (50u)u. Let x' E n-'(y). (Ux{x'))T

3

(Ux{x'))ou

Then Uxlx') 2

3

xfix{x)

and

(x~x{x'))ou > xaH(F)x{x'u).

By the normality of

H(F), xaH(F) = xH(F)u = xAH(F)u = xFa = xF, (where A AH(F) = F, by theorem 15). (Ux{x'))T

3

(UX{X'))OU

2

Thus

xH(F)ux{x'u)

3

xFx{x'u),

SO

= B(X,x);

General Structure Theorems

(Ux{x') IT n

-1

3

XFOUX{X'U),

(y)x{x'u).

of n

-1

which, by the RIC property, is equal to

That is,

(y)x{x'u}

221

-1

(y)x{x'u) c (Ux{x'))T.

I[

is R(n)

(recall that

n

But the orbit closure

is open) and this completes

the proof. 28. Theorem. Suppose, in addition to the assumptions of theorem 27,

r

residual subset

Proof. Let

VE = [x E n

E

-1

of n-l(y)w, such that if x

(y)wl(x,x')T -1

(y)w.

is

z

"E

net" in R(n)

E

= R(n)

T such that = R(n),

(U,x{x'})t2

E, z2.

(i.e., if z'

VE

z

J

J'

Clearly VE

E

Now let

R(n), d(z',z 1 <

J

-1

-1

so there

6 > 0, w e write

for some z 1 .

E:

J

(ylw. By theorem 27,

29. Corollary. Let

U1

of U

and

is an open U2 c U1 and t2 E T such that

1

3

U 2

3

...

3

Un.

Then if

r, Ix,x')T =

r

=

(j = 1,. . . ,n)

J

Clearly. we have

for j = 1 , . . . , n. Therefore, if

E

(ylw. First,

{zl,. . . ,zn) be an

Continue this process obtaining U

and if x

is an

zl. Now, again by theorem 27,

(U1x{x'))tl

is dense in n-l(y)w.

in n-'(y)w

(x,x')T = R(n)

so there is a (relatively) open subset

relatively open sets with U (Unx{x'))t

r,

If A c XxX, z E XxX and

be a relatively open set in n

(Ulx{x'))T

so

dense in R(n)l.

E

if d(z,z') < 6, for all z' E A.

A

(Ux{x'))T

E

We show it is also dense in II

we introduce some notation.

Let U

there is a

X, y = n(x'),

E

> 0, and let

open subset of n

tl

x'

i s a metric space. Then if

that X

n

x

n=1,2,.. . ,

E

V1,

-n

Un c U, x

r

E

VE '

is residual

R(r).

X be an incontractible minimal flow. Then X

is

weakly mixing if and only if it has no equicontinuous factor. Proof.

If X

is incontractible and has no equicontinuous factor, then

the map

X+1

is RIC and so satisfies the hypotheses of theorem 27, so

222

Chupter I 4

X+1

is a weakly mixing extension, which is to say that

X

is a weakly

mixing flow. On the other hand, if a minimal flow is weakly mixing, then (always) it has no equicontinuous factor. From theorem 27 we immediately obtain a structure theorem for arbitrary minimal flows. 30. Theorem. Let

X

X be a minimal flow. Then a proximal extension of

is a R I C and weakly mixing extension of a strictly PI flow.

Consider the PI tower for X.

Proof.

Then Xm

is a proximal extens on

X, and is also a RIC and weakly mixing extension of the strictly PI

of

,. flow Y The PI tower and theorem 30 yield another proof of the Furstenberg structure theorem for distal minimal metric flows. Suppose that distal minimal flow with X metric. hence RIC,

so

X

is distal,

The homomorphism X + 1

the first stage of the PI tower (applying theorem 21) just

reproduces X + l .

It then follows by induction that all the proximal

extensions in the PI tower are isomorphisms and all the flows Xa isomorphic with X. n : Xm+Ym m

point XxX

z =

In particular, Xm = X.

is weakly mixing.

(x',x'')

in R(nm)

Since X, = X

is metric, there is a

with dense orbit in R(nm).

is minimal, and R(am) = A,

an isomorphism, and X = Xm = Ym

are

Now by theorem 27,

is distal, hence pointwise almost periodic, 3

R(nm) =

is a

But, since

is minimal, so

the diagonal. Therefore n,

is

is obtained as a succession of

equicontinuous extensions and limits. Note that we could obtain a proof of the Furstenberg structure theorem in general (i.e.. X not metric) from these considerations if we knew that a distal topologically ergodic flow is minimal ( s o R(nm) minimal and, as above,

II

co

is an isomorphism).

is

This is indeed the case,

General Structure Theorems

223

but the proof is evidently beyond the scope of the techniques developed here.

For a proof, see R. Ellis, Pacific J. Math, 7s (19781, 345-349. )

We now apply the general structure theorem for minimal flows to obtain an intrinsic characterization of PI flows. We say that a minimal flow X

W is a closed invariant subset of XxX

has property ( # ) if, whenever

which is (topologically) ergodic and has a dense subset of almost periodic points, then W 31.

Theorem. Let

X

is minimal.

be a metric minimal flow. Then X

is PI if and

only if it has property (#I. We require two lemmas. In the first lemma the flows involved are not assumed to be minimal. 32. Lemma. R:

X+Y

X and Y be flows with metric phase spaces, and let

Let

be a homomorphism. Suppose Y

is ergodic with a dense set of

almost periodic points. Then there is a closed invariant subset

X such that n(Xo)

=

xo

of

Y, and Xo is ergodic with a dense set of almost

periodic points. Proof. Let

Xo

be a minimal (with respect to inclusion) closed

invariant subset of X such that n(Xo) = Y.

(The existence of Xo

follows by a direct application of Zorn’s lemma.) Let let

U be a non-empty open invariant subset of Xo.

Xo. Let Xb = Xo of

Xo,

so

U.

\

Then Xb

n (X’) = n(X6) 0

0

It follows that n,(U)

f

no =

nixo and

We show that

fi

=

is a closed invariant proper subset

Y. That is, U contains a fiber no-1 (y).

is a neighborhood of y, for if {yn)

is a net

-1

with y +y, xn E no (yn) and (a subnet of) xn+x* then n -1 x* E no (y) c U so xn E U for n 2 n0 and yn = nO (xn E no(U) for

n

2

no.

Then int

dense, no(c) = Y

A

0

(U) is non-empty open and invariant, so

and

fi

= Xo.

n,(U)

is

224

Chapter 14

has a dense set of almost periodic points, it is

To show that Xo

sufficient to show that every open invariant non-empty subset U contains an almost periodic point.

Let

as in the first part of the proof.

If yn E Y

is almost periodic.

xn

E

As above,

is a net of almost

xn-+x*

with

H

O

(xn = yn

-1

no (y) c U, so

E

U for n large.

33. Lemma. Let

Then R ( n )

II:

be a R I C homomorphism of minimal flows

X-jY

has a dense set of almost periodic points.

Proof. Let F = '!7(Y,yo), n ( x 1 = yo, 0

Let

-1 no (y) c U,

Y such that

E

then there are xn E Xo

periodic points with y +y, n and xn

y

of Xo

(x,x')

E

R(n), n(x) = n(x') = yop,

are nets {fJ)

and

if')

J

in F and

-1

so

II

so

{t5)

x f't .+x'. Since (x f .t , x f't 1 E R(n) O J J O J J O J J proof is completed.

(yap) =

x,x'

E

xoFop, for p E M.

x0Fop. Then there

in T with

f t +x and O J J is almost periodic, the x

Now we turn to the proof of theorem 31. Suppose first that the minimal flow X

is strictly PI. We show that property ( # )

under all the extensions by which X

is obtained.

If

is preserved

II: 2 4 2 '

is a

homomorphism of minimal flows which is either proximal or equicontinuous, then II

lifts property ( # I .

(For, if W c 2 x 2

is ergodic with a dense

set of almost periodic points, then W' = (nxn)(W) has the same property, so

if 2'

satisfies ( # I ,

W'

is minimal.

contains a unique minimal subset, almost periodic points implies W

so

Now, if

TI

is proximal, W

the existence of a dense subset of

is minimal.

If II

W is a union of minimal sets, and the ergodicity of W

is equicontinuous,

implies that

W

(13 < 01 be a collecP tion of minimal flows, as in the definition of a PI flow, suppose that is minimal).

each Z

B

Let

u

satisfies

be a limit ordinal, let

(#I, and let 2 =

VZ

B'

(2 )

If W c 2x2

is ergodic with

General Structure Theorems

dense almost periodic points, then each W

225

= (n

XIC )(W) has the same B P is minimal. It is immediate that W = VW so W property and so W P B’ is minimal. Now if X is PI, then there is a strictly PI flow X’

B

which is a proximal extension of X. and X x X )

lemma 32 (applied to X ’ x X ‘ Now suppose X

Then X‘

has property ( # I and

is a minimal flow which is not PI. Then the flow X,

occurring in the PI tower for X

is also not PI, since it is a proximal

extension of X, and the homomorphism x w : Xm+Y, mixing.

X has property ( # I .

implies that

Thus the relation R(n,)

is RIC and weakly

is ergodic and has a dense set of

almost periodic points (lemma 33). Since R(n,)

is obviously not

minimal (it contains the diagonal as a proper subset) X, satisfy ( # I , and so

does not satisfy ( # I .

X, a proximal factor of Xw,

34. Corollary. A metric minimal flow X

does not

for which proximal is an

equivalence relation is PI. Proof. Let

W be a topologically ergodic subset of XxX. Since X is metric, W

(and hence W)

contains a dense orbit, W =

z.

enveloping semigroup of X has a unique minimal right ideal

W

=

3 has a unique minimal set zI. Hence if W

then W \ ZI

is a non-empty open subset of W

Now the

I,

so

is not minimal,

=

which contains no almost

periodic points. Let

3 be a collection of minimal flows. Then, as in the disjoint-

I

ness chapter, let

F

disjoint from all

X

E

denote the collection of minimal flows which are

3. We write T’f, fD

and W M

for the class of PI,

distal, and weakly mixing minimal flows, respectively. 35. Theorem.

If the acting group T

I

Proof. M

3 I), so

chapter).

Now suppose X

(Pf). c

I I)

= Whf

is abelian, then

(T.9)

I

= WM.

(theorem 26 of the disjointness

is weakly mixing and let B(X) = A.

We first

Chapter 14

226

note that if D

then D does not contain Gm.

For, if D

flow (theorem 23) and by theorem 25 U ( D )

as a factor, so

X'

then L A c

is normal in G I ,

is a PI flow, and

D

f

G,

is a P I

has a non-trivial equicontin-

(and therefore X)

factor. Now, if L = AC,,

If Y

then U(D)

G,

Since A c D, a proximal extension X'

incontractible).

(since G,

3

and

T is abelian every minimal flow is

uous factor (recall that if

U(D)

G with A c D

is a t-closed subgroup of

of X

has

has an equicontinuous

is a t-closed subgroup of G

L and L

B = B(Y),

3 GW.

then B

3

Therefore AGm = G.

G,,

so

AB = C,

and X

is disjoint from Y.

Recall that a flow X xo

(that is,

is point distal if it contains a distal point

P ( x o ) = {xo}).

We are going to prove a structure theorem

for point distal minimal flows. It is possible to give a shorter proof than the one presented here, (see the appendix to Glasner's book), but we

will develop some interesting concepts along the way. As

is the case with most of the notions we have introduced, point

distal can be relativized. A homomorphism n : X-+Y distal if there is an xo E X proximal to x -1 I[

such that there are no other points

in the fiber determined by

( n ( x 1') n P ( x , ) = { x o } . )

is called point

xo.

(That is. is a n

In this case, we say that xo

0

distal point. 36. Lemma. Let $:

Y+Z

X,Y, and

Z

be minimal flows and let

be homomorphisms with n

n : X+Z,

open and point distal. Then R

F,

*

has a dense set of almost periodic points.

*

(x,y) E R n, borhood of x. Since n

Proof. Let

that

+(V) c n(U).

and let z = n ( x ) = Jl(y).

Let

U be a neigh-

is open there is a neighborhood V

Let W = int *(V);

W

*

0

of y such

by theorem 15 of chapter 1,

General Structure Theorems

and W c n ( l J ) . Let D be the set of

D

and invariant, D

0

f

let

y' E V

such that

(x',y')

E

Let

Now

= n(x')

x'

E

E

D

(x',y')

A

U

Since

n-'(W),

A

and

is an almost

w e J(M)

and let

and since x'

Then z'w = z'

y'w = y'.

= n(x').

(let z' = $(y')

periodic point

distal points of X.

I

is dense in X.

$(y')

227

such that

D, x'w = x'. 1 Since

the proof is completed.

(UxV) n Rn,*,

37. Theorem. An open point distal homomorphism is RIC.

Proof. Let

show that if # : Y+Z

is a proximal homomorphism, then

disjoint, o r , equivalently R

R

*

It is sufficient to

be open and point distal.

I : X-2

$

R

is minimal (theorem 18).

and $

are

By lemma 36,

has a dense set of almost periodic points. Therefore, it is

I,

* R *

sufficient to show R are minimal sets in

has a unique minimal set. and let x,x'

E

X, y

E

Suppose N

Y with

and N'

(x,y) E N,

11,

(x',y) E N'. and y

Let

x' = xq. Then *(y) = n(x')

(x',yq) = (x,y)q E N

are proximal. Then

(x',y), (x'yq)

= rr(xq) = *(yq

and

(x',y)

E

so

Y9

N'

with

proximal. Therefore N = N'.

We apply theorem 37 to show that a metric point distal minimal flow is PI. Since Xm = X v Ym, the homomorphism n m : Xm+Ym

is point

distal. Now let x*

be a

I,

distal point.

is metric by theorem 22) there is an x (x,x*)T = R(n-1.

y = n,(x)

But

-1

E I

m

(nm(x*))

w E J MI

nm(x*w) = y, (x*,x*w) E P so

Xm

such that

(x,x*) is an almost periodic point.

= nm(x*) and

R(nm) = (x,x')T = A,

By theorem 28 (recall that

(If

such that xw = x, then yw = y,

x*w = x*.) Therefore x = x*,

the diagonal. Hence

I

m

and

is an isomorphism and we

have proved 38. Theorem. A metric point distal minimal flow is PI.

Chapter I4

228

39. Corollary. A point distal flow is incontractible. If it is metric,

it has a non-trivial equicontinuous factor. Proof. The first statement follows from theorem 37.

The second

statement follows from theorem 38 and corollary 25.

Now we develop a construction which is of independent interest. Let n : X+Y

be a homomorphism of minimal flows, xo E X Let Y* be the quasi-factor of X

yo = n(xo).

and

-1 Y* = [ n (yo)opIp E MI.

with x u = x defined by

Xu = X v Y* = ( ( ~ ~ , n - ~ ( y ~ ) o=u ) T

Let

-1 [(xop,n (y0)op)Jp E MI.

Let

tion maps and T : Y*+Y

be the homomorphism defined by

n u : X*+Y*,

Q:

X*+X

be the projec-

-1

7

--

that is, there is a net

(yOp)tn+{n

-1

(yo)op)

{tn)

=

r(n-'(yO)op)

yop. Recall (theorem 16 of the disjointness chapter) that proximal

0

0

is highly

T

in T such that

*

in the Hausdorff topology of '2

.

Equivalently, every non-empty open subset of Y* contains a fiber then x E

(x,;) E Xu

Obviously, i f

sufficient for a point in XxY*

.;

In fact, this condition is

to be in Xu.

To show this, we first

require a lemma.

40. Lemma. Let X be a minimal flow, let x0 E X with xou = xo let 7 : M-+X p E 7

-1

(xo)oq

if and only if 7

Proof. We first note: -1

(xo)oq c 7

-1

7

7

7

-1

(x,)ou.

-1 (xo)oaob c Let

if p,q E M

with

Js J

J s J'

E

-1

MI. Then, if

p,q E M,

(xo)oq.

q E 7-'(x0),p,

where

then

rJ E 7

-1

(x,)

and

-1 -1 Then 7 ( ~ ~ 1 o r .+sr (xo)oq J j -1 c 7 (xOrJ 1sJ = y-'(x O 1sJ -+T-'(x 0 lop. Now let

also

(xo)or

(p

(xo)op = 7

0

is a net in T

D =

-1

(x lop. For q = lim r

{sj)

-1

~(p) = xop

be defined by

with

s

J

+p.

-1 If a,b E D, then Doab = 7 (x,)oab = -1 -1 7 (xo)Ob c 7 (xo)ou = D, atld ab = uab

9 = [Do
M with

Do<

c D1.

and

E

7

-1

and

(xo)oab c D.

A Zorn's lemma argument shows

General Structure Theorems

3 contains an (inclusion) minimal element C = D o r .

that

it follows that r = ur

Dor = C c D, then Coc c C.

For

respectively and

lim Dord Coc

t

E

c = lim d.T

T.+r,

lim Coc t

jj

p

J

E

nets in D and

{t }

J

Coc = Dorc c D

But

(since r,c

T

D)

E

with t.+p

J J

= lim Ct

j

(xo)oq

Next we show that the For, suppose

constitutes a decomposition of M. and cj

J

E

C.

Then Coq =

Thus if r = Cop' n Cop, Cop' = COr = Cop.

= Cop.

[Coplp E MI = [DorpJpE MI = [Doqlq -1

C,

E

Jj

= C.

q = lim c t

Cop, so

But

{d.},

C we have COC = C.

E

collection [Coplp E MI E

D. Next, we show if c

3. It follows from the defining property of C that Coc = C.

Summarizing, if c

q

Since

Coc = lim C0d.t =

so

J

E

with

J j'

J c lim Dr.+Dor

J J

so

229

-1

if and only if

MI =

E

-1

(xo)op =

[ r-1 (xo)oqlq

Thus

MI.

E

(xo)oq and the proof is

completed. E

51,

It is sufficient to show x0q

f 71

41. Theorem. Xu = [(x,;)

XxY*lx

E

is open and u

A*

is highly

proximal. Proof. -1

(yo)oq =

r~

X = MI. u

y =

A

-1

A

-1

For, if this is proved, and A

(yolop if and only if

(note that lemma 40 is this result for the case

(y,)op

(yo)op so

-1

-1

(yo)oq =

A

-1

-

(yo)op.

and

-1

AT = 6 ,

r : M+X,

-1

(x,y) =

(x,;)

E

X*

then x 0

(yo)op = [q

-1

-1

0

with r ( p ) = xop. 6 ( p ) = y p,

8: M+Y

and by lemma 40 6

with x E y, x = x q, ,-,

(xoq,n (yo)oq) E Xu. As we have noted, if Now, let

-

(x,y) E XxY*

E

MI6

-1

(yo)oq = 6

-1

E

Y.

so

(yo)op].

-1

Then E (yo)op = r ( 8 (yo)bp) = r(Iq E MI6 (yo)oq = 6-1(yo)opl) = [r(q)16-1(Yo)"q = s-l(yo)opl c Ir(q)Irs-l(yoloq = 7s-1 (yo)opl = -1 [xoqln (yo)oq = A

-1

A

-1

(yobpl c

-1 (yo)op = [xoqln (yo)oq =

A

-1

A

-1

(y0)op.

(yo)op]

In particular, as desired.

Using this representation of Xu it is an easy exercise to show

Chapter 14

230

that

is open.

I*

Since

T

is highly proximal, it follows that

(P

is

highly proximal. The definition and characterization of X*

has a certain formal

resemblance to our earlier construction, in which an arbitrary homomorphism is lifted to a R I C extension (lemma 19 and theorem 2 0 ) .

The next

theorem shows that if the given homomorphism is point distal, the two constructions are in fact the same. 42.

Theorem. Suppose n: X-+Y

minimal flows, xo S(Y,y0).

E

is a point distal homomorphism of

X with x0 u

Then xOFop =

H

-1

and y0 = n(x,).

= xo

(yo)op, for p E M.

F =

Let

Therefore (in the

notation of theorems 20 and 411, X' = X*, Y' = Y*

and the various

homomorphisms are the same. In particular, the homomorphisms

0

and

8'

in theorem 20 are highly proximal. -1

It is sufficient to show that n (yo)ou = x0Fou. By theorem 41, -1 -1 -1 n*-l(n (y0)ou) = "x,n (yo)ou)~x 6 n (yo)ou~. NOW n* is point

Proof.

distal and by theorem 41, it is open, so by theorem 37, n*

-1 Then (since B ( Y * , n (yo)ou) = F),

n*

-1

-1

is RIC.

-1

( n (y0)ou) = (xo,n (yo)ou)Fou.

Comparing first coordinates of these two representations of n*-'(n-l(y0)w),

= x0Fou. we obtain ~r-~(y~)ou

Now consider the PI tower of a point distal minimal flow X. first note that all the vertical arrows are point distal.

We

This is clear

when we lift to a R I C extension (in the notation of the PI tower construction, from n projection from X

B+ 1

P

is the and n' since X = X v Y P P+1 P P+1 B to YP+l, and is also obvious when an to n')

equicontinuous extension is inserted, as well as under passage to inverse Thus we may apply theorem 42 and conclude inductively that

limits.

*

XB+l = Xp, Ys+l = 2;; and that

0

B

and

8'

P

are highly proximal.

Since

General Structure Theorems

23 1

the composition of highly proximal extensions is highly proximal, the homomorphism from X,

to X

is also highly proximal.

Thus we have proved the first part of the following structure theorem for point distal minimal flows. 43. Theorem. Let

X be a point distal minimal flow. Then

In the canonical PI tower for X,

(i)

the proximal homomorphisms are

highly proximal. Suppose X

(ii)

are almost

1:1,

is metric. Then the (highly) proximal extensions

and an almost

extension of X

1:l

flow. Precisely, there is an ordinal number flows {Ya}, (a 5

r)),

Yo = 1, Y

such that

extensions of the trivial

1:l

a succession of equicontinuous and almost

can be obtained as

and a family of minimal

7)

is an almost

1:l

7)

extension of X, Y is an equicontinuous or almost 1 : l a+1

Ya, and for a a limit ordinal, Ya

p < a

(Ya =

extension of

is the inverse limit of Y

for

B

The ordinal number

r)

is finite or countable.

The proof of (ii) is a consequence of the following two lemmas, the first of which is purely topological.

44. Lemma. Let X and Y be compact metric spaces and let

r

be continuous. Then there is a residual subset y E

r,

is open at all points of

K

45. Lemma. Let

71:

flows, and let y

E

Then

71

-1

X+Y

71

of

H:

X+Y

Y, such that for

(y).

be a highly proximal homomorphism of minimal

Y such that

K

is open at all points of

-1

TI

(y).

is almost one-one. Hence a highly proximal extension of a

metric minimal flow is almost one-one. Proof of Lemma 44. Note that only if Now; let

K

-1

,

K

is open at all points

(regarded as a map from Y

c > 0, and let

to

X

2 1,

-1 (y) if and

71

is continuous at

y.

NE (y) be the mfnimum number of open c balls

Chapter 14

232

-1

needed to cover

TI

N (y')

(Indeed, if U

&

-1

TI

5

N&(y).

(y).

(y') c U for y'

is sufficiently close to y, then

If y'

is open with n

near y . )

rE

Let

-1

(y) c U

denote the set of y

rz

open. We show that

has no interior. For suppose y

there is a sequence y +y

with N,(yn)

sufficiently large yn

rc&

n

int

E

y' = yn, obtaining y" E

rz

< Nc(y).

sufficiently near y*, N&(;)

= I, =

r

E

int

rCE'

Then

But for n

with N&(y") < NE y'). with NE(y*) = 1.

nr,

Y such

and we can app y the same argument to

process, we obtain y* E l-z

contradiction. Thus

E

Clearly rE is

sufficiently near y, Nc(y') = NE(y).

that for all y'

then

so

y* E

re,

Iterating this

But f o r all and we have reached a

is residual. Finally, if y E

r,

-1

n

is

n -

n

continuous at y. sequence yn+y

m*

< 2' I-1

(If and x

TI-^ E

is not continuous at y, then there is a -1 -1 71 (y) such that d(x,n (y,)) 2 p > 0. If

it is easy to see that

N1(y,) -

m

< Nl(y), for all n sufficiently m

large. 1 Proof of Lemma 45. If n p

E

M with yp

= y,

singleton, since n

is open at all points of

we have

-1

TI

(y) = n

n

-1

(y),

then if

-1 yp) = TI-1 (y)op which is a

is highly proximal.

Part (iil of theorem 43 has a valid (and obvious) converse which satisfies the conclusion is point d i s al.

--

any flow

The Ellis "two circle"

minimal set discussed in chapter 1 is a highly proximal extension of an equicontinuous cascade and clearly is not point distal. J. van der Woude (Pac. J. Math. 120 (1985) 453-467) has given an intrinsic characterization of the "HPI" flows

--

(those PI flows for which all the

proximal extensions are highly proximal) which is analogous to the characterization of PI flows given in theorem 34. The structure theorem for metric point distal flows was first proved

General Structure Theorems

233

by Veech (Amer. J. Math. 92 (19701, 205-242) under the additional assumption that the set of distal points is residual. Indeed, this assumption is redundant, as we now show.

(This was first proved by Ellis

(Trans. Amer. Math. SOC.186 (197311,203-218).

X be a metric point distal minimal flow. Then the

46. Theorem. Let

set of distal points in X

is residual.

Proof. The proof depends on three lemmas.

If X

is a flow, we write

DX for the set of distal points in X. 47. Lemma. Let n : X+Y

be a homomorphism of minimal flows with

metric phase spaces which is either almost is residual. Then DX Proof. Suppose n at which n

is

A

-1

(XI =

n

1:l. Then

A

-1

n=1,2,.. .

E

Z,

r

(Wn).

-1

with Wn

DY

A

-1

(W 1

-1

X =

r

is open, and

n

n Dy

n

-1

(1) is dense

( X ) is residual. Moreover, n E

is also

open and dense in Y, so

-1

(.Y)

c DX

P, then (n(x),n(x')) = -1

n(x') = y, and x' = { n

is distal, n

Suppose

be the set of points of Y

is residual, so

since

n

r

Let

x = { n-1 (y)} and (x,x')

(y,n(x')) E P, so If n

1:l.

n Wn , n=1,2,.. .

(since it is invariant), for if y

o r distal.

is residual.

is almost

residual. Then Z =

1:l

(y)) = x.

(Dy) = DX, and an argument almost exactly like

the one just given shows that

DX is residual. (This case does not

require that the spaces be metric.) Now applying lemma 47 to the lower level of the PI tower for X,

obtain that

Xm = Ym has a residual

set o f distal points.

the process is countable, so that at a limit ordinal

(Recall that

there are only a

countable number of flows involved. Hence the property of Dx

be i ng

U

residual is preserved.)

we

Finally, we must show that the set of distal

Chapter 14

234

points of X

itself is residual. This follows immediately from the next

two lemmas. 48. Lemma. Let K : X-+Y

be a homomorphism of minimal flows. Then

n(DX ) c Dy. Proof. Let x

DX and y = n(x).

E

If y'

y' = yv for some idempotent v, and

#

y with

R(XV) =

yv = y'.

(y,y')

E

P, then

But then

(x,xv) E P, and this contradicts x E DX. 49.

Lemma. Let

flows. Let K

I[:

X+Y

If a

extension of minimal

1:l

be a residual subset of X such that

the points at which n

Proof.

be an almost

is 1:l.

is a residual subset of Y.

Then R ( K )

is a closed nowhere dense subset of X, then

is closed and nowhere dense in Y. For, if 13 then there are t i , . . . , tm E T with Y =

.

a* = at.U.. Uat,, 1

n ( a * ) = Y, su

j

is closed nowhere dense so

/3 = n I a )

has non-empty interior,

ptiU ... UBt,.

Then, if

K c a* and therefore K is

nowhere dense, which is a contradiction. Now X \ K c

a

K is a subset of

Y \ R(K)

C

u

n(a.1 j=1,2,.. .

u

aj , j=1,2,. ..

and

n(K)

where

is

residual. Now we present some examples of point distal flows. We first describe a general method for obtaining almost one-to-one extensions of minimal ambits. Let

(Y,T) be a minimal flow and let yo

E

Y. Let f

be a bounded continuous real valued function defined on the orbit y0 T.

(It is

not

assumed that f can be continuously extended to Y =

and,

indeed, that is what makes the following construction interesting.) Let

F : T+R

be defined by F(t) = f(yot).

Since F

is bounded, it has a

continuous extension, which we still call F, to BT, so Define an equivalence relation

- on

f3T

by

p-q

F : PTjIR.

if yop = yoq and

General Structure Theorems

F(pt) = F(qt),

t

for all

235

-

T. it is immediate that

E

X = PT/-

invariant equivalence relation on PT. Let

and let

the equivalence class determined by the identity e.

X,

x0

Then T

be

acts on

(X,T) is a flow with X = 3 and there is an ambit homomor0

so

TI:(X,xo)-+(Y,yo) (if x0 p = x0 q

phism

T

Now define

C(X)

E

F ( p ) = F(q)

so

is a closed

by

T

and

then p-q

T(xOp) = F(p).

yop = yoq).

so

(If x0 p = xoq, then p-q

is well defined.) Note that if t E T,

?(x t) = F(t) = f(yot). 0 Let

E be the set of y

extended to y.

Y such that

E

(Note that

y

E

E

f cannot be continuously

if and only if

lim f(z)

does not

=+Y ZEY0T exist. 1 50. Lemma.

E = [yly = yop = yoq and F(p)

(i)

(ii) n Proof.

-1

( E l = [xithere is x'

and

J

lim f(y t .) O J

(ii) p,q

E

some p , q ~ P T l .

with T ( x )

f

T(x')l.

in T

{s }

J

with y t .+y, O J

If y

lim f(y s 1.

f

y

E

E and let x

-1 E TI (y)..

PT with y0p = y0q = y and F(p) f

xoq

T(xoq) = F(q) 51. Lemma.

Similarly, if x

x0q, y = a(x) = n(x')

F(q),

E

by (i).

Y,

TI

is one-to-one at A

E

=

9.

T(xop)

*

?(XI

So if (say)

yop = yoq

E

singleton) if and only if yT

so

T(xoq). f

T(xoq),

is in the right hand side,

E

and y If y

so

yop =

Then by (i) there are

and n(x 0 p) = n(xoq) = y.

we can let x' = xoq. = x p, x' = 0

*

E, then and

so

0.i

Let

E

y s.+y O J

We may suppose t .+p, s +q, J J and F(p) = lim f(y0 t. i ) f lim f(y 0s. i = F(q).

Hence xop

x

F(q),

Clearly the right side is contained in E.

(i)

there are nets {t.)

yoq = y

f

-1 ETI (n(x))

y

and

(i.e.,

F(p) = T(xop)

-1 TI

(y)

is a

f

Chapter 14

236

Proof.

If yt

E

E for some t

( i i ) of lemma 50.

x,x'

-1

=

Since f

x p

= In

Suppose yt t E for all t E T ,

(y). Then x = x0 p, x' = xoq

x

p-q, so

-1

T, then In-l(y)l

for p,q

E

(yt)I 2 2 by

and let /3T, and yop = yoq

for t E T by ( i ) of lemma 50, and

Since yt t E, F(pt) = F(qt)

y.

=

n

E

E

= xoq = x'.

is continuous on yOT, yoT n E = 4

and

is minimal. Thus n lemma 51. Therefore X = 3 0

n

:

-1

(yo) = {xo}

by

is an almost

X+Y

one-to-one extension (and is therefore highly proximal). If y

Y

E

-1

In

is such that

then we say that the orbit of y

t

E

n -1 (y)

T with t

= {x+,x-}

#

e.

+ -

p ,p

E

+

#

{-l,l}

is E

E,

-1 K

F

is also

{-l,l}

so since

+

+

-

- . Thus {x+,x-}

does not contain any more points.

0

Now yoq =

yap+

=

y.

and

some t

f

f

F(qt),

Let

+

so we need only show F(p+t) = F(qt) e, F(p+t)

f

We show x* = xoq = x

F(qe),

y

E

E, there

= n(x0p-) = y,

T(xop-),

x* E n-'(y), o r what is the

+

F(p+e) = F(p 1 = 1 = F(q) = for t

f

e.

But if, f o r

then, since y0qt = y0 p+t = yt, yt

( i ) of lemma 50, and this contradicts the assumption that #

4

and to complete the proof we need to

c n-'(y),

x q = x* and suppose F(q) = 1. same thing q-p'.

+

x = xop , x = xop-; since ?(xOp 1

x

t

n E =

(y) consists of two points,

BT with F(p+) = 1, F(p-1 = -1 and n(x 0p+)

show that n-l(y)

for

Et

and that

with T(x+) = 1, T(x-1 = -1.

by lemma 50. Let

x

f

Then if y

Proof, Clearly the range of are

$1

f

is "split."

52. Lemma. Suppose the range of

for

(equivalently, if yT n E

(y)l > 1

e. The same argument works if F(q) = -1

E

Et n E

E, by

= $

and so the proof is

completed. We apply these considerations to the case of a minimal cascade If y

E

Y, we write O(y)

for the orbit of y. O(y) = {Sn(yI},z.

(Y,S). Then

237

General Structure Theorems

if yo

E

Y and f

is a continuous map from O(y )

to

0

A =

{-l,l},

we

obtain, as in the previous discussion, an almost one-to-one extension (X,R) of

If E

(Y,S),I I : X-Y.

is the set of

O(y) n E

TI

-1

(yo). We assume E

I K -1 (y)I X to K

n

-1

= 2 for y E

A

such that

(y) = {x+,x-),

$;

f

f

asymptotic.

$

Sn(E) n E = $ f o r

and

I K -1 (y)I

> 1

if

n

f

0.

so

E. Moreover, there is a continuous map T from for n E Z.

T(Rn(xO)) = f(Sn(yo)), then T(x+)

(If Rn'(x+)+z+, k

#

T(x-1

but

-

x+

Rn'(x-)+z+

If y E E,

for

?(Rn(x+) = T(Rn(x-)) and x

are doubly

for some sequence {ni}

lniI-%m, then T(R ( z ) ) = T(R ( z 1 )

with

Y such that f

since O(yo) n E = $, In-l(y0)I = 1, {xo} =

0. The latter equality implies that

f

E

then if y E Y,

cannot be continuously extended to y', and only if

y'

k

for all integers k, and

it follows (lemma 51 and lemma 50 (ii)) that z

+

Now we construct a minimal distal (in fact Z

-

= z .)

extension of

)

2

(X,R).

53. Theorem. With the assumptions and notation of the previous discus-

Y' = Xxh and let

sion, let

S'(X,E)= (R(x),T(x)s). Proof. that

Then

s'

be the homeomorphism of Y'

(Y',S')

is minimal.

is not minimal, there is a continuous g : X+h

If Y'

defined by

such

This is essentially a consequence of theorem

g(R(x)) = T(x)g(x).

10 in the "distal flows" chapter, but we sketch a proof for completeness.

Note that

(Y',S')

extension of

(X,R). Suppose (Y',S')

y1 = (x~,E)E Y'. (x2,c') E such e ' ,

If x2

E

is not minimal, and let

X there is a unique

00.(By minimality of

X =

and if there are two, then both

0 0 . 3ut then

of Y'

is pointwise almost periodic, since it is a group

and

E' E A

such that

there is at least one

(x2,1) and

(x2,-l) are in

is preserved by the automorphism (x,1) H l x , - l )

= Y'.)

Define g(x2) =

E'.

It is easily checked that

Chapter 14

238

g

is continuous and satisfies the required functional equation. Thus to show that Y’

such g exists. such that g

If there is such a g ,

assumes both

is closed and K {x+,x-),

is minimal it is sufficient to show that no

with

1 and -1

K be the set of y

let on

-1

TI

Y (since yo Q K). Let y

#

?(x+)

T(x-1

#

(y). E.

E

It is clear that K Then

and T(Rn(x+)) = T(R”(x-))

If y Q K, then g(x+) = g(x-1, g(R(x+)) = T(x+)g(x+) g(R(x-11,

and

S(y) = n(R(x+)) = n(R(x-))

T(Rn(x+)) = T(R”(x-))

Y

E

is in K.

TI-'(^)

=

for n

#

0.

T(x-)g(x-) =

#

Since

for all n > 0, we have Sn(y) E K for all

n > 0. Since the forward orbit of any point in a minimal flow is dense,

K is closed, we have K = Y, a contradiction. Then y

and g(xf) #

f

g(x-), T(R-’(x+))

= T(R-lfx-)),

= g(R-’(x-))

T(R-’(x-))g(x-)

so

and similarly g(R-n(x+))

Therefore S-”(y) E K for all

E

K,

-1 + -1 + gfR (x 1) = ?(R (x ))g(x+) f

g(R-n(x-)).

n > 0, and again we conclude K = Y.

Hence no such function g can exist, and Y‘

is minimal.

In the minimal cascade just constructed, the proximal relation is not an equivalence relation. Indeed, if y E E with then we show that for

E

E

A,

and positively proximal with

-1 TI

(y) = {x,x‘),

(x,E) is negatively proximal with

(x’,-E).

(x.c)) (Thus ((x’,~),

E

(x’,~),

P,

((x,~),(x’,-E))E P, but, since (Y’,S‘) is a group extension of ( X , R ) , ((x’,&), (x’,-E)) Q P. 1

If n > 0, then an easy induction shows that S‘”(x,c) = (R”(x),T(R”-’(x)). the same with x’

k

T(R (x’)) if k for n

#

. .T(R’(x))?(R(x))T(x)E)

in place of x.

Now

*

and Sln(x’,~) is k f(x’) but T(R (XI) =

0 (by lemmas 50 (ii) and 51, since Sn(E) n E = @ n 0). Hence the second coordinates of S (x,c) and Sn(x’,-c) #

are the same, and since x and x’ a r e positively (and negatively) proximal under R then

(x,c) and

(XI,-&)

are positively proximal.

General Structure Theorems

239

For n < 0, Stn(x,&) = (Rn(x),T(Rn(x) . . .T(R-lx)&) n coordinates of S' ( x , E ) and

(x',E)

and S'n(x',~)

are the same, and so

xA

funct ion

(x,E)

are negatively proximal.

is a topological space and A c Y

If Y

so the second

y E Y

is continuous at

boundary of A).

then the characteristic

if and only if y st aA

(Y,S) is a minimal cascade and A

Hence, if

Sn(aA) n aA = Q, whenever n

subset of Y such that

tion discussed above applies to

If

int aA = +,

f

is a

0, the construc-

(so the range of

= xAIU(yo)

now {0,1} rather than { - l , l } ) for any yo O(yo) n aA = Q,.

(the

E

f

is

Y such that

a simple application of the Baire

category theorem shows that there always are such orbits. An important example is obtained when Y the reals modulo one) and S

is the circle (regarded as

is an irrational rotation S(y) = y+a. Let

A be an interval on the circle say A = [O,@I

such that a and

rationally independent (this guarantees that Sn(aA) n aA = $ The cascade

(X,T) obtained from

(a,@). If

sion,

-1

TI

of aA)

71:

X+Y

if n

0).

f

(Y,S) is called a Sturmian flow of

is the homomorphism defined in our dlscus-

(y) consists of one point, unless y

in which case

are

/3

-1

R

E

Za u (@+Za) (the orbit

(y) consists of two points

which are positively and negatively asymptotic.

TI

-1

(y) = {x+,x-}

In the notation of our

construction, T(x+) = 0 , T(x-1 = I . In the "classical" Sturmian minimal flow, as defined by Hedlund (Amer.

J. Math.,

(19441, 605-620) A = [O,al

irrational ( s o p = a).

Thus it is not the case that Sn(aA) n aA = Q,. -1

However, it is still true that not in the orbit of 0 then so

In-'(O)l

show that

12. x* =

+ x .

and S(y) = y+a, with a

In -1

In

Suppose that Suppose

(y)l = 1 or 2. Clearly, if y

(y)l = 1.

x* E n

-1

(0)

As above,

{x+,x-) c n-'(O)

and (say) T(x*) = 1.

Rki(xo)-+x+, Re'(xo)+x*.

is

Then

We

Chapter 14

240

T(Rk'(xO))+T(x+)

1, T(Re'(xo))-+T(x*)

=

= T(Re'(xO))

T(Rk'(xol)

similarly x (Sei(x 1 ) = 1, A 0

A

= [O.a].

That is, we have

yo+kia,y0+liaE A.

e

k

so

we may suppose

so

T(Rki(xO)) = rA(Sk'(x 01 )

But

= 1.

= 1,

and S '(yo)

S '(yo)

y0+k.a+O, 1

y0+e.a+O 1

=

1 and

are in

and

It follows that y0+k.a and yo+Lia approach 0 1

from the right. Therefore yo+a+ki a and y0+a+t.a approach a 1 the right for n

*

so

T(R(x+)) = T(R(x*))

from

0. As above, T(Rn(x+)) = T(Rn(x*)l

=

Thus, in this flow a single orbit is

0,1, and hence x* = .'x

split, and the inverse image of a point in this orbit consists of a pair of positively and negatively asymptotic points. In some cases, the almost one-to-one extension constructed above can be realized as a subflow of the shift flow on is a minimal flow and

y

f

A

the same with y and y'

4,

and let f = x

AIYOT

:

with x

*

x';

u(x0p)

*

F(qt) f

for some t

u(xoq).

yOpt E int A

If yop

by

dx,p)(t)

we show that

dx)

P * q. Then y0p

with x = xop, x' = xoq, so F(pt)

E

Y with

y't E int AC

and yo

E

Y with y0 T

n

(or

aA

=

(X,T) be the almost

Let

1).

construction, define u : X+{0,1IT x,x' E X

Let

(Y,T)

(Y,T) defined by f. Using the notation of our

one-to-one extension of

Let

int A

E

interchanged). y0T+{O,

Suppose

Y such that if y,y'

is a subset of

then there is a t E T with yt

y'

T

{0,1} .

E

*

f

= T(xopt) = F(pt).

*

dx').

Yoq

or

Let y0p

=

p,q yoq

E

/3T but

T. In the latter case certainly yoq. C

and yOqt E int A

let

.

Let

t

T such that (say)

E

{t

J

}

and

{s }

J

be nets in T

t +p, sj+q. Then, for j 2 j y t t E A, y s . t E ,'A F(pt) = J 0' 0 j O J lim x (y t t) = 1, F(qt) = lim x (y s t) = 0 and again F(pt) * F(qt), A OJ A OJ

with

so

u(xop)

f

Note that

u(x,q). u(x,)(t)

= T(xot) = f(yot) = XA(yot).

X (identified with u(X))

That is, an orbit of

can be calculated directly in terms of "data"

General Structure Theorems

from Y, without recourse to f3T. point whose orbit is not split

24 1

Indeed, the same is true for any

-- that

is, any

y

(BAIT.

Z !

These conditions are satisfied when the minimal cascade equicontinuous. Recall that

Y can be given the structure of a

monothetic group. That is, Y additively) and S

is a compact group (which we write

is translation by a g

in Y, S(y) = y+g. We suppose that A

then Sn(y') z

0

A

E

A

E

and let

z

E

(and the same with y and y' Sn'(y)

int A

E

with

Sn'(y)+zO

0. (Dynamically this

1

and by interchanging y and y' zo+(y-y') E A.

Since z

0

E

A

int A,

E

interchanged).

Let

or y+nig+z 0

Then y'+nig

Since A

y'+n.g = y+n.g+(y'-y)+zo+(y'-y).

A and

#

Y such that whenever Sn(y)

we are using the condition A = cl(int A)). 1

Y with

is a closed subset of

is not invariant under any non-identity automorphism of the

A

(Y,S).1 Now suppose y,y'

flow

Y whose multiples are dense

E

A = cl(int A) and that A+z f A whenever says that

(Y,S) is

E

A

(here

and

is closed, zo+(y'-y)

E

A,

in the above argument, we have

is arbitrary, it follows that A+(y-y')

=

y-y' = 0, or y = y'.

so

The subset

A

=

[O,pl

of the circle K

used to construct the

Sturmian minimal flow obviously satisfies this condition. Thus this flow

z.

may be regarded as the orbit closure of w = (XA(na)) in {0,1)

This

is how the Sturmian flows are usually defined. These considerations can be used to analyze the Morse minimal set. Recall from chapter 1 that the endomorphism by

nw(n) = w(n)+o(n+l)

set Ho. chapter 0's

and

Since x(wl) ,

of Cl = { O , l ) '

maps the Morse minimal set Mo

= n(w2)

if and ofily if w2 = u;

' denotes the automorphism of l'sl,

H

then Mo

equicont nuous extension).

i-2

Ho

onto a minimal (where, as in

obtained by interchanging

is a Z2 extension of Ho We show that

defined

(hence

H

is an

is an almost one-to-one

Chapter 14

242

(hence proximal) extension of the "adding machine" flow

(D

,TI

2

--

on the dyadics. Recall that if wo =

the equicontinuous

. . . 01101001. . .

is

k the defining point of Mo,

then f o r

n > 0, wo(n) = xci(mod 2)

i=o

k

c c i 2 i is the base 2 expansion of n. Now regard n as a point

where

i =O

of D2, n = (co, cl,.,..,ck,O,O ,... 1

(ck= 1, c n = O f o r

n > k).

follows easily that if ho = n(wo) then, for n > 0, ho(n) = 1 first

e such that ce =

0 is even and

(xo,xl,.. . )

e

0 if this

D21 such that x -0 is even

e-

e

1

Then it is easily checked that cl(int A) = A, a A = {(l,l,l,l A+z = A

if the

is odd. Now let

all xi=l o r the first E

it

... I},

and

if and only if z = 0. By the above discussion XA(n) = Ao(n)

for n t 0. Equivalently, if r : D -+D2 2

phism t(x) = x+(l,O,O, ... 1,

Ho = O ( h0 1

then

X*(T

n

is the generating homeomor(0))

= ho(n).

is an almost one-to-one extension of

D2

.

This proves that Thus the Morse

minimal set is an equicontinuous extension of an almost one-to-one extension of an equicontinuous flow, and is therefore point distal. The examples of proximal extensions which we have given

so

far have

all been highly proximal (and thus almost one-to-one in the metric case). We now develop a general procedure for obtaining proximal (but not highly proximal) extensions of an arbitrary minimal cascade all skew products o n X = ZxY

(where Y

( 2 . ~ 1 . These are

is a compact metric space).

In particular, these provide examples of metric flows which are PI but not point distal. This construction is due to Glasner and Weiss (Israel

J. Math, 3 (19791, 321-336). If N

is a compact metric space, let H(N) denote the space of all

243

General Structure Theorems

self homeomorphisms of

N, provided with the metric

(Thus a sequence gn+g -1

if gn+g

in the topology determined by -1

and gn +g

uniformly on N.)

d

if and only

It is easy to see that

R(N) is a complete metric topological group. Y be a

(2,v) be a compact metric infinite minimal cascade, let

Let

compact metric space, and put homeomorphism m i d group of H ( X )

X = ZxY. We also write v for the

of X, v(z,y) = (v(z),y).

consisting of those G

Rs(X) be the sub-

Let

which fix the first coordinate.

Such a G determines (and is determined by) a continuous map

z into

R P ( Y ) , so

~(z,y= ) (z,gz(y)).

Let

subgroup of R(Y), let Ts be the subgroup of zt-+gz

only if G(z,y) = (z,gz(y)), Y (v) = [G-~OCIG E

r

where

of

2

into

gz E rl.

If

r

is a

RsQs(X) whose elements

r

(so

G E Ts

i f and

Finally, put

rSi.

The next two theorems show that for appropriately chosen subgroups of R ( Y ) ,

"most" T E Ip (v) are minimal homeomorphisms of X

r

(X,T) is a proximal extension of form T(z,y) = (v(z),h (y)) Z

( 2 , ~ ) . Note that such a

T has the

so we do indeed have a skew product action.

X to

is not almost one-to-one.

54. Theorem. Let

(Y,T) be a minimal flow, where

connected subgroup of R ( Y ) . such that Proof.

If U

r

for which

Each fiber is homeomorphic with Y, so obviously the extension of Z

of

~(v) = [G-~OCIG e R~PS(X)I.

-1 -1 Note that for G E Rs(X), G oC(z,y) = (v(zl,gv(z)gz(y)).

arise from continuous maps

z"gz

r

is a pathwise

Then there is a residual subset R

(X,T) is minimal, for all T

is a non-empty open subset of

E R.

X, let

of

Chapter 14

244 m

EU

u Ti ( U )

[T E Yr(u)l

=

It is clear that EU

= XI.

is an open subset

i=1

3y(o,

of

X, and X

and that i f

= nEU ,

{U.) 1

is a countable basis for the open sets of

then T E

is a minimal homeomorphism of

X

i if and only if T E R.

Hence it is sufficient to show that EU

7, for every non-empty open subset

in

To show that E

is dense in

U

G-luG

E

G E

Ts,

and since R(X) is a topolo-

it is easily verified that GEUGel = EGU,

gical group, it follows that CEUG-' =

U of X. Now, if

is dense

ECu.

F,it is sufficient to show that

Eu or, equivalently, that

u E GEUG

-1

E. Since G and

=

G-lU

U are arbitrary, for this, in turn, it is sufficient to show that

-

u E EU, which is a consequence of the following lemma.

55. Lemma.

d(cr,G-'uG)

<

If c > 0, there is a G

E

Ts such that G-'&

be non-empty open subsets of 7. and Y

respectively such that WxV c U. Since f Y , T ) finite subset of

r,

n-1 hi(V) = Y. Let i=l n

u

= [O,l]

to

d(h-'h ,id) < tl t2

which we denote by

6.

is minimal there is a

{ho,hl,.. . .hn-l) such that

-

n

n

t-ht

be an extension to a continuous map from

r.

Let

E.

(The existence of such a 6

6

> 0 be such that

definition of the metric d).

5m <

EU and

E.

Proof. Let W and V

I

E

Since the space Z

W such that the sets

Let

It 1-t 2I <

6

implies

follows easily from the

m be a positive integer such that

is infinite, there is an open subset A

A,dA),

. . . , um- 1 ( A )

of

are pairwise disjoint. Let

which is homeomorphic to a Cantor set and let m- 1 6 : K--+I be a continuous onto map. Define 6 on c r i ( K ) by i =o i 6 ( z ) = 6 ( d i ( z ) ) if z E u (K) and extend 6 to a continuous map of be a subset of A

u

K

General Structure Theorems

all of

2

to

I.

Define 8:Z + I

by

245

m- 1 co(ui(z)). If

8(z) =

i =o z E K, ii(ui(z)) = i i ( z ) ,

e(z) = i(z),

so

and 8(W)

=

I.

g : Z+r

Let

n -1 (z,gzhi (y))

-

=

n

(z,he(z)h~~zl(y)) = (z,y). Since hi(y) n

(z,y) E C(WxV) c G(U). that

SInce

Zx{y) c C(U) u u ( C ( U ) u

arbitrary, and therefore

(Z,u)

...

V,

E

we have

is minimal, there exists k such

k u u ( C ( U ) 1.

O i u (C(U)) = X,

u

Note that

y

is

and the proof of the lemma is

i =O completed. 56.

Theorem. Let

r

be a pathwise connected subgroup of X(Y) such

that if y1,y2 E Y, there are neighborhoods U respectively such that for every diameter

(h(V u U)) <

such that

E

( 2 , ~ ) for all

Yr(u)

of

T

E

R.

Y and

be a finite open covering of YxY, such that there is an h N

E.

with

Then there is a residual subset R

> 0 and every i,

(h(Ui u V,)) <

r

2, and, for U,V non-empty open subsets of

( i = 1, . . . , N)

{U XV i i

for every

E

of y1 and y2

> 0, there is an h E

(X,T) is a proximal extension of

Proof. Fix zo

Let

E.

E

and V

Then, if R =

E

r

with diameter

m

n n EU i=l n=l

i

,v ,-, 1 1 n

and

T

E

W,

the

Chapter 14

246

extension

(X,T)-+(Z,o) has the property that all points in the fiber zo

determined by

are proximal. Since

( 2 , ~ )is minimal, this implies

that the extension is proximal. To show R

is residual, we show that

(clearly it is open i n

EU,V,& ( U , V open as above) is dense in P r ( ( r ) ) ; for this it is sufficient to show that

H

E

Ts.

Now, if H E Ts

is such that

0

c

H(z,y) = (z,hZ(y))

-1 -1 implies d(hz (yl),hz (y,))

d(y1,y2) < 6

then EhZ (U),hZ (V1.6 y E U u V,

is defined by

HEu,v,&H-',

E

Q

HE

u,v, cH - ~ (for, if

T

<

and

z

for

c,

> 0

6

E

2,

EhZ (U),hZ ( V 1 . 6 '

E

0

0

0

and if k

for

is the integer such that diameter

1

hZ (U) u hZ (V) < 6 , then for Y E U u V, 0 0 (H-'TH) k (zo,y) = H-1 TkH(zO,y) = H-1Tk (zo.hzo(y)). and it follows from [Tk(z,,y')Iy'

E

6

the choice of

and the definition of

diameter [(H-lTH)k(~o,y)ly E U u V l <

E,

H-l that s o ' HTH-'

E

us v, & 1.

E

Thus i t

is sufficient to show that u E Eu, v, & ' The proof is an immediate consequence of the following lemma (whose statement and proof are in part similar to those of lemma 5 5 ) .

57. Lemma. If ~ , >8 0 there is a G and d(o,G-l&C)

E Ts

such that

with diameter

(hl(U u V)) < c ,

E

u, v,&

and put

2

.5

h = hl = identity. As in the previous proof, let 0

continuous extension of

I to

implies d(h-'h ,id) < 6 , tl t2 let A

E

< 6.

Proof. Let hl c T

2 < q,

G-luG

let

r,

let

I-J > 0 such that

z,,

such that A , a ( A ) ,

are pairwise disjoint, let K be a subset of A 0

E

be a

Itl-t2


n be a positive integer such th t

be a neighborhood of

to a Cantor set with z

t-ht

K, let

:

K+ I

...,on-1 ( A )

which is homeomorphic

be a continuous onto map

2 47

General Structure Theorems

with e"(z 1 = 0

1 z,

define

e"

on

m- 1 ui(K) by i=O

u

G(ui(z)) =

;(z)

for z

E

K,

m- 1

extend

to all of 2 ,

and put

'c"

e(z) = -

8a ( z ) ) .

n

Put gz = he(z)

i =O

and

G(z,y) = (z,gz(y)).

Then exactly as in lemma 54, d(G-'uG,u) < 6.

k O(u ( z 1 )

Choose k so that

0

( 2 , ~ ) is minimal and

e

is sufficiently close to one (recall

is onto) so that

-1 g k a (2,)

-1 k k -1 identity map. Then G u G(zO,y) = (u (z,),g u

is close to the hl(y)),

(2,)

and it

2

follows from the defining property of hl that G-luG E EU,V,E.

2 Thus, if

r

is a subgroup of H(Y) which satisfies the conditions of

theorems 54 and 56, and if T sets of

is in the intersection of the residual sub-

Yr(u) from these two theorems, then (X,T) is a minimal flow

which is a proximal extension of

r

( 2 , ~ ) .An example is obtained by tak

as SL(2,R) (the group of real 2x2 matrices of determinant 1

acting on Y = P ,

the projective line (the set of lines in the plane

through the origin). we take

1)

I Since 'F

is homeomorphic with the circle, then

( 2 , ~ )as an irrational rotation of the circle, our construction

provides a minimal action on the 2 torus which is a proximal extension of an irrational rotation. We conclude by showing how some of the examples developed earlier can be used to obtain a weakly mixing minimal (hence not PI) flow. The following simple and ingenious construction is due to Karl Petersen and Leonard Shapiro (Transactions Amer. Math. SOC. 177 (19731, 375-389). (X,T) be a minimal cascade such that there is a continuous map to a 2 point set (which for convenience we write as

Let

u of X

(1,211 and such

that there is a pair of doubly (i.e., positively and negatively) proximal points x and y

for which u ( x ) = 1 and u(y) = 2, but

Chapter 1 4

248

for n

u(Tn(x)) = u(Tn(y))

f

0. The Sturmian flows certainly satisfy

these conditions. We use the function u

to construct a new flow

(X,T) with respect to u).

primitive of 0

Xn = u-l(n),

(so

X

=

0 0 XI u X ) 2

and let

(Xu,Tu) (called the

For n = 1,2, let

X21 be a homeomorphic copy of

0 1 Let Xu be the disjoint union Xi, with 'p: X24 X 2 a homeomorphism. of X and Xi and define the homeomorphism TU of Xu by

T ' ( x )

=

I

T(x),

0 if x E XI

cp(x),

if x

E

0 X2

~('p-l(x)), if x

E

x2. 1

Thus TU is obtained from T by "introducing a delay. "

Schemat

cally, we have the following picture (where we consider a point for which T(x)

x

E

2

and T (x) are in Xi).

T(x) = T'(x)

X

(TUI3(xI = T2 (x)

I

Clearly the cascade

(XU,TU) is minimal.

In order to show it is

weakly mixing, we show it has no non-constant continuous eigenfunctions. (Recall, from the chapter on the equicontinuous structure relation, that a minimal cascade is weakly mixing if and only if it has no non-trivial equicontinuous factor, and from the Furstenberg structure theorem chapter, a minimal cascade has a non-trivial equicontinuous factor if and only if it has a non-constant continuous eigenfunction.)

To show that the cascade (Xu ,Tu )

is weakly mixing, we require the

249

General Structure Theorems

following lemma. 58. Lemma.

u u (X ,T )

if and only if there is a continuous function f : X+K circle) such that

f(T(x)) =
Proof. Suppose g =
g(T'(x)) x

E

0

If x

E

(the unit

for x E X.

is a continuous eigenfunction on

xu,

Let f be the restriction of g to X c Xu.

X1, Tu(x) = T(x)

<

has a continuous eigenfunction with eigenvalue

E

If

=
X, and f(T(x)) = f(T'(x))

u 2 u 2 X2, T(x) = (T ) (x), and f(T(x)) = g((T 1 (XI) 0

=

<2g(x) =


Xu by

satisfies f(T(x)) =
g(x) = f(x)

Then, if x E Xy, g(T'(x))

= f(T(x))

=
define g if x if x

,g(~~(x))= g(cp(x)) =
=

<2f(cp-'(x))),

x

E

E

1 X2.

E

0

X2,

1 x2.

= <2f(cp-1(x)),

so in all cases

g(Tu(x)) =
u u (X ,T )

has no non-constant continuous

eigenfunctions, it is sufficient to show that if f : X+K continuous with f(T(x)) =
then

<=

1

and from the minimality of

is

(for then (X',Tu)

it follows

is constant).

Now, if x and y are the doubly proximal points whose existence we are assuming, recall that u(x) = 1, u(y) = 2,

and u(Tn(x))

= u(Tn(y))

Chapter 14

250

., . f(T2(y))

f(T"(y1)

f(y)

f(T(x))

--ma-= f(T(y)) f( T ( y )1

f(Tn(y)l f(y)

-2

f (Tn(y) 1

5 E=<.T7jT.

f (Tn-'(y) 1 Now let n tend to infinity along the net corresponding to the positive proximality of x

and y and obtain - - - or
f(x)

But, for n < 0, we have

f (Tn(xl)

f(y) , f (Tn(y)l

proximality of x and y, f(x) = f(y)

and

using the negative

so

<=

1.

This completes the

proof. If

(X,T) is a Sturmian flow of type

the minimal flow (XU,TU) is prime

--

(a,P)

it can be shown that

that is, it has no non-trivial

factors (H. Furstenberg, H.B. Keynes, and L. Shapiro, Israel J. Math. ( 1973 1,

14

26-381.

Exercises. 1.

The following considerations show that a factor of a strictly PI flow

need not be strictly PI. Let X be a minimal flow for which proximal is a closed (hence an equivalence) relation.

Suppose X+X/P

is a prime

extension (i.e,,it is not possible to interpolate a flow Y such that X-+Y-+X/P

with both homomorphisms non-trivial).

Suppose 2

distal extension of X whose maximal distal factor 2d Xd = W P , and 2' clearly 2

be a factor of 2

is strictly PI.

Show that 2'

example of flows X,Z, and 2' Mathematical Systems Theory,

2. Let

(X,T) and

with Z+Z'

is a

coincides with

proximal. Then

is not strictly PI. (An

as above was given by L. Shapiro, (19711, 76-88).

(Y,T) be minimal flows. Then X and Y have a

common almost one-to-one extension if and only if there are points

xo

E X

and yo E Y and an "orbit isomorphism" of xoT

is, a homeomorphism

(p

from xoT t o

to yoT

(that

y0T such that (p(xot) = y,t).

General Structure Theorems A PI

3.

25 1

flow can have uncountably many minimal right ideals in its

enveloping semigroup. Let

X denote the space of all rays in R2

emanating from the origin, and let T = SL(2,R) act on X in the usual fashion, and clearly A

on IR2

immediate that

(X,TI

s a

Z2

T acts on X

and that this action is transitive.) Now (Y,T) where Y

extension of

X,T) is P I .

ideals.

set of p E E(X) number)

Show that

(For any flow ( X , T ) such that

is the space of lines

(Y,T) is a proximal

has uncountably many minimal right a non-empty subset of X, (where I I

lXopl = 1

the

denotes cardinal

is closed and invariant, so if this set is non-empty, it

md

lXopl = 1 for some p E I ,

if X

0

I is a minimal right ideal

Thus if then

lXoql = 1 for all

= [ a , a + n ) is a half closed arc in

then there is a p f

E(X)

and Xo

contains a minimal right ideal.

xp

acts

preserves lines; it is

through the origin, and T acts as above. Since flow,

(A E T

t X0p). 1

E

E(X) such that

X

q E I.

Now

(regarded as the circle),

IXopl = 1,

and if x E X, x

ct

X

0'

This Page Intentionally Left Blank

253

Appendix I Nets Frequently the simplest and most intuitive proofs of theorems in metric spaces are accomplished by means of sequences (for example, the characterizations of the closure of a set and of the continuity of maps between metric spaces).

As is well known, sequences do not "suffice" to

describe the topology of spaces which are not metrizable (or at least first countable)

- the elementary theorems alluded to above are not valid

in general. Fortunately, there is a theory of convergence in general topological spaces which allows one to use arguments which are very similar to those involving sequences in metric spaces.

This is the

theory of nets (sometimes called generalized sequences) which we now develop. For more details and further discussion, consult General Topolonv by J.L. Kelley. Recall that a sequence is a function on the positive integers IN. N

property of

which is important for sequences is the possibility of

choosing "large" integers and m 2 n'.

The

- if n,n'

E N

m

there is an m E N with

To define a net, we replace H by a directed set

2

n

- a

partially ordered set which has the corresponding property.

A partially ordered set is a set D equipped with a reflexive transitive relation 5 directed

set

property:

(as usual, we write d t d'

if d'

5

d).

A

is a partially ordered set with the following additional

if dl,d2 E D there is a d E D with d 5: dl and

d 2 d2'

Of course, the positive integers and real numbers, with the usual ordering, are directed sets. An example of a different nature is the collection of neighborhoods of a point in a topological space, ordered by

Appendix I

254

inclusion. (If X

the collection of neighborhoods of

U

V.

3

If U1.U2

x

is a topological space and

E

X we write N x f o r

E

x; if U,V E Nx, U

Nx then obviously U1 n U2

2

U

1

5

and

V means Uz,

Nx

so

is indeed a directed set. A similar argument shows that any neighborhood V

base

at

x is directed by inclusion.)

A net is a function whose domain

{xi}. if the directed set D is understood

€or a net, o r just

{‘i’iED

is a directed set. We will write

o r if it is not necessary to specify it. { xi

’

i ED

If

{Xl’ieD

A

E

E A

i

for

every k E D

for all

is a net i n a set X and

i t m,

i

is a set, we say that E

D.

is a subset of

A

X, we say

if there is an m E D such that

is eventually in A

that

xi

if x

is a net in A

If A

is fresuently in A

and we say {xi)iED

if for

k such that x E A. m A subset F of the directed set D is called cofinal if f o r every

m

E

there is an rn

D there is a p

E

2

F with p

2

m.

(It is easy to see that a cofinal

subset of a directed set is also directed by

in X is frequently in the set A subset F of D such that the net

Thus the net

2.1

{xiIiED

if and only i f there is a cofinal is in

{xi)iEF

A.

X be a topological space, and let {Xi’iED be a net Then {xi}iED is said to converge to x E X (we write lim x Now let

x.-+x) 1

if, for every U

i.e., there is an m Note that if V

E

E

Nx the net

D such that xi

{ xi

E

i = x

’

i ED

E

V

U for all

we choose

xv

Or

is eventually in U; i t m.

is a neighborhood base at x, and V

inclusion, then if for each V

in X.

E

is directed by

V, {xv)vEV

is a net

with x +x. V

1.

Theorem. Let

X be a topological space, let

and let x E X. Then x E with x +x. 1

A

be a subset of X,

if and only if there is a net

{xi)

in

A

Nets

Proof. Suppose such a net eventually in U.

255

{xi) exists. Let

Therefore U n A

0

f

U E

and x

E

Nx,

A.

{xi)

so

is

Now suppose x

E

A.

Let the neighborhood system Nx be directed by inclusion and let xu

E

for U

U n A,

converges to

E

Nx. Then clearly {xu)

is a net in A

which

x.

2. Theorem. The topological space X

is Hausdorff if and only if every

net in X converges to at most one point ("limits are unique"). Proof.

If X

is Hausdorff, then it is clear that a net cannot have two

different limits.

(If U and

V.)

Suppose X

y but

U nV

be eventually in both U and there are x,y E X

with

x

f

V are disjoint open sets, a net cannot

f

is not Hausdorff. Then

0

for all

U

E

Nx, V

E

N

Y. NxxN directed by (U,V) 2 (U',V') if Y For each (U,V) E NxxN choose x U 3 U' and V 3 V'. Y u,v E U n V. Then {x } is a net with xu,,,---+x and xU,"+y. Consider the Cartesian product

u,v

3. Theorem. Let

Y and x {xi}

E

X and Y be topological spaces, f a map from X to

X. Then f is continuous at x if and only if, whenever

is a net with x.+x 1

{f(x. 1)

the net

converges to

1

f(x).

Proof. The proof of necessity is an easy modification of the usual proof for sequences in a metric space. To prove sufficiency, suppose V neighborhood of

f(x).

is not a neighborhood of

If f-'(V)

for every U E Nx, there is an xu {XU'U~NX

is a net with xu+x,

E

but

U

-1

with

f(xU)

CC

xu g f V

so

(V).

x,

is a

then

Then

f(xu)-+f(x).

Now we define subnet. The definition is somewhat involved, but as we shall see from its consequences, it provides the "right" generalization of subsequence. Let

be a net in the set X.

A net

{y 1

J JeE

in X

is a

Appendix I

256

subnet

of

i f t h e r e is a map

{xi}

x o N ( j ) (j E E l

such t h a t i f

p t n, N

that if

w e s a y t h a t t h e map

E

Thus as

t rn.

P

m

N

N : E+D,

with

j H N .

J

D

t h e r e is a n

n E E

p

g e t s l a r g e s o does

y

=

J

x

Nj

=

with t h e property N

In t h i s c a s e

P'

is c o f i n a l .

Of c o u r s e , a subsequence of a sequence is an example of a subnet = E = N,

(D

For a subnet i n g e n e r a l , t h e

w i t h t h e usual o r d e r i n g ) .

directed s e t s

and

D

need not be t h e same.

E

I n p a r t i c u l a r , a subnet

of a sequence need not be a subsequence. If

e a s y t o s e e t h a t every subnet a l s o converges t o

Let

is s a i d t o be a c l u s t e r p o i n t of

neighborhood of Theorem.

4.

{xi)

1

Proof.

If

The n e t

{xi)

has

x

X

of

is f r e q u e n t l y i n every

as a c l u s t e r p o i n t i f and o n l y i f

x

x.

is a c l u s t e r p o i n t of t h e n e t

x

A point

x.

has a subnet which converges t o

{x.}

X.

{xi}

if

t h e n i t is

x.

be a n e t i n t h e t o p o l o g i c a l space

{x } i

x.+x 1

is a n e t i n a t o p o l o g i c a l space w i t h

{Xi}iED

{xi}iED, l e t E be t h e

d i r e c t e d set d e f i n e d by

E = [ ( i , U ) l i E D. U E Nx (Note t h a t t h e r e l a t i o n (i,U),(j,V)

E E,

then

L

with

xe

let

2

k

d e f i n e a map

(p :

does indeed d i r e c t

5

x

i

E

E U n V,

E+D

by

i n c r e a s i n g and c o f i n a l ( i f (i,U)

xk on

E

E

W.

El,

E

and If

U, x so

J

let

E V;

i 2 j

c p ( i , U ) = 1 5 j). Now, i f

so t h e subnet d e f i n e d by

= x (p

E Ul

since i f

k E D

with

k 1 i,j

and

(j,V).)

I t is clear t h a t

D let

( j , U ) 5 (k,W),

E,

(L,U n V) 2 (i,U)

cp(i,U) = i. j E

xi

with

J

with

W E Nx, E U c W

converges t o

xi let

cp

E D

so

with

(by d e f i n i t i o n o f

x.

Now

is

E U E Nx,

k

and

I

Nets

Conversely, suppose

E+D

cp :

{xi} has a subnet which converges to

W

define the subnet. Now, let

show that there is an i with cp(e) with

257

2

k

(since cp

f t e and

and x E W i

E

xp(f 1

Nx and let k

E

D with i 2 k

and

x

i

is cofinal, such an e

W. Let

E

i = cp(f)

as required. Hence x

E

E

W.

so

Let

D. We must

E

Now let

exists).

D,

x.

Let

e

E

E

f

E

E

i = cp(f) 2 cp(e) t k

is a cluster point of the net

{xi}.

is compact if and only if every net in X has

5. Theorem. A space X

a convergent subnet. It is sufficient, by theorem 4, to show that compactness of X

Proof.

Suppose X

equivalent with every net having a cluster point. and let

{xi)i ~ Dbe a net in X. For

F = fi

let

m

m

E

Then the collection {HmlmED satisfies the finite

rn'

t 0.

If x

n Fm,

E

mED

point.

it is easy to see that

Let

Then B

B 1,B2

E B,

Now, let

d

is a cluster point

be a family of closed sets with the finite intersection B

be the collection of finite intersections of sets in

also has the finite intersection property; in fact, if then B1 n B

2

E B.

It is sufficient to show that

B

8 be directed by inclusion (B

choose xB E B,

point of this net.

so

Let

# 0.

B' E B

Since B'

arbitrary, we have z E

t 0.

I(

5

B'

if B

3

B';

since B

is indeed a directed set).

is

If

{xBIBEB is a net. Let z be a cluster and let

cluster point, there is a B" E B That is B" n V

nB B€B

closed under finite intersections, E

x

Conversely suppose that every net in X has a cluster

{xi).

property and let

B

By compactness

{Fm'mED'

maD

of the net

d.

is compact,

D, let Hm = Ix. li 2 ml, and 1

intersection property, and therefore so does

nF

is

3

V

NZ. By the definition of

E

with B" 2 B'

B", B' n V

= B" and so

f

nB B€B

0.

such that

xB,, E V.

Since V E NZ

is

f 0.

There is another theory of convergence in topological spaces, using

Appendix I

258

filters, which is equivalent to the one we have developed here. A filter

SF

in a set X is a non-empty collection of non-empty subsets of

X

which is closed under finite intersections and contains every superset of each number of

3. If X is a topological space, and x

is said to converge to

X, then N x ,

is obviously a filter and a filter SF

the set of neighborhoods of x, in X

E

x

if

3

3

Nx.

From an esthetic point of view, filters are perhaps preferable to nets. Filters are intrinsic to the space, whereas nets require the introduction of something outside the space, namely a directed set? However, we have chosen to use nets in this book because of their striking similarity to sequences, with which most readers are familiar.

259

Appendix I1 Uniform Spaces In this book, we have made extensive use of the language of uniform spaces, which are generalizations of metric spaces. The reason for this is not generalization for generalization's sake, but rather that the

class of metric spaces is not a suitable category for the development of our theory. As we have seen, even if one's main interest is a flow on a compact metric space, one is ed to the consideration of the enveloping semigroup which is a subspace of an uncountable product of copies of the phase space and therefore is n general not metrizable.

A uniform space is "like" a metric space in that it gives rise to a topology in which it is possible to compare neighborhoods at different points.

However (unlike metric spaces) the class of uniform spaces is

closed under the formation of arbitrary products. Moreover, a compact Hausdorff space admits a unique uniformity which generates its topology. In this brief appendix, we will develop those aspects of the theory of uniform spaces which are necessary for the topics in this book.

Many

proofs are omitted or sketched. The one substantial proof included is the uniformizability of compact Hausdorff spaces mentioned in the preceding paragraph. For a more complete treatment, see Chapter 11 of to pol on.^

for Analysis

by A. Wilansky (Ginn and Co., reprinted by Krieger).

A uniformity on a.set

X is a collection of subsets of XxX

each of

which contains the diagonal and satisfying some additional properties. As a first approximation, one may think of a uniformity as a collection

of neighborhoods of the diagonal.

(Of course, this is not meaningful

until a uniformity and the topology derived from it are defined.) Before

260

Appendix 11

giving the definition, we introduce some notation for subsets of If a c X ~ Xlet

a-' = [(x,y)l(y,x)

E

a ] . If a-'

to be symmetric, note that if p c XxX, p n p-' define a "multiplication" for subsets of = [(x,z)l(x,y) E a, (y,z) E

aof3

contain A,

then a08

f

0.1

XxX:

13 for some

x

u a(x). X€S

is said

If a , @ c XxX define

(If a and

We will frequent y w r te fl (aofl)or

=.

2

for

f3

/3ofl.

ao(floa).

X and a c XxX, let a(x) = [y E X l ( x , y ) E a1

E

S c X, a(S) =

and

then a

is symmetric. We also

y E XI

Note that this multiplication is associative: If

= a

XxX.

and i f

Clearly y E a(x) if and only if x

E

a-'

Y)

aop(x) = a(p(x)).

Now for the precise definition. A uniformity or uniform structure on a set X

is a collection U

(i)

The diagonal

(ii)

If a

A

c a

of subsets of XxX satisfying for every a E 11.

and p c XxX with

E U

( i i i ) If a,a' E U

/3 3 a, then

U.

If a E 21.

(v)

If a E 'U there is a 6 E 'U such that

an "index.

'I

E 'U.

then a n a' E 21.

(iv)

The pair

p

then

a-1 E

= pap c a .

(X,U) is called a uniform space. A member of 'U is called (Other terms in use are "entourage" and "connector." 1

Properties ( i )

- (iii) can be succinctly summarized as

filter of supersets of A."

"U

is a

Property (v) is the most important. It is

essentially a substitute for the triangle inequality in metric spaces, as we will see later. Just as in the case of a topology, it is sometimes more convenient to specify a collection of subsets which generates a uniformity. A uniformity base on X

is a collection V

satisfy (i)

A c a for every a E V

of subsets of XxX

which

Uniform Spaces

If a,a'

(ii)

Y, then there is a f3

E

(iii) If a E V

If a

(ivl If Y

7 E E

E

Y with

with f3 c a n a ' .

V 7

c a-'.

Y with p 2 c a.

is a base in this sense, then the collection U

of supersets

is evidently a uniformity.

of sets in Y If U

there is a

Y there is a p

E

261

is a uniformity, then the set Y of symmetric indexes in U

(Y = [ a n a-1 la

E

'111)

is a base for 21.

Thus we may assume (and

frequently will, without explicit mention) that an index is symmetric.

(X,u)

If

is a uniform space then 21 defines a topology (the uniform

topology) as follows. A subset G of X there is an a

'U such that

E

are open and x

E C

and a (x) c G2

so

2

1

A

G2,

If x E X then a(x)

(For example, if G1 and

then there are a l , a 2

E

n a2, a(x) c G1

G2.

if a = a

A

is a neighborhood of x

topology. To see this let G = [y E Xl@(y) c a(x) Then x 6

2

c f3.

E

G c a(x). Then if

If p(y) c a ( x ) ,

z E 6(y), 6 ( z )

E

G

It is immediate that this

a(x) c C.

process does indeed define a topology.

is open if for every x

let

6

E

'U

1 ' 1 with

c2

acl(x) c C1

1

in the uniform for some f3

E

U1.

be symmetric such that

c p(y) c a(x). Therefore G is open.

It follows that every index a of 21 is a neighborhood of the diagonal A

in the uniform topology. In general, it is not the case

that every neighborhood of the diagonal is an index nor is it the case that in a topological space the set of neighborhoods of the diagonal constitutes a uniformity. Moreover it is possible that two distinct uniformities on a set generate the same topology. However, as we shall see below, in a compact Hausdorff space, the collection of neighborhoods of the diagonal is a uniformity generating the topology, and is the only such uniformity.

The "prototypical" example of a uniform space is, of course, a metric

Appendix II

262

(X,d). A base for the uniformity is the collection of sets aE

space ( E

> 0)

where a =

Property (v) holds since

this is just a rephrasing of the triangle inequality.

c ac;

ac,20aE,2

(x,y)ld(x,y) < & I .

is a

Note that the proof given above, that in a uniform space a(x) neighborhood of

x, is a paraphrase of the elementary proof that “open

balls” are open sets in a metric space. Note also that if k > 0 the metric on X as d.

kd defines the same uniformity

This is a formalization of the idea that all the metrics

kd are “essentially the same.“

If

(X,d) is a pseudometric space

axioms for a metric space except that then the sets a

&

fi.e., d

satisfies all the

d(x,y) = 0 does not imply x = y)

as above still define a base for a uniformity. ( A uniform space

this case the uniform space is not Hausdorff. is Hausdorff, o r separated, if

na=

A.

In (X,U)

It is immediate that this is

a d

equivalent to the uniform topology being Hausdorff.)

It can be shown that a uniform space is pseudometrizable (there is a pseudometric which generates the uniformity) if and only if it has a countable base (Wilansky, p. 227). Another class of examples of uniform spaces is provided by topological groups.

In fact, if T

is a topological group there are three (in gen-

eral distinct) natural uniformities on T. Let of neighborhoods of the identity e. hU = [(s,t) E TxTls-lt

[hUIU the

E A’]

left and

and

If U

E

N denote the collection

N , let -1

Ul.

E

U)

and let pu = [(s,t)lst

[puIU E

Nl

both define bases for uniformities on T,

E

The families

rinht uniformities respectively. The two sided uniformity

has as a base the collection of intersections [hU n pvJU,V E X I .

In

general, these uniformities are distinct; however, they are equivalent in the sense that they generate the same topology (which is, of course, the

Unifam Spaces

topology of the group).

If T

263

is compact, these uniformities coincide.

Uniform spaces are the natural setting for the study of uniform continuity and uniform convergence. If

X+Y

spaces, a map

f

:

there is an a

E

U such that

(Y,V) are uniform

(X,U) and

is uniformly continuous if, for every (3 (fxf)(a) c 8.

homeomorphism if it is a bijection and both

The map

f

-1 f and f

V

E

is a uniform are uniformly

continuous. If X from X

is a set

b

no

E

21

and all

fn converges to

there is an no x

E

E

D such that

f

a net of maps

(f(x),fn(x))

[;la E V ]

E

for

a

X. Indeed, in this case a collection of maps 9

= [(f,g) E YxYl(f(x),g(x))

define

X if

uniformly on

to Y can be made into a uniform space as follows.

from X

tion

(Y,V) is a uniform space {fn’ncD

Y and f : X+Y,

to

for every a n

and

E

a for all x

E

XI.

If a

E

V

The collec-

forms a base for a uniformity, and convergence in the

topology defined by it is obviously uniform convergence as defined above. One readily proves the usual theorems

- for example, a uniform limit

of continuous maps is continuous, a continuous function defined on a compact space is uniformly continuous (with respect to the unique uniformity defined below), a uniform space is compact if and only if i t

is complete and totally bounded (once the appropriate terms are defined in the obvious manner).

The proofs are direct paraphrases of the

corresponding proofs for metric spaces.

(Xi,Ui)( i

Let

X

=

nX

E

Y) be a family of uniform spaces and let

The uniformity on X

(the product uniformity) is defined as

iEY i‘

If F = { i l ,..., im}

follows. and a i.

E

Ui (j = 1, . . . ,m),

let

J

J

= [(x,y)

#

al,.. .

is a finite subset of the index set 3

“rn

E

XxXl(xi ,yi 1

J

J

E

ai

J

( J = 1,..., mil.

A base

Y for

Appendix II

264

the product uniformity is the collection of all such sets

. . 'am

(for all finite subsets F of 9). It is immediate that the topology defined by the product uniformity is the (topological) product of the uniform topologies on the X

i'

It follows that an arbitrary product of pseudometric spaces is uniformizable and therefore a subspace of such a product is as well. (We omit the obvious definition of subspace uniformity.) In fact, the converse holds.

If

(X,U)

X

is a uniform space, then

is uniformly

homeomorphic to a subspace of a product of pseudometric spaces (if (X,U) is Hausdorff replace "pseudometric" by "metric"1.

Closely related is a

topological characterization: a topological space is uniformizable if and only if it is completely regular. We will not prove these theorems (see Wilansky, p. 230, ff) although we use the first one in the proof of lemma 2 in the "Joint continuity theorems" chapter. However, in the applications of lemma 2 in that chapter, the space is in fact compact Hausdorff and the embeddability of such spaces in a product of metric spaces is well known and easy to prove. Now we state and prove the theorem on the uniformizability of compact Hausdorff spaces. Theorem. Let

X be a compact Hausdorff space. Then NA, the collection

of neighborhoods of the diagonal, is a uniformity which generates the topology. Moreover U '

NA is the only compatible uniformity (that is, if

is a uniformity which generates the topology of

Proof. If

fl,...,fnE C(X)

[(x,y)llfi(x)-fi(y)l

<

E,

for

is an open neighborhood of A. (xo,yo) & W,

and c > 0 let i = 1,2 , . . . , n].

Let

there is an f E C(X)

W

E

01

X

then U = NA).

fl'* .. ,fn,c

--

Clearly af

N A with W open.

such that

If(xo)-f(yo)

> 1.

Let

Uniform Spaces Uf = [(x,y) E XxXlIf(x)-f(y)l open cover of

In this way, we can construct an

By compactness, there are fl,...,fn E C(X) such

XxX\W.

XxX\W c Uf u.. . uUf . 1 n

that

> 11.

265

Then af

l,. . . ,fn,1/2

c W.

Since

a c a it follows immediately that fl’.. . ,fn,e/2 fl’.. . ,fn,e

NA

is a

(Obviously N A generates the topology of X. 1

uniformity.

U be a uniformity on X which generates the topology. To

Now let

U = NA’ it is sufficient to show: if V E N A then 6 c V for some 6 E U. Now, if x E X, there is a a E U with a (xlxa (XI c V. X X X prove

c a U with px symmetric and 132 x x’ Then N X c u p_(x) so by compactness X c Up, (xi). Let 6 = n px . i=l i i=1, . . . , N i X€X * Now, if (y,y’)E 6 then y E 3/, (xi) for some i with 1 5 i 5 N and

For each a

X

let 6,

E

i Y’

E

6(y) c 13,

(y) so

y’ E

E

8, (xi) c ax (xi) i

PX (xi) c i

i

Y

2

i

so

ax (xi). Also i

(y,y’) E ax (xi)xax (xi) c V. i i

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