Extensions of Minimal Transformation Groups 940099561X, 9789400995611

Table of contents :
Contents
Preface to the English Edition
List of Symbols
Introduction
Chapter I. Topological Transformation Groups
§(1.1): Basic Definitions
§(1.2): Recursion
§(1.3): Relations
§(1.4): The Ellis Semigroup
§(1.5): Pointwise Almost Periodic Transformation Groups
§(1.6): Distal and Equicontinuous Transformation Groups
Chapter II. Minimal Transformation Groups
§(2.7): The Enveloping Semigroup of a Minimal Transformation Group
§(2.8): Almost Periodic and Locally Almost Periodic Minimal Sets
§(2.9): Distal Minimal Transformation Groups
§(2.10): Transitive Distal Transformation Groups and Nil-Flows
§(2.11): Topological Properties of Minimal Sets
Chapter III: Extensions of Minimal Transformation Groups
§(3.12): The Basic Theory of Extensions
§(3.13): Equicontinuous and Stable Extensions
§(3.14): The Structure of Distal and Almost Distal Extensions
§(3.15): Groups Associated with Minimal Extensions
§(3.16): Group Extensions
§(3.17): Relations Between Group Extensions and Equicontinuous Extensions; A Strengthened Structure Theorem for Distal Extensions
§(3.18): A Method for Constructing Minimal Sets
§(3.19): Disjointness
Chapter IV. Extensions and Equations
§(4.20): Shift Dynamical Systems
§(4.21): Extensions Associated with Certain Classes of Equations
§(4.22): Almost Automorphic Extensions and Synchronous Solutions of Differential Equations
§(4.23): Extensions with Zero-Dimensional Fibres and Almost Periodic Solutions
§(4.24): Linear Extensions of Dynamical Systems and Linear Differential Equations
Bibliography
Subject Index

Citation preview

Extensions of Minimal Transformation Groups I. U. Bronstein Academy of Sciences of the Moldavian Soviet Socialist Republic Institute of Mathematics and Computer Center

SUTHOFF & NOORDHOFF 1979 Alp hen aan den Rijn The Netherlands Germantown, Maryland USA

Copyright© 1979 Sijthoff & Noordhoff International Publishers bv, Alphen aan den Rijn, The Netherlands A II rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 90 286 0368 9

Original title: "Rassirenija minimal'nyh grupp preobrazovanii" published in 1975 in Kisihev Edited by Dr. S. Swierczkowski

Printed in The Netherlands

CONTENTS

PREFACE TO THE ENGLISH EDITION

vii

LIST OF SYMBOLS.,,

... viii

INTRODUCTION

1

CHAPTER I

l•l 1•2 1•3 1•4 1•5 1°6

TOPOLOGICAL TRANSFORMATION GROUPS

9

BASIC DEFINITIONS RECURSION, RELATIONS, THE ELLIS SEMIGROUP,,, POINTWISE ALMOST PERIODIC TRANSFORMATION GROUPS DISTAL AND EQUICONTINUOUS TRANSFORMATION GROUPS

13 15 23 32

9

36

CHAPTER II

MINIMAL TRANSFORMATION GROUPS

2·7 2•8

THE ENVELOPING SEMIGROUP OF A MINIMAL TRANSFORMATION GROUP ALMOST PERIODIC AND LOCALLY ALMOST PERIODIC MINIMAL SETS , .

V

45

45

49

vi

2•9 2·10

2 ·11

CONTENTS

DISTAL MINIMAL TRANSFORMATION GROUPS ,,, TRANSITIVE DISTAL TRANSFORMATION GROUPS AND NIL-FLOWS TOPOLOGICAL PROPERTIES OF MINIMAL SETS ..

68 76 83

CHAPTER III EXTENSIONS OF MINIMAL TRANSFORMATION GROUPS 3•12 3•13 3•14 3•15 3•16 3•17

3,18 3,19

THE BASIC THEORY OF EXTENSIONS ,,, EQUICONTINUOUS AND STABLE EXTENSIONS THE STRUCTURE OF DISTAL AND ALMOST DISTAL EXTENSIONS GROUPS ASSOCIATED WITH MINIMAL EXTENSIONS GROUP EXTENSIONS RELATIONSHIPS BETWEEN GROUP EXTENSIONS AND EQUICONTINUOUS EXTENSIONS; A STRENGTHENED -STRUCTURE THEOREM FOR DISTAL EXTENSIONS A METHOD FOR CONSTRUCTING MINIMAL SETS,, DISJOINTNESS , , ,

107 107 118 129 147 166

185 198 215

CHAPTER IV EXTENSIONS AND EQUATIONS 4·20 4'21 4·22 4·23 4•24

SHIFT DYNAMICAL SYSTEMS EXTENSIONS ASSOCIATED WITH CERTAIN CLASSES OF EQUATIONS ALMOST AUTOMORPHIC EXTENSIONS AND SYNCHRONOUS SOLUTIONS OF DIFFERENTIAL EQUATIONS,,. EXTENSIONS WITH ZERO-DIMENSIONAL FIBERS AND ALMOST PERIODIC SOLUTIONS,, LINEAR EXTENSIONS OF DYNAMICAL SYSTEMS AND LINEAR DIFFERENTIAL EQUATIONS ,,,

232 232 238 245 260 274

BIBLIOGRAPHY

293

SUBJECT INDEX

317

PREFACE TO THE ENGLISH EDITION

This edition is an almost exact translation of the original Russian text. A few improvements have been made in the presentation. The list of references has been enlarged to include some papers published more recently, and the latter are marked with an asterisk. THE AUTHOR

vii

LIST OF SYMBOLS

A(X ,T)

2•7.3

c2 [ ( Y, T, p) , G, h] C9, [ (Y ,T ,P) ,G,h]

3•16.6 3•16.12

E = E(X,T,n) Ey

1•4.7 3•12.8 3•16.4 3•16.12

Ext[(Y,T,p),G,h] Extt((Y,T,p) ,G,h]

M = M(X,T,n) M(R)

1•3.3 2•9.4

P = P(X,T,n) Px

1•3.3 2•8.9

Q = Q(X,T,n) Q''' = Q>'.. :T + G defined by t>.. = nt (t e T) is a homomorphism of the group Tonto G; (6) if x e X, then nx is a continuous function. If ACX and BC T, then the set (Ax Bh = {xtlx eA,t eB} will be, for brevity denoted by AB. Similarly, if t e T, then instead of {antlaeA}, we shall write At. If xeX, then xB denotes the set {xn t It e B}.

1•1.2

1•1.3 LEMMA: The following assertions hold: ( 1) : If A C X, t e T, then At = At; (2):

If Ac X and BC Tare compact subsets, then AB is also compact;

(3): If A and Bare compact subsets of X and T, respectively,

and Wis a neighbourhood of AB, then there exist neighbourhoods U of A and V of B such that UV c W. Proof: (1): This assertion is true, since nt:x +Xis a homeomorphism. (2): The set AB is compact as an image of the compact set A x B under the continuous map n. ( 3) : Let a e A and t e B. There exist neighbourhoods Ua,t of a and Va,t oft such that Ua,tVa,t CW. Choose a fini.te set FC A with AC U [Ua,tlaeF] Ut. Let Vt= ri [Va,tlaeF], then UtVtC W (teB). Thus, for each element t e B there exist neighbourhoods Vt of t and Ut of the set A, such that UtVt CW. Since the set Bis compact, we may choose a finite set EC B such that BC U [VtlteE]. Put V = U [VtlteE] and U = ri [Vt It e E] • Then U is a neighbourhood of A, V is a neighbourhood of B and UV C W. I

=

1•1.4 LEMMA: Let X,Y be unifomt spaces, let ~:X + Y be a continuous map and let A be a compact subset of X. Then

TRANSFORMATION GROUPS

11

for each Se U'[Y] there exists an index a e 'U[X] such that xmp C xqiS for au X eA.

The proof is left to the reader. 1•1.5 LEMMA: Let X be a uniform space, let a et.C[X], and let A and B be compact subsets of X and T, respectively. Then: (1): If t e T, then there exist a Se 'U[X] and a neighbourhood V of the identity e e T such that xStV C xta for aU x eA; (2) : There exists an index S e U[X] such that xSt C xta for aU x e A, t e B.

Proof: (1) follows immediately from Lemma 1•1.4. The second assertion follows from (1). I The property expressed in (2) is sometimes called UNIFORM INTEGRAL CONTINUITY.

1•1.6

Let Ac X and Sc T. The set A is called an S-INVARIANT SET if AS CA. T-invariant sets are, for brevity, called INVARIANT SETS. By definition, the empty set is invariant. It is easy to verify the following propositions: (1) if the set A is invariant, then X....___A, A, intA are also invariant; (2) the intersection (union) of a family of invariant subsets of Xis invariant. Let x e X and SC T. The set xS is called the S-ORBIT of x. The set xT will be simply called the ORBIT (or TRAJECTORY) of the point x. It is clear that the orbit xT is the smallest invariant subset of X, containing the point x. If y e xT, then xT = yT. The set xT is the smallest closed invariant set containing x. If Y is an invariant subset of X, then the transformation group (X,T,n) induces on Ya topological transformation group, which will be denoted by (Y,T,n). A non-empty closed invariant subset MC Xis called a MINIMAL SUBSET if M does not contain a proper closed invariant subset. Note that a set M is minimal iff M = xT for every point x e M. 1•1.7 THEOREM: Any non-empty compact invariant set ACX contains a minimal subset. Proof: Let I be the family of all non-empty closed invariant subsets of A. I t ¢, since A e I. Further, I is partially ordered by inclusion: A1 a,; A2, iff A1 C A2. If '1 is a linearly ordered sub-family of I, then the intersection B of all sets from 1 is non-empty, because A is compact. Clearly, Bis invariant. Hence, BeI. By Zorn's Lemma, I contains at least one minimal element M. It is clear that Mis a minimal subset of A. I

1•1.8

Let {CXi,T,ni)lieI} be a family of transformation groups indexed by a set I and let X = II [Xi Ii e I] be the (Cartesian) topological product of the family {Xi Ii e I}. Let us define a map n:XxT-+ X as follows: if x = {xilieII and teT, then

12

TOPOLOGICAL

(x,th = {xp:rlieI}, It is easy to verify that (X,T,1t) is a topological transformation group. We shall say that (X,T,1t) is the CARTESIAN PRODUCT OF THE FAMILY and denote it as: (X,T,1t)

=

1I[- X/R by: (A' t )p = A TI t

(AeX/R,teT).

It is easy to verify that (X/R,T,p) is a topological transformation group and that the canonical projection cp:X + X/R is a homomorphism of (X,T,n) onto (X/R,T,p). In fact, it is evident that the identity axiom and the homomorphism axiom are satisfied. To verify the third axiom, let A e X/R, t e T and let U be a neighbourhood of (A,t)p = Ant inX/R. The set Ucp- 1 is open in X, since cp is continuous. If x eA, then xnt eATit C Ucp- 1 . We can choose a neighbourhood V of x and a neighbourhood W of the element t such that (V,W)n C Ucp- 1 . Since the map cp is open, the set Vcp = {BIB e X/R, B () V ,j: ¢} is a neighbourhood of the point A e X/R. Furthermore, if Be Vcp ands e W, then Bn 3 c Ucp-1, and therefore (B,s)p =Bn 3 e U. Thus (X/R,T,p) is a transformation group. Let A eX/R and xeA. Then xcp = A and xntcp = Ant, since xnt eAnt. Hence xntcp = xcppt (x eX,t eT), i.e., cp is a homomorphism of (X,T,n) onto (X/R,T,p). Suppose now that Xis compact, Risa closed invariant equivalence relation on X and cp is the canonical projection of X onto X/R. We shall show that (X/R,T,p), where pis defined as above, is a topologcal transformation group. It will suffice to verify the continuity axiom. Let A e X/R, t e T and let U be a neighbourhood of the point (A,t)p =Antin X/R. The set Ucp- 1 is open in X and Ant C Ucp-1. The set A is closed and therefore compact. By Lemma 1•1.3(3) there exist neighbourhoods V of the set ACX and W of the element t e T such that (V,W)TI c Ucp-1. Since the relation R is closed, V contains an R-saturated neighbourhood V1 of the set A. By definition of the quotient topology, V1cp is a neighbourhood of the point A e X/R and (Vn,W)p c U. Thus (X/R,T,p) is a topological transformation group. The above considerations show that cp:X + X/R is a homomorphism of (X,T,n) onto (X/R,T,p).

17

TRANSFORMATION GROUPS

Conversely, let X be a compact space and let cp:X + Y be a homomorphism of (X,T,n) onto a transformation group (Y,T,a). The set R = {(x1,x2)Jx1,x2eX,xH = x2cp} is a closed invariant equivalence relation and, moreover, (Y,T,a) is isomorphic to (X/R,T,p). Thus, in the class of transformation groups with compact phase spaces, the notion of a homomorphism is in fact equivalent to the notion of a closed invariant equivalence relation. Let (X,T,n) be a topological transformation group and let (XxX,T,n) be the Cartesian product of the transformation group (X,T,n) with itself. By definition, the map n:(XxX) xT + (XxX) is given by ((x,y).,t)n = (x,t)(nt x nt) = (xnt,ynt) ((x,y) eXx X, t e T) • A relation R is invariant iff the set RC Xx X is invariant under (Xx X,T,n). Let X be a compact space, let (X ,'l_', n) be a transformation group and RC Xx X be a relation. By R•• we will denote the smallest closed invariant equivalence relation containing R. 1•3,3

Let (X,U) be a uniform space and let (X,T,n) be a transformation group. Let us iptroduce the following defin-

itions. Two points x and y of X are called PROXIMAL under (X,T,n) if for every index a e U[X] there exists an element t e T such that (xt ,yt) e a. Two points, which are not proximal are called DISTAL. The set of all proximal pairs of points is called the PROXIMAL RELATION and is denoted by P(X,T,n) or, simply, by P. Two points X and y of X are said to be REGIONALLY PROXIMAL under (X,T,n) if for each index a e U[X] and for each neighbourhood U of x and each neighbourhood V of y we can choose points x1 e U, Yl e V and an element t e T in such a way that (x1t ,y1t) ea. The set of all regionally proximal pairs of points is called the REGIONALLY PROXIMAL RELATION and is denoted by Q(X,T,n) or, simply, by Q, Let L = L(X ,T, n) denote the set of all pairs (x ,Y) e Xx X such that for every index a e U[X] there exists a syndetic subset A C T, satisfying the condition (xt,yt)ea for all teA. The set L(X,T,n) is called the SYNDETICALLY PROXIMAL RELATION. By M = M(X ,T, n) we denote the set of al 1 pairs (x ,Y) e Xx X such that for every index a e U and for every neighbourhood U of x and every neighbourhood V of y there exist points x1 e U, Yl e V and a syndetic subset ACT such that (x1t,y1t) ea for all t eA. The set M(X,T,n) is called the REGIONALLY SYNDETICALLY PROXIMAL RELATION.

By K* we denote the set of all compact subsets of the group T. The following equalities are valid:

n [aTJa eU]; = n [aTJa e U]; L = n u n a.Kt; aeU KeK''' teT p

(3. 1)

Q

(3.2) (3. 3)

TOPOLOGICAL

18 M =

n u n

(3.4)

a.Kt.

a.eU Ke.K.* teT

We prove, for example, the second formula. If (x,y) e Q and W is a neighbourhood of (x ,y) e Xx X, then there exist neighbourhoods U of x and V of y such that U x V c W. For these neighbourhoods and each index a. e U there exist points xi e U, Yl e V and an element teT such that (x1t,y1t)ea.. Put x:z= xit, Y2 = yit; then (x1,y1) e U x V c W and (x1,y1) = (x2r 1 ,y2t- 1 ) e a.T, i.e., wn a.T ¢ for every neighbourhood W of (x,y) and each index a. e U. But this just means that

+

(x ,y) e n [a.TI a. e U]

.

Conversely, suppose that the last formula holds. If U is a neighbourhood of x, V is a neighbourhood of y and a. t.C[X], then there exists a pair (x1 ,y1) e a.T such that Xl e U, Yl e V. Since (xi ,Yl) e aT, there exist a pa;i.r (x2 ,Y2) ea and an element t e T for which x1 = x2t, Yl = Y2t, Hence (xit- 1 ,yit-l) ea, from which it follows that (x ,y) e Q. The proof of the other three formulae is analogous. It is immediate that

e

t,C LC MC QC XxX,

!:;CLCPCQcXxX,

(3.5)

where t, = {(x,x)lxeX}. The relations P,Q,L,M are invariant, reflexive and symmetric. Let us investigate some properties of these relations. We prove first two lemmas. 1•3.4 LEMMA: Let T be a group and let A1,A2,K1,K2 be subsets of T such that T = A1K1 = A2K2. Then

Proof: For each t e T there exist elements a2 eA2 and k2 e K2 such that t = a2k2. Choose elements a1 e A1 and k1 e K1 for which a2 = a1k1, Then t = a1k1k2 and a1 = a2k1 -le Ai r"IA2K1-l. I 1•3.5 LEMMA: Let T be a group, n be a positive integer and A1, •• ,,An,K1, •• ,,Kn be subsets of T such that T = AiKi (i = 1, ... , n). Then

n i-1 n )-1)) nn Ki,

n [ T = ( Ai ,z,=1

Kj

J=O

,z,=1

where Ko= {e} and e is the identity of T. Proof: The statement is true for n = l. Suppose that it holds for n = p. We shall prove that it is true also for n = p + l. Since T = Ap+lKp+l, we get by our assumption that

TRANSFORMATION GROUPS

19

It follows by the preceding lemma that

n

p+l [

[i-1

i=l

j=O

= (

Ai

IT

Kj

)-1) )p+l

IT

Ki.

1

i=l

1•3.6 THEOMM: I f the space Xis compact, then Lis an equivalence relation on X. Proof: We need only to show that L is transitive. Let (x ,y) e Land (y,z)eL. We shall prove that (x,z)eL. Let ae'U[X]. Choose S e U:[X] so that s 2 C a. Then there exist subsets Al C T and K1 C T such that K1 EK•'•, T = A1K1 and (x ,Y )Al '= { (xt ,yt) It e A1} C s. Since the set K is compact, the set K1- 1 is also compact. Since the space X is comfact, for S e U[X] we can choose an index yeU:[X] such that yK1- CS (see Lemma 1•1.4(2)). From (y ,z) e L it follows that there exist sets A2 C T and K2 C T for which K2 e K•'•, T = A2K2 and (y ,z )A2 C y. Then (y ,z )A2K1 -l C S. Let A = A1 nA2K1 -l; then (x ,Y )A C S and (y ,z )AC S; therefore (x,z)Ac a. By Lemma 1•3,4, T = AK1K2. Since K1 and K2 are compact sets, the set K1K2 is also compact. Hence A is a syndetic set. Thus (x ,z) e L. I

1•3. 7 LEMMA: I f u eL, then uTC L. Proof: Let we uT and a e U:[X]. Choose an index Se U[X] such that BC a. There exist sets AC T and Kc T such that Ke K•'•, T = AK and uA Cs. We shall prove that weL. It will suffice to prove that wtK-lna 4 ¢ for all t eT. Suppose that to the contrarywtK-lna =¢for some teT. ThenwtK-lni3 =¢,i.e., wtK- 1 C (XxX)......_S, Since the set (XxX)......_S is open in XxX and tK- 1 is compact, there exists a neighbourhood U of w such that UtK- 1 n S = ¢. Since we uT, there exists an element s e T with us e U. But st e T = AK, and hence stK- 1 n A ¢; therefore ustK- 1 n uA f ¢ and moreover ustK- 1 n S 4 ¢, since uA C S. But this contradicts the definition of U. I

+

1•3.8 LEMMA: I f Xis compact, z eXxX and zTc P, then z eL. Proof: Let a e U[X] . For each point we zT C P there exists an element tw such that wtw ea, and therefore Uwtwc a for some neighbourhood Uw of the point w in X x X. Since X is compact, Xx X is also compact; hence we can choose a finite subset F of zT so that zTCU[Uw]weF]. Let us define K = {tw- 1 \weF}, A= {titeT,ztea}; we shall prove that T = AK. Let teT. Since

TOPOLOGICAL

20

zte,zTc U[UwlwcF], there exists a point fe,F such that zte,Uf. Hence zttf E a; therefore ttf EA and t E AK. I · 1•3.9 THEOREM: Let X be a compact space and let (X,T,n) be a transformation group. Then (1) L == {zlz €XXX zTc P};

(2) p

L iff z

E

P implies zT C P;

(3) If P is closed in Xx X, then P

L and P is a closed

invariant equivalence relation. Proof: This follows immediately from the preceding two lemmas.

I

1•3,10 LEMMA: Let n be a positive integer, let X1, .. ,,Xn be compact spaces, (X1 ,T), ... , (Xn ,T) transformation groups, let (xi,Yi)EL(Xi,T) and let aicU[Xi] (i == 1, ..• ,n). Then there is a syndetic subset A C T such that (xi ,Yi )A C ai for each i (i == 1, .. . ,n).

Proof: There are subsets A1 C T and K1 € K'h(T) which satisfy the conditions T == A1K1 and (x1,Y1)A C a1, Since Xz and K1-l are compact and (x2 ,Y2) E L(X2 ,T), there exist subsets A2 C T and K2 € K,',(T) such that T == A2K2, and (x2 ,Y2) € A2K1 -l C a2. Proceeding by induction, we get subsets A1, ... ,An,Kl, ... ,Kn of the group T which satisfy the conditions Kie,K,'' Q.

Proof: Clearly (1) implies (2). Let P::J Q. Then P = Q, since always Pc Q, and therefore P is closed in Xx X. By Theorem l • 3. 9, P = L. Therefore the equalities (1) follow from the formulae (3.5).

I

Suppose that Xis compact and let R1,R2 be closed invariant equivalence relations. If R1 c R2, then the canonical projection A of X/R1 onto X/R2 is a homomorphism of (X/R1,T) onto (X/R2,T). Let {(Xi,T)lieI} be a family of transformation groups with compact phase spaces and for each i e I let Ai be a homomorphism of (X ,T) into (Xi ,T). Suppose that the maps Ai (i e I) separates the points of X, i.e., for each two distinct points x1 and x2 of X there is a Ai such that XlAi X2Ai, Define a map A:X + TI[Xil i e I] by XA = {XAi Ii e I}. It is easy to verify that A is an isomorphism of (X,T) onto an invariant subset of the Cartesian product TI[(Xi,T)li eI]. A transformation group (Xo,T) is called a TRIVIAL TRANSFORMATION GROUP if Xo consists of a single point. Let (P =(P(X,T) be a property of transformation groups that is preserved under isomorphisms. The property(P is called an ADMISSIBLE PROPERTY if it satisfies the following three conditions: 1·3.15

+

CONDITION (1): The trivial transformation group (Xo,T) has

the property '•(T), then S(A) is a closed syndetic subgroup, and therefore

TRANSFORMATION GROUPS

57

the quotient group T !S(A) is compact. T/S(A) by

Define a map p: (T /S(A)) x T

+

(S(A)T,t)p = S(A)Tt,

(t,T eT).

It is easy to verify that (T/S(A),T,p) is a minimal equicontinuous transformation group. 2 • 8. 26 LEMMA: Let (X ,T, n) belong to the class c,\ let x e X and let A eA'''(T). Then the map 1/lx:T/S(A) + X/R(A) defined by (S(A)T)1)!x = xS(A)T

= xAT,

(T

e T) ,

is an isomorphism of (T/S(A),T,p) onto (X/R(A),T,n). Proof: It is easy to see that 1/lx is continuous and pto1/lx 1/lxont (t eT). We shall show that 1/lx is injective. For if xAT1 = xAT2, then XT1T2- 1 exA, that is, T1T2- 1 eS(A), and therefore S(A h1 = S(A h2. Since the space T /S(A) is compact and the space X/R(A) is Hausdorff, 1)1 is a homeomorphism.I In what follows, F will denote the neighbourhood filter at the identity e e T. 2•8.27 LEMMA: If the transformation group (X,T,n) belongs to the class c,·, and V is a neighbourhood of the identity e e T, then there exists an index a e U[X] such that (x ,Y) ea implies

xA C yAV,

yA C xAV.

Proof: The set { v,·, IVe F}, where

v,·,

= {(S(Ah,S(A)t)IT- 1 te V}

is a uniformity base of the compact topological group T/S(A). Let z e X. Since 1/lx:T/S(A) + X/R(A) is a homeomorphism, there exists an index a e U[X] such that xA = zA , yA = zAt and (S(A), S(A )t) e whenever (x ,Y) e a. Clearly a is the required index. I

v,·,

2•8.28 THEOREM: If the transformation group (X,T) belongs to the class c,·, and A e A1•(T), then R(A) is an invariant equivalence relation which is both open and closed.

Proof: By Lemma2•8.24, R(A) is a closed invariant equivalence relation. Let 13 e U[X]. Since X is compact, there exists a neighbourhood Ve F such that (x ,xt) e 13 whenever x e X and t e V. By Lemma 2•8.27, given V we can find an index aeU[X] such that (x,y) ea implies xA C yAV and yA C xAV. Then xA C yAl3, yA C xAl3, whenever (x ,y) e a. I 2: 8. 29 LEMMA: If (X ,T, n) be longs to the class c1, and Ak e A•' Q1u =ti= {(x,x)lxeX};

(2): If Sis a limit ordinal, S,.;; !&, then Qs =

n Qy;

y G{ ::> • .. ::> Gk

::::>

Gk+l = {e}.

If k = 0, the group G is commutative, A= {e}, therefore the assertion is trivial in this case. Suppose that our claim has been proved for all connected simply connected nilpotent Lie groups with the length of the lower central series smaller thank (k > 0), and suppose that G is a group with the length of its lower central series equal to k. Clearly the group N = Gk is contained in the centre of G. Then G1 = GIN is a connected simply connected nilpotent Lie group, HnN is a syndetic subgroup of N, the groups NHIH and Nl(HnN) are isomorphic, and HNIH is a syndetic subgroup of G1, Moreover, the length of the lower central series of G1 is equal to (k - 1). The group AIN is the commutator of the group G1. Since the flow (GIAH,lR,a) is minimal, the flow (GIANH,lR,cr) is also minimal. Hence by the inductive assumption, the set GINH is minimal under the flow induced by (GIH,cp). We shall prove that (GIH,cp) is also minimal. By Corollary 2•10.8, the flow (GIH,cp) is distal. It follows from Theorem 1•6.4 that the set M = {H(tcp)jteJR} is minimal under (GIH,cp). Denote Mn (NHIH) by S. Since the group N is contained in the centre of G, every translation by an element of the group N is an automorphism of the transformation group (GIH,cp). Hence it follows that Sis a closed subsemigroup of the compact group NHIH. Since every minimal right ideal of the semigroup S contains an idem-

TRANSFORMATION GROUPS

81

potent, namely the identity of NH/H, we conclude that Sis a subgroup of NH/H. Let us prove that S coincides with NH/Hand, consequently, that (G/H,rp) is a minimal transformation group. Suppose that, to the contrary, Sis a proper subgroup of NH/H. Without loss of generality we may then suppose that Sis the identity subgroup. Since the flow on G/NH is minimal by the inductive hypothesis, the flow on Mis mapped isomorphically onto the minimal flow (G/NH,rp). It follows that for each element geG there exists an element (N()H)b of the group N/(NI\H) such that Hg(Nn H)b

= HgbeM.

We shall show that (NnH)b is uniquely defined. Suppose the contrary to be true. Then for some element g e G there are elements h1 e N and h2 e N such that Hgb1 = Hgb2, but (NnH)b1 (Nn H)b2, Set bo = b1 - 1b2, Then Hgb2 = Hgb1bo e MnMbo, Since the set Mis minimal and N is a subset of the centre of G, the set Mbo is also minimal. Therefore Hgb2 e MnMbo implies Mbo = M, and thus Hbo e M, i.e. , Hbo e S and Hbo H. This contradicts our assumption that Sis the identity group. Let us consider the map iJi :G/H-+ (G/NH) x (N/NnH), where (Hg)iJi = (NHg,(NnH)b) and let us choose the element (NnH)b in such a way that Hg(NnH)b = Hgb e M. It follows from the above that iJi is well defined. We shall show now that iJi is bijective. If (Hg1)iJi = (Hg2)iJi, then NHg1 = NHg2, i.e., Hg1 = Hg2b for some beN. If the element (NI\H)b1 is such that Hg1b1 e M, then also Hg2bb1 e M. Therefore the second coordinate of the point (Hg2)iJi is equal to (NnH)bb1, From the hypothesis (Hg1H = (Hg2)iJi it follows that (N()H)b1 = (NI\H)bb1, i.e., beNnH. Therefore Hg1 = Hg2b = Hg2. Thus iJi is injective. Now we shall prove that iJi is continuous. If the net {Hgil i e I} converges to an element Hg e G/H, then {NHgi Ii e I} -+ NHg. For each i e I there exists an element (NnH)bi such that bi e N and Hgibi e M Ci e I). Let (NI\H)bo be the second coordinate of (Hg)iJi, that is, HgboeM. We shall prove that {(Ni\H)bilieI}-+ (NI\H)bo, Suppose the contrary holds. Since the group N/(NI\H) is compact, there exists a subnet {(Ni\H)bj lj e.'f} of the net {(NnH)bilieI} such that {(NnH)bjlje.'f}-+ (NnH)b1, where b1e N and, moreover, (NnH)b1 (NnH)bo, Since Hgibi eM, we get Hgb1 e M; hence the second coordinate of the point (Hg)iJi is equal to (NnH)b1, This is impossible because iJi is well defined. Thus the map iJi is continuous and bijective. Since G/H is compact, iJi is a homeomorphism of G/H onto (G/NH) x (N/(NnH)). Since NI (N n H) is a finite-dimensional torus, its fundamental group is commutative. Byhypothesis, His a discrete subgroup of G, hence His the fundamental group of the space G/H (Pontrjagin [1]). Since the length of the lower central series of G is k and Hn G{_ is a syndetic subgroup of G{_ (i = 1, ... , k) , the length of the lower central series of His also equal to k. Similarly NH/N is the fundamental group of the space G/NH and the length of the lower central series of the group NH/Nisequal to (k -1). Thus the length of the lower central series of the fundamental group of the space G/H is k, but the length of the lower central

+

+

+

82

MINIMAL

series of the fundamental group (NH/N) x (N()H) of (G/NH) x (N/N()H) is equal to (k - 1). This is impossible, because the spaces G/H and (G/NH) x (N/N()H) are homeomorphic, as was shown above. This contradiction shows that (G/H,~) is a minimal flow.I 2•10.12 EXAMPLE: Let G be the group of all matrices of the form 1

a

b

0

1

C

0

0

1

where a,b,c are real numbers, and let H be the set of all those matrices in G which have integer entries. Clearly G is a connected simply connected nilpotent Lie group and H is a discrete syndetic subgroup of G. Each one-parameter subgroup of G (i.e., each continuous homomorphism ~:JR+ G) is of the form

t~

=

yt + ;l:aSt

1

at

0

1

St

0

0

1

2

where a, S and y are real numbers ( t eJR) . We can conclude from Theorem 2•10.11 that the flow (G/H,~) is minimal iff the numbers a and Sare rationally independent. Indeed, A= [G,G] is the subgroup of G consisting of all matrices of the form 1

0

b

0

1

0

0

0

1

and the flow induced by (G/H,~) on the adjoint two-dimensional torus T = G/AH is minimal iff the numbers a and Sare rationally independent. By Corollary 2•10.6,theflow (G/H,~) is distal for every one-parameter subgroup ~:JR+ G. Thus whenever a and Sare rationally independent, (G/H,~) is a distal minimal flow. Moreover, the flow (G/H,~) is not equicontinuous in this case. Indeed, the fundamental group n1(G/H) of the space G/H is equal to H, which is not commutative. If the minimal flow (G/H,~) were equicontinuous, the nilmanifold G/H would be, by Theorem 2•8.5, homeomorphic to a commutative topological group, and hence the fundamental group n1(G/H) would be commutative. Thus the flow (G/H,~) is not equicontinuous.

TRANSFORMATION GROUPS

83

REMARKS AND BIBLIOGRAPHICAL NOTES Theorems 2•10.4,5 were proved by Wu [4) and independently by Bron~tein [11]. Corollaries 2•10.6,8 were first obtained by Keynes [1]. Theorem 2•10.11 is due to Gree (see L. Auslander, Green and Hahn [1]). The proof of this theorem given here is due to Bron~tein [2,5]. Example 2•10.12 is taken from the book by L. Auslander, Green and Hahn [1].

§(2•11): TOPOLOGICAL PROPERTIES OF MINIMAL SETS We shall investigate in this section various topological properties of minimal sets and we shall discuss their situation in the phase space, We shall consider several concepts of homogeneity and relations between them, as well as between homogeneity, distality and equicontinuity. We ·shall state some sufficient conditions for the fundamental group of a minimal set to be nontrivial, We shall also discuss the kind of spaces which can support a minimal transformation group. The exposition will be illustrated by several examples, 2•11,l THEOREM: Let (X,T,n) be a topological transformation group and let ACX be a minimal set. Then the following assertions hold: (1): A is open in X iff intA

4 ¢;

(2): If intA 4 ¢, then A is a union of components of the space X; (3): If intA 4 ¢ and the group Tis connected, then A is a component of X; (4): If intA

4¢

and Xis connected, then A= X.

Proof: (1): Let x s intA and ye A. Since A is minimal, we have Hence there exists an element ts T with ynt s intA. Since nt:x +Xis a homeomorphism, the set B = (intA)nt-l is open in X and, moreover, y s BC A. Thus the set A is open in X. The converse is obvious, (2): Let intA 4 ¢, By (1) the set A is open in X. But A is also closed in X, Hence (2) is true. (3): Let intA 4 ¢ and let T be connected. If x e A, then xT = A and consequently A is connected. Therefore (2) implies that A is a component of X. (4): This statement follows at once from (2) ,I yT = A,

2•11.2

Let n be a positive integer. A space Xis called an if it is connected and each point of X has a neighbourhood homeomorphic ton-dimensional Euclidian space, Let X be an n-dimensional manifold. It is proved in dimension theory (see: Hurewicz and Wallman [1]) that if ACX, then n-DIMENSIONAL MANIFOLD

MINIMAL

84

dimA ~ n, where dim denotes the inductive dimension, and moreover dimA. = n iff int A+¢. 2•11,3 THEOREM: Let X be an n-dimensional manifold and let ACX be a minimal set of (X,T,n) such that A+ X. Then dimA ,,; n - 1.

Proof: Suppose the contrary: dimA = n. Then intA +¢and since Xis connected, we get X = A by Theorem 2•11.1(4), but this contradicts A+ X. Consequently, dimA,,; n - l,I 2•11.4

Let n be a positive integer. A compact metrizable space X of dimX = n is called a CANTOR MANIFOLD if X cannot be presented as a union of two non-empty closed subsets x1 and x2

such that dim(X 1 nx2 ) ,,; n - 2. Note that a Cantor manifold is not necessarily a manifold in the above sense. It is not difficult to see that a Cantor manifold is connected and moreover each n-dimensional Cantor manifold is of dimension n at each of its points. By the Hurewicz-Tumarkin Theorem (see: Hurewicz and Wallman [7]), each n-dimensional compact metrizable space contains an n-dimensional Cantor manifold. 2•11.5 THEOREM: Let X be a finite-dimensional metrizable compact space, Ta connected group and (X,T,n) a minimal transformation group. Then Xis a Cantor manifold. Proof: Let n = dimX. Suppose to the contrary that Xis not a Cantor manifold. Then there exist closed non-empty subsets A and B of X such that X = AUE and dim(Ai\B),,; n - 2. By the HurewiczTumarkin Theorem there is a subset CCX such that dimC = n and C is a Cantor manifold. Put E = {tit e,T,Ct CA} and F = {tit e,T,Ct CB}. Since A and Bare closed in X, the sets E and Fare closed in T. For each t ~ T, the set Ct is homeomorphic to C, and hence Ct is a Cantor manifold. Since dim(Ai\B)

~

n - 2,

dimCt = n,

we get either Ct CA or Ct CB. In other words, T = ElJF, and By hypothesis, the group T is connected, hence either T = E or T = F. Suppose, for example, that T = E. Then CT CA, hence CTC A. Therefore CT+ X, but this contradicts the fact that (X,T,n) is minimal.I En F = ¢.

2•11.6

If the hypothesis that Tis connected is omitted from Theorem 2•11.5, the above conclusion cannot be drawn. This is demonstrated by the following example. Let A be a closed rectangle in the plane, defined,by

A= {(x,t)la,,; x,,; a+ ~,b,,; t,,; b + h}, where~> O, h > 0. Let us denote by A* the union of the following three rectangles:

TRANSFORMATION GROUPS

85

Ao= {(x,t)\a c. x c. a + 51 l,b c. t ( b + 21 h}, {(x,t)\a +

A2 = {(x,t)\a +

l5

3

l ( x (a+ 5 6.,b ~ tc.b+h},

iL(

x (a+ l,b +

½h

( t c. b + h}.

Define by induction a sequence of sets Xn (n = 0,1, ... ) . Let = {(x,t)\o c. x c. 1,0 c. t , 1}, X1 = X*o. Then X1 is a union of three rectangles, which we shall denote by S(l,O), S(l,1), S(l,2):

Xo

S(l,O)

= {(x,t)\o

(

X

~

1 1 5'0 ( t ( 2 } '

S(l,1)

{ 0. Choose n large enough so that the breadth of each rectangle S(n,j) (0 ~ j ~ 3n - 1) is smaller than s/2. Letko be such that a 2 = (x 2 ,t 2 ) eS(n,ko). There exist numbers m and jo such that m ~ n, o ~ jo ~ 3m - 1, S(m,jo) C S(n,ko) and moreover lt2 - ti < s/2, whenever a= (x,t) eS(m,jo). Observe that a1 eS(m,R-o) for a certain number to. It follows from the definition of the map F that there exists a number r such that (x3,t3) = a1F'r'eS(m,jo) C S(n,ko). Then lt3 - t2I < s/2. Since the breadth of the rectangle S(n,k 0 ) is smaller than s/2, we get lx 3 - x2I < s/2. Therefore p(a2,a1Fr) < E. Since a1 and a 2 are arbitrary points of the space X, we conclude that Xis a minimal set. It is clear that the dimension of Xis equal to Oat the point (0,0) eX and it is equal to 1 at (½,½) eX. Therefore X is not a Cantor manifold. Observe that Xis a locally almost periodic minimal set and that two points of X are proximal iff they belong to the same vertical segment (the proof is left to the reader). If S(n,j) c S(m,k), then S(n,j)Fn C S(m,k)Fm; therefore A

B

+ ¢.

2•11.7

A TRANSFORMATION GROUP (X,T,n) is called:

(1) TOPOLOGICALLY HOMOGENEOUS if the space X is homogeneous, i.e. , given two points x ,Y e X, there exists a homeomorphism ~:X + X with x~ = y; (2) ORBITALLY HOMOGENEOUS if for each two points x,y e X there exists a homeomorphism ~:X + X, x~ = y, which preserves orbits (i.e., (zT)~ = (z~)T for each point z eX); (3) DYNAMICALLY HOMOGENEOUS (or, simply, homogeneous) if for each two points x ,Y e X there exists an automorphism ~ of the transformation group (X,T,n) sending x toy (i.e., a homeomorphism ~:X + X such that x~ = y and (znt)~ = (z~)nt for each z e X, t e T) ; (4) h-HOMOGENEOUS if for each two points x ,y e X there exists an orbit-preserving homeomorphism ~:X + X, x~ = y, which is homotopic to the identity map. (5) HARMONIZABLE if there is a dynamically homogeneous trans-

TRANSFORMATION GROUPS

87

formation group (X,T,n*), which has the same orbits as (X,T,n) does. Let X be a topological space, (Y,p) a metric space, {~nln = 1,2, ... } a sequence of continuous maps of X into y and X EX. We say that {~n} CONVERGES TO A MAP ~:X-+ y UNIFORMLY AT THE POINT x if for each s > 0 there exists a neighbourhood U of x and a positive integer N such that p(y~,Y~n) < s for all y EU, n ;;, N. 2•11.8

2•11.9 LEMMA: Let X be a topological space, let Y be a metric space, and let {~nln = 1,2, ... } be a sequence of continuous maps of X into Y which converges to a function ~ :X -+ Y at each point x EX. Let x0 denote the set of all points x EX at which {~n} converges to~ uniformly. Then Xo can be represented as the intersection of a countable family of open dense subsets of X.

Proof: Let A( d denote the set of all points x EX such that p(y~,Y~m) < s for all points y from some neighbourhood of x and for all m, greater than some positive integer. It is clear that

n A(l/n). n=l 00

for each s >

O

the set A(s) is open in X and Xo =

Lets> 0. It is enough to show that the set B = X'-A(s) can be represented as a union of a countable family of closed nowhere dense subsets. Set 8 = s/5. For each positive integer n, let Cn denote the set of all points x EX such that p(x~ ,x~m) < 8 whenever m ;,, n.

By hypothesis, X =

B n Cn, then B =

have B

0

n=l B, and hence B

LJ

LJ

LJ

Cn; therefore B = (Br'ICn). n=l n=l Dn. Since the set B is closed we

O

Dn ::i Dn. It remains to show n=l n=l that the sets Dn, or equivalently, the Dn are nowhere dense in X. Let n be a fixed positive integer. Suppose to the contrary that Dn contains some non-empty open subset U C X. Let x EU and p ;;, n. Then there exists a positive integer q;;, n such that p(x~,x~q) < 8. There is also a point y E Ur'IDn C Cn with p(x~p,Y~p) < 8 and p(x~q,Y~q) < 8. Then p(x~,x~p) ~ p(x~,x~q) + p(x~q,Y~q) + p(y~q, y~) + p(y~,Y~p) + p(y~p,X~p) < 58 = s. Thus p(x~,x~p) < s whenever x.;U and p;,, n. By definition.!. U C A ( s) = X ...___B, and hence Un B = ¢. On the other hand, U C D C B, hence Un B ¢, because U is a non-empty open set. We have arrived at a contradiction. This completes the proof. I ::i

+

2•11.10 THEOREM: A homogeneous transformation group (X,T,n) with a compact metrizable phase space is equicontinuous.

Proof: We need only to show that the closure of the set G = {ntltET} in C(X,X) is compact (Section §(1·6)). Let {tnln = 1, 2,. . . } be any sequence of elements tn ET. We shall prove that

88

MINIMAL

{ 11 tn} contains a uniformly convergent subsequence. Without loss of generality we may suppose that

A.illi

Let xo e X. . xo11 tn exists

and is equal to Yo e X. Let x be an arbitrary point of the space X. There exists an automorphism g eA(X,T,11) with xog = x. Then y 0 g = limx 011tng = limx 0g11tn = limx11tn. Thus the sequence {11tnl n = 1,2, ... } converges at each point xeX. Let h:X-+ X be the limit function. The space Xis compact and therefore complete. By the preceding lemma and Baire's category theorem, there exists a point x1e X such that {11tn} converges to h uniformly at x1. Let us show that { 11 tn} -+ h uniformly on X. Let x e X and e: > 0. There exists an element g eA(X,T,11) with x = xig. Choose a number o > 0 such that ~(z1g,z2g) < e: for all z1,z2 e X with p(z1,z2) < o. Since {11tn} converges uniformly to hat x1, we can choose a neighbourhood U1 of x1 and an integer N > 0 such that p(zh, z11tn) < o for all z e U1 and n > N. Denote U1g by U. Since x = x1g and g:X-+ Xis a homeomorphism, U is a neighbourhood of x. If yeU and n > N, then yg- 1 eU1 and therefore p(yhg- 1 ,y11tng- 1 ) = p(yg-lh,yg-l11tn) < o. Then (ye U ,n > N).

(11.1)

Thus for every point x e X there exists a neighbourhood U of x and a number N such that (11.1) holds. Since the space Xis compact, it follows that the sequence {ntn} converges to h uniformly on

X.I

2•11.11 THEOREM: If the space Xis compact and the group T is commutative, then every equicontinuous minimal transformation group (X,T,11) is homogeneous.

Proof: By Theorem 2•8.5 the transformation group (X,T,n) is isomorphic to a transformation group (S,T,p), where Sis a compact commutative topological group, (s,t)p = s(tcp) (s eS, t eT) and cp:T-+ Sis a homomorphism, satisfying the condition~= S. Let p,qeS. The map ijJ:S-+ S defined by sijJ = qp-ls (seS) is an automorphism taking p to q.l The hypothesis that Tis commutative cannot be dropped, as shown by the example of the rotation group of the two-dimensional sphere. 2•11.12 THEOREM: Let X be a compact metric space a:nd let T be a commutative group. A minimal transformation group (X,T,11) is homogeneous iff it is equicontinuous. Proof: This follows from Theorems 2•11.10,11.I 2•11.13 LEMMA: Let X be a compact space, (X,T) a distal transformation group and E = E(X,T) the enveloping semigroup.

TRANSFORMATION GROUPS

89

Then (E,T) is a homogeneous minimal transfoY'ITlation group. Proof: By Theorem 1•6.3, the enveloping semigroup is in fact a group. Hence it follows from Lemma 1•4.8 that (E,T) is a minimal transformation group. Let p ,q e E. The map qi :E -+ E defined by scp = qp-ls (s e E) is continuous and bijective. From this we easily deduce that qi is an automorphism of the transformation group (E,T) sending p to q,I 2•11.14 LEMMA: A homogeneous transfoY'ITlation group (X,T) with a compact phase space is distal.

Proof: Since Xis compact, it follows from the Theorems 1•1.7 and l • 5. 4 that X contains an almost periodic point xo EX. Since the transformation group (X,T) is homogeneous, all the points x e X are almost periodic. Moreover, the transformation group (Xx X, T) is pointwise almost periodic. Therefore it follows from Theorem 1•6.3 that (X,T) is distal. I 2•11.15 THEOREM: A minimal transfoY'ITlation group (X,T) with a compact phase space is distal iff (E(X,T),T) is a homogeneous transfoY'ITlation group.

Proof: Let (E,T) be homogeneous. Then (E,T) is distal by Lernma 2•11.14. There is a homomorphism o£ (E,T) onto (X,T), hence (X,T) is distal (Corollary 1•6.7.). The second assertion follows from Lemma 2•11.13,I 2•11.16 THEOREM: Let (X,T) be a distal minimal transfoY'ITlation group with a compact phase space and suppose that at least one of the following conditions is satisfied: (a):

The group E(X,T) is commutative;

(b): The space E(X,T) is metrizable.

Then (X,T) is equicontinuous. Proof: If (a) is satisfied, then the multiplication in the compact group E(X,T) is continuous in each variable separately. Hence it follows from Lemma 2•8.2 that (X,T) is equicontinuous. If condition (b) is satisfied, then the transformation group (E,T) is equicontinuous by Lemma 2•11.13 and Theorem 2•11.10. Hence it follows that (X,T) is also equicontinuous (Corollary 1•6.16). 2•11.17 LEMMA: A minimal transfoY'ITlation group (X,T,n) with a compact phase space is homogeneous iff the map cpx:E(X,T,n) -+ X, where PC+!x = xp (p EE), is injective for each x e X.

Proof: If (X,T,n) is homogeneous, then it follows from the Theorems 2•11.15 and 1·6.3 that E(X,T,n) is a group. Let xeX and p e E be such that pcpx = xp = x. We shall show that p is the identity of the group E.

Indeed, if {ntili EI} is a net that

converges to p, then {xn ti Ii e I} converges to xp = x. For each pointy eX, there exists an automorphism qi EA(X,T,n) with xcp = y.

90

MINIMAL

Thus

and hence yp = y (ye X). But this just means that p is the identity of E. If P~x = q~x, then xpq-1 = x. Then, as it was shown above, p = q. Hence ~x is injective. Conversely, let ~x be injective. The semigroup E contains at least one minimal ideal, say I. Since xI = X, there exists an element p e I with P~x = x. From the injectivity of ~x at the point e we conclude that e = p e I. Hence I = E; consequently E is a group (Corollary 1•4.13). By Theorem 1•6.3 the transformation group (X,T) is distal, and from Lemma 2•11.13 we deduce that (E,T) is homogeneous. Since ~xis bijective, it is an isomorphism of (E,T) onto (X,T). Thus the transformation group (X,T) is homogeneous.I 2•11.18 Given a minimal transformation group (X,T,n), a point Xe Xis called a CHARACTERISTIC POINT if for every net {ti}, ti e T, the condition {xti} -+ x implies {yti} -+ y for all ye X. 2•11.19 THEOREM: A minimal transformation group (X,T,n) with a corrrpact phase space is homogeneous iff X has at least one characteristic point. Proof: If Xis a homogeneous minimal set, then all of its points are characteristic. Conversely, let x e X be a characteristic point. It is easily seen from Definition 2•11.18, that if the equality xi; = x holds for some element I; e E(X ,T), then I; is the identity map of X. Hence, the map ~x=E-+ Xis injective at the point e e E. For the same reason as in the proof of the second part of Lemma 2•11.17, Eis a group; hence (X,T) is distal and (E,T) is homogeneous. We shall prove that ~xis injective. In fact, if xi;= xn, then x/;n-1 = x and therefore l;n-1 = e, that is, I;= n. Thus ~x:(E,T)-+ (X,T) is an isomorphism.I 2•11.20 THEOREM: The class of all distal minimal transformation groups with compact phase spaces and a fixed phase group T contains a universal transformation group (Do,T) which is unique up to isomorphism. Proof: It is easy to verify that the class of all distal minimal transformation groups with compact phase spaces is admissible, and hence the required result is a consequence of Theorem 2•7.12. I 2•11.21 THEOREM: The universal distal minimal transformation group (Do,T) is homogeneous. Proof: It can be seen from the proof of Theorem 2•7.12 that (Do,T) is isomorphic to (E(Do,T),T), but the latter transformation group is homogeneous by Lemma 2•11.13.I 2•11.22 Since equicontinuity of a transformation group is preserved under homomorphisms (Corollary 1•6.16) and there

91

TRANSFORMATION GROUPS

exist minimal distal flows which are not equicontinuous (Example 2•9.19-20), the transformation group (Do,JR) is not equicontinuous. From Theorem 2•11.16 we may thus conclude that the space E(Do,JR) = Do is not metrizable. This shows that the metrizability condition put on the phase space in Theorem 2•11.10 cannot be dropped. 2•11.23 LEMMA: Let X he a compact space, (X,T,n) a transformation group, ~:(X,T,n) + (Y,T,p) a homomorphism and let (Y,T) be minimal. If the map~ is a local homeomorphism, then (X,T,n) is pointwise almost periodic.

Proof: Let x e X, y = x~. Since each point of Y is almost periodic, there exists an idempotent v e E( Y ,T) belonging to some minimal right ideal K of E(Y,T) and such that yv = y. By Theorem 1·4.21(9), we can choose a minimal right ideal IC E(X,T) and an idempotent ueI so that uS;; = v, where S;;:E(X,T) +E(Y,T) is the homomorphism of the enveloping semigroups induced by~- We shall prove that xu = x. Suppose the contrary holds. Choose a net taeT, such that {nta} +u inE(X,T). Then {xta} +xu, and therefore {yta} = {x~ta} = {xta~} + (xu)~ = x~(uS;;) = yv = y. On the other hand {xta} + xu, {xuta} + xuu = xu and (xuta)~ = (xta) ~ = yta· Since xuta xta and {yta} + y, this contradicts the injectivity of~ in every neighbourhood of the point xu. Thus xu = x for an element u e I; consequently x is an almost periodic point. I {t 0: } ,

+

2 • 11. 24 Let Y be a connected locally pathwise connected space, p:

X +Ya covering map, Ga connected Lie group and (Y,G,p) a transformation group. Then there exists no more than one transformation group (X,G,n) covering (Y,G,p) (i,e., p:(X,G,TI) + (Y,G,p) is a homomorphism). If G is simply connected, then the covering transformation group actually exists. Let us prove this assertion. Define a map o :Xx G + Y by (x,g)o

=

(xp,g)p,

(x e X ,g e G).

Let xo e X, Yo = xop. Since TI 1 ( G) O, the image of the homomorphism o~:TI1((X,xo) x (G,e)) + TI1(Y,yo) induced by o is contained in the image of the homomorphism P.,.:TI1(X,xo) + TI1(Y,yo). By the covering theorem, there exists map TI :X X G + X such that ii;op = Ci. It is easy to verify, that (X,G,n) is the required transformation group.

a

2•11.25 LEMMA: Let G be a connected Lie group, Ya compact pathwise connected space, (Y ,G) a minimal transformation group, p:X +Ya covering map and let (X,G) he the covering transformation group. If Xis compact, then (X,G) is a minimal transformation group. If Xis not compact, then X does not contain non-empty compact invariant subsets.

Proof: Since (Y,G) is minimal and G is connected, the space Y is also connected. The space Xis connected by the very definition of a covering map.

MINIMAL

92

Let X be compact.

Since pis a local homeomorphism, each point

x e X is almost periodic (Lemma 2 • 11. 23) . Furthermore, each fibre of p:X-+ Y is finite and Xis connected; therefore Xis a minimal

set. Now suppose that Xis not compact, but contains a non-empty compact invariant set. Then X contains a compact minimal set M. If the interior of M with respect to Xis non-empty, then X = M by Theorem 2•11.1, but this leads to a contradiction. Thus intx M = ¢. Let m e M. The point mp e Y has a connected open neighbourhood V such that each component of the preimage vp-1 is open in X and if we restrict p:X-+ Y to vp-1, we get a homeomorphism. Let U be the component of vp-1 containing the point m. Then unM is a relative neighbourhood of min M. Since Y is compact and minimal, it follows that W = inty(U(1M)p ¢. In view of the fact that Plu:U-+ Vis a homeomorphism, ¢ unwp-1 c unM c M. This contradicts intxM = ¢.I

+

+

2•11.26 THEOREM: Let X be a compact connected locally pathwise connected space, Y a ZocaUy pathwise connected and semilocally simply connected space, Ga connected Lie group and let (X,G) and (Y,G) be transformation groups, where (Y,G) be minimal. Let p: (X ,G) -+ (Y ,G) be a homomorphism, xo e X and Yo = xoP· Then PJ.(111(X,xo)) is of normal index in the group 111(Y,yo) (here p,,. is the homomorphism of fundamental groups induced by p). "In particular, if the group 111(Y,yo) is infinite, then 111(X,xo) O.

+

Proof: Without loss of generality we may suppose that G is simply connected (it is enough to replace G by its universal covering group). According to the theorem on the existence of covering maps, we can construct a covering map q:(Z,zo)-+ (Y,yo) such that

Let (Z,G) be the transformation group which covers (Y,G) by q. It follows from the covering map theorem that there exists a unique continuous map r:(X,xo)-+ (Z,zo) with roq = p. Since Xr is a compact invariant subset of Z, Lemma 2•11.25 implies that Z is compact. But it is clear that Z is compact iff the group q,,. ( 111 (Z,zo)) p 1/111(X,xo)) is a subgroup of normal index of the"group 111(Y ,Yo)· 1

=

2•11.27 Let T be a topological group, Sa closed syndetic subgroup of T, Ka topological space and let (K,S,p) be a transformation group. Define two topological transformation groups (KxT,S,a) and (KxT,T,r) as fo11ows: (k,t)a 8 = (kp 8 ,s

(k,t),r

=

-1

(k,tr),

t),

(k e K, t e T, s e S) ,

(keK,teT,reT).

TRANSFORMATION GROUPS

93

Let (KxT)/S be the quotient space of orbits of (KxT,S,o). It is easy to verify that the actions o and T commute, and therefore T induces a transformation group ((KxT)/S,T,rr). In this case we shall say that (K,S,p) is a GLOBAL SECTION of the transformation group ((KxT)/S,T,n). Observe that there exists a homomorphism q:((KxT)/S,T,rr)-+ (T/S,T,p), where (St,r)p = Str (t,re T), namely, [(k,t)S]q = [{(kps,s- 1 t)JseS})q = St (keK,teT). Since Sis a syndetic subgroup of T, the quotient space T/S = {StJteT} is compact. The transformation group (T/S,T,p) is, of course, minimal. 2•ll.28 THEOREM: Let (X,T) be a transformation group, Sa closed syndetic subgroup of T, Ka closed subset of X and assume that the foUowing conditions are satisfied: (1): KT = X;

(2): KS C K; (3): If k e K and kt e K, then t e S.

Then (K,S) is a global section of (X,T). Proof: We must show that the transformation group (X,T) is isomorphic to ((KxT)/S,T,rr). By condition (1), each point xeX can be represented in the form x = kt, where k e K, t e T. This can be done in different ways, but if x = k1t 1 , where k1 e K, t1 e T, then k1 = ktt 1 -1. Using (3) we conclude that s1 = tti- 1 e S. Thus k1 = ks1, t1 = s1- 1 t. Hence, the map f:X-+ (KxT)/S, defined by xf = (k,t)S

= {(ks,s -1 t) Is GS},

for x = kt, k eK, t eT, is well defined. defined by

[(k,t)S]f-l = kt,

The inverse map f-1 is

(k e K, t e T),

and it is clearly continuous. We shall prove that f is also continuous. Let {xa}, xa e X, be a net that converges to some point x.;X. By (1), Xa = ka.ta., where ka.eK, ta.eT. Since the subgroup Sis syndetic, there exists some compact subset N satisfying T = SN. Therefore, ta = sa.na., where na. G N, Sa. GS. Denote .Q,a. = ka.sa.. By condition (2), we have .Q,a e K. Without loss of generality we may suppose that {na} -+ n eN. Then {.Q,a} = {kasa} = {xana- 1 } -+ xn-1 .Q, and .Q, e K because K is closed. Hence x = .Q,n and therefore {xa.f} = {(ka.,ta.)S} = {(.Q,a.,na.)S}-+ (.Q,,n)S = xf. I

=

2•11.29 THEOREM: Let X be a compact space, (X,T) a minimal transformation group, H a syndetic normal subgroup of T, xo X and let xoH = K X. Then there exists a subgroup S of T such that S:::) Hand (K,S) is a global section of (X,T).

+

Proof: Define S = { t Jt e T ,Kt C K}. Then S :::) H and S is a closed syndetic subsemigroup of the group T. By Lemma 2•8.17,

e

MINIMAL

94

Sis a subgroup of T. It follows from the definition of S that KSC K. Since T = SN for some compact subset N of T, we have KT = KSN C KN C X, and hence X = KT C KN C X; consequently, X = KT. Further, if k e K and kt e K, then 7ZH = ktH = xoH = K, because K = xoH is an H-minimal set (Lemma 2•8.20), i.e., (xoH,H) is a minimal transformation group. Thus

Kt= xoHt = kHt

=

kHt

ktH = K,

and this means that t e S. We have thus proved that al 1 the conditions of Theorem 2•11.28 are satisfied. Hence (K,S) is a global section. I 2•11.30 THEOREM: Let X be a connected locally pathwise connected space, Ka closed connected subset of X, Ta connected Lie group, Sa non-trivial discreet syndetic subgroup of T and let (X,T) be a transformation group. Suppose that moreover the conditions (1-3) of Theorem 2•11.28 are satisfied. Then the fundamental group of the space Xis nontrivial: 1t1(X) t o.

Proof: It follows from the restrictions put on T and S, that the canonical projection of Tonto T/S is a covering map. Moreover 1t1(T/S) =St 0. Furthermore, X = (KxT)/S is the total space of a fibre bundle with base T/S and fibre K, associated with the principal fibre bundle T ~ T/S. Hence, we have the exact sequence of homotopy groups (Hu (1], p. 214):

Since the fibre K is a closed connected subset of a locally arcwise connected space X, 1to(K) = O. Therefore the map

is surjective.

Thus 1t1(X)

t

0.

I

2•11.31 COROLLARY: Let X be a compact locally pathwiseconnected space and let (X,JR) be a minimal, but not totally minimal, flow. Then 1t1(X) t 0.

Proof: By hypothesis, there exists a subgroup SC JR, S = a,z, a t O, such that the cascade (X,aZ3) is not minimal. Let x e X and K = xS; then Kt X. By Theorem 2•11.29 we may without loss of generality suppose that (K,S) is a global section of the flow (X,m.). To conclude the proof we refer to Theorem 2•11.30. I 2•11.32 THEOREM: Let (X,d) be a compact metric locally arcwise connected space, and (X,JR,1t) be a minimal flow with a global section. If the flow (X,E.,1t) is h-homogeneous, then it is dynamically homogeneous (and consequently almost periodic).

Proof: By hypothesis, there exists a homomorphism~ of the

95

TRANSFORMATION GROUPS

flow (X,lR,n) on some periodic flow (Sl,JR,p). Without loss of generality we may suppose, that the period of (Sl,lR,p) is 1. Let xo e X be an arbitrary but fixed point. Let us show that xo is a characteristic point. To verify this, we may restrict ourselves to consider only points Yo e X with xocp = y 0 cp. Let E be an arbi trary positive number. There exists a homeomorphism h:X + X which is homotopic to the identity map, preserves orbits and sends xo to YO· The map hocp is homotopic to cp:X + s1 , therefore the induced homomorphisms of fundamental groups coincide, i.e.

(hocp ),.,

=

cp;,.

( 11. 2)

Let zo = x 0 cp. The fundamental group n1(S 1 ,zo) is isomorphic to the group LZ. If 0 is a loop at the point xo, then 0h is a loop at YO = xoh e X. Let [0] be the class of loops homotopic to the loop 0; then [0] en1(X,xo) and [0cp], [0hcp] en1(Sl,zo). By (11.2) [0cp]

=

[0]cp 1,

=

[0] (hocp )1,

= [0hcp].

( 11. 3)

Since X is a compact set, given E > 0 there exists a number 6. > 0 such that 6. < 1/3 and d(z,znt) < E/2 for each zeX and te(-6.,L). Put S = {xlxeX,xcp = zo} and U = {xntlxeS,ltl < L}. Since U is an open subset of X and the space X is locally pathwise connected, we can choose a small enough number ex> O so that ex< E, S(yo,ex) C U and every two points from S(yo,ex) can be joined by a path situated in U. Choose a number L' >Osuch that L' < 6./2 and d(z,znt) < ex/2 for all z eX and t e (-6.' ,L' ). The open subset {xntlxeS,te(-6.' ,L' )} will be denoted by V'. There exists a number ex' > 0 with ex' < ex/2 and S(yo,ex') CU'. Since his a homeomorphism, S(xo,S)h C S(yo,ex') for some S > 0. There exists a number S' > 0 such that S' < S and every two points of S(xo, (3' ) may be joined by a path lying in S(xo,S). Further, choose a number 6." > O such that L" < 6./2 and d(z,znt) < (3' /2 for all z eX, te(-6.",L"). We get an open subset U" = {xntlxeS,ltl < L"}. Finally, there exists a number o > 0 with o < S' /2 and S(xo,o) C

U". Suppose that d(x 0 ,x 0 nt) < o for some teJR. Thenxonteu", hence t = m +µ,where mis an integer and lµI < L". Since d(xo, x 0 nt) < 8 < (3', there exists a path i joining x 0 to x 0 nt and lying in S(x 0 ,s). By the choice of L" and owe have that d(x 0 nt,x 0 nm) = d(xont,xontn-µ)


0 and every point Yo EX there exists a number o >Osuch that d(xo,xont) < o implies d(yo,Yont) < E. Therefore x 0 is a characteristic point. By Theorem 2•11.19, the flow (X,JR,n) is homogeneous, and consequently almost periodic (Theorem 2•11.10).1 2•11.33 COROLLARY: A minimal harrnonizable flow with a compact metrizable locally pathwise connected phase space is

TRANSFORMATION GROUPS

97

either almost periodic or admits no global sections (i.e., it does not admit a homomorphism onto a periodic flow). Proof: By Theorems 2•11.10 and 2•8.5, a harmonizable flow is h-homogeneous. 2•11.34 COROLLARY: The distal minimal flow on the three-dimensional torus described in 2•9.19-20 is not harmonizable. 2•11.35 We will now give an example of a topological homogeneous, but orbitally non-homogeneous, minimal flow. Let G be the group of all real matrices of the form a (11.6)

1 0

Subsequently we shall write for the sake of brevity (a,b,c) instead of (11.6). Let N denote the subgroup of G consisting of all matrices with integer elements. The homogeneous space x0 = GIN = {Nglg e G} is a compact three-dimensional nil-manifold and N is its fundamental group. Each homeomorphism¢ of x0 onto itself, for which N¢ = N, induces an automorphism

i. e. , an automorphism ¢,,: N -+ N. Every homomorphism ~O of JR into G is of the form (t eJR),

where a, Sandy are arbitrary real parameters. ism ~o:lR-+ G induces a flow (Xo:lR,rc):

(Ng,t)rc

=

Ng(t~o),

The homomorph-

(Ng e Xo, t eJR) .

If a and Sare rationally independent, then (Xo,JR,rc) is a distal minimal (but not an equicontinuous) transformation group (see: Example 2•10.12). We prove next several auxiliary results. The identity (o,o,o) of the group G will be denoted bye. 2•11.36 LEMMA: For every homeomorphism¢ of Xo onto itself carrying the point Ne GIN = Xo to a point Q Nq, there exists a unique homeomorphism ~=G-+ G withe~= q and N(g~) (Ng)¢ (g e G).

Proof: This follows from the theorem on fibre maps (Hu ( 1], Theorem 2•16.2), if we take into account that G is the universal

98

MINIMAL

covering space of Xo-1 2•11.37 LEMMA: Let¢ and~ be homeomorphisms of Xo onto itself such that N¢ = N~ = N. If the induced homomorphisms ~*:N + N and ~*:N + N coincide, then there exists a bounded map !o:G + G such that gqi = (gl/))(g!o) for aU g e G, where the maps qi and 1/J are chosen according to Lerrma 2•11.36 with eqi = el/)= e.

Proof: If k eN, then N = (Nk)¢ = N(kqi), hence kqi eN. Similarly, kl/)eN for all keN. Let keN. Consider the map -:G + G with the properties stated above. Let g EG and t EJR. There exists a number, EJR with ((Ng)l\t)F ((Ng)F)IT'. Then

MINIMAL

100 N((g(t[f!o))f) =

N(gf)(T[f!Q).

Thus, for each t eJR there is an element kt e N with kt = (gf) (T[fJQ)((g(t[fJo))f)- 1 . By the continuity of the right side of this equality, kt is a continuous function defined on the real line and taking its values in the discreet group N. Therefore kt is a constant. Since T = 0 whenever t = O, we get kt= ko = e for all teJR. Thus (g(t[f! 0 ))f = (gf)(T[fJQ), Since gf = (g[f!)(g>-) (ge G), we have (11.9)

Recall that [fl is an automorphism of G, i.e., [f! has the form (11.7) and thus (11.8) is satisfied. Let g = (x,y,z) and gA = (x>-,YA,zA). Since >-:G + G is bounded, the functions xA,YA,zA are also bounded. Comparing the co-ordinates of the left and of the right sides of (11.9), we get three equalities of which we need only the following two:

After reduction we get

(11 ,9')

which holds for all t e:JR, Therefore, dividing the first equality by the second and taking the limit as t + we get 00 ,

i.e.,

(11.10)

Since a is a transcendental number, equality (11.10) holds only when m1 = n1 = O, m2 = n1, Recalling the hypothesis (11.8), we find that = - 1.

Consequently, either 1,

or

(11.11)

TRANSFORMATION GROUPS

101

(11.12) Now we shall show that the homeomorphism F:X 0 + Xo can always be chosen in such a way that the condition (11.11) is satisfied. Indeed, define a map wo:G + G by

(x,y,z)wo

=

(-x,-y,z),

(x,y,z elR).

It is easy to verify that wo is an automorphism of the group G. Since Nwo = N, wo induces a homeomorphism '¥ 0 :X 0 + Xo, One can show directly that (g

e G, t elR) .

Hence, '¥0 is orbit-preserving. The homeomorphism'¥= '¥ooF also preserves the orbits and it carries N to Ngo, It is clear that W = woof satisfies the condition (11.11). Let S 1 =~and let (:iZ + h,t)p = :;z + h + t. Then (S 1,R,p) is a periodic flow of period 1. Define a continuous map ~0 :Xo + s 1 by (N(x,y,z))~o =:iZ + y (x ,y ,z elR) . It is easy to verify that ~O is a homomorphism of (Xo,lR,n) onto (S 1 ,R,p). The map ~O induces a homomorphism of the fundamental groups:

where

(m,n,p)~o,,,

=

n,

Cm ,n ,p e:iZ) .

Let Ng 0 be an arbitrary point of Xo with the property that g 0 = (x,y,z) satisfies the condition y e:iZ. By the choice of the homeomorphism'¥, we have

where Pl and P2 are some integers. Therefore ('¥0~0),., = ((!lo),., and it follows from the proof of Theorem 2•11.32 that the flow (Xo,lR,n) is almost periodic. But this is impossible because the fundamental group n1(Xo,N) = N is not commutative. I 2•11.41 REMARK: It is not hard to verify that (11.9') implies that either,= tor,= -t. In the second case we replace the homeomorphism F by '¥ooF and we conclude that (Xo,lR,n) is not orbitally homogeneous, because it is not dynamically homogeneous. Thus we obtain a new proof of Theorem 2•11.40 which does not depend on Theorem 2•11.32. 2•11.42 We give an example of an orbitally homogeneous, but not harmonizable minimal transformation group with 'two-dimensional time' .

102

MINIMAL

Let x0 be the above space GIN and let T be the Cartesian product JRXlR. Define a transformation group (Xo,T,o) as follows. If (t,1) eT and g = {x,y,z), we set (Ng)o(t, 1 ) = N(x + at,y + St,z + xSt + yt + !aSt 2 + 1). In other words, (Ng)o(t, 1 )

=

(Ng)(t~o)(o,0,1).

If the numbers a and Sare rationally independent, then the transformation group (Xo,T,o) is minimal. Since Tis commutative, whereas the group IT1(Xo,N) = N is not commutative, we conclude that (X 0 ,T,o) is not harmonizable. Let go be an arbitrary element of G. Define a map ¢:Xo + Xo by (Ng)¢ = Nggo Cg E G). Clearly ¢ is a homeomorphism which maps N onto Ngo. It remains to verify that¢ maps orbits of (Xo,T,o) into orbits. Let go= (xo,Yo,zo) and g = (x,y,z). Then (Ng(t~o)(o,0,1))¢ N(x,y,z)(t~o)(o,o,1)(xo,yo,zo) N(x,y,z)(x 0 ,y 0 ,z 0 )(t~ 0 )(o,o,1'), where 1'

1 + Yoat - xoSt.

Hence

2•11.43 The following interesting question arises in the study of minimal transformation groups: which topological properties characterise a space X which is a phase space of some minimal transformation group (X,T,n)? This problem is far from being solved even if T = lR or T = ~. In the next subsections we present some partial results obtained in this area. 2•11.44 Let Af[fl be a c""-smooth closed (i.e., without boundary) connected compact manifold and let 3 1 = {E; IE; Ea:, IE; I = 1}. We shall show that there exists a diffeomorphism F of the manifold Mm x 3 1 onto its elf such that the cascade (Mill x 3 1 ,F) generated by Fis minimal. Provide Mill with a Riemannian metric of class C00 • Let Difr (Mill x 3 1 ) denote the set of all C00 -diffeomorphisms of the manifold Mill x 31 onto itself. For each positive integer k let Pk(F1,F2) denote a metric in Difr(Mill x 31) which measures the proximity of the k-jets not only of F1 and F2 but also of F1- 1 and F2- 1 . Define a periodic flow (Mill x 3 1 ,JR,o) as follows: (x,E;,t)o

= (x,E;)ot

= (x,c;e2nit),

Cx

EMm,c; E 3 1 ,t GlR).

TRANSFORMATION GROUPS

103

Let q be any positive integer. notations: 8q = {e2nitl 0

We introduce the following

1/q}'

8,

{e211itlo

liq =MIDxiSq,

LI' q

MIDxc'q,

~

t

llk,q = llqok/q,



o, ... ,hn -

n

1), the closure of each of which is diffeomorphic to the disk Dm. Write

104

MINIMAL

(r

=

0,1, ... ,hn - 1;,Q,

=

0,1, ... ,kn - 1).

Observe that G,Q, , r C Gr x oNth n• h nk n q n C fJ.o' 'n q . Clearly, the set G,Q,,r is diffeomorphic to ][)m+l and G,Q,,r()Gm,s

= ¢

+

whenever either ,Q, t m or r s. We choose hn open sets Ei,r (r = 0,1, ... ,hn 1) inside of each set Ei in such a way that (a): Ei r()E,Q,

'

'

8

= ¢,

+ s;

whenever r

(S): E,Q,,r is diffeomorphic to nm+l (r

=

0,1, . .. ,hn - 1).

We shall now use the following proposition, which follows from results obtained by Palais [1] (see: Anosov and Katok [1]). THEOREM: Let om be a connected open smooth manifold of class C~. Let Ei and Gt (t = l, .•. ,n) be two families of open subsets of om whose closures are C -diffeomorphic to them-dimensional ball nm, and moreover: 00

(r

f

s).

Then there exists a C -diffeomorphism S:om + om, which agrees with the identity map outside some compact set N1 Com and which maps E,Q, onto Gi (,Q, = l, .•. ,n). 00

We apply this theorem to the manifold tJ.o,qn and to the families of subsets Ei,r and G,Q, r (t = 0,1, ... ,kn - l;r = 0,1, ... ,hn - 1). Since the map Sis the'identity outside some compact N1 C tJ.6,qn' it can be extended to tJ. 0 q , and then to Mm x sl as follows: ' n

(x e D.r q ,r = 1, ... ,qn - 1) . ' n

Such an extension ensures that Sand oan commute. An+l·

Denote S by

Since Fn+l tends to Fn as Sn+ o, condition (2n) will be satisfied if ,Q,n is chosen to be a large enough multiple of hn. It remains to verify condition (3n). First we observe that

and therefore

TRANSFORMATION GROUPS

105

B -1 Fn+l sqn = BnoAn+100 s6nqn oAn+l -1 on

(s = 1,2, ... ).

Let (x,y) eMmxsl and (y,i;) = (x,y)BnAn+l· There exists a number r such that y e Gr. If .Q,n > 3hn, then for each .Q, (Q, = O,l, .. ,,kn - 1) there is a number s(.Q,) with (y,i;)o 8 (.Q,)SnqneG.Q, , r· Hence, (y,i;)o 8 (£)SnqnAn+l-l eE.Q,,r C

E.Q,,

Therefore the finite set

r -= {( x,y )BAn n+1° s(.Q,)SnqnA n+l -1[ .Q, satisfies the condition S(f,Yn+l)

~

to q . ' n

0 , 1, ... , kn - 1}

It follows by the

choice of Yn+l that (3n) holds. By (2n) the limit F = n-+-oo lim Fn exists and Fe Diff"'(Mm x s 1 ). Conditions (2n) and (3n) imply that (Mm x s 1 ,F) is a minimal cascade. 2•11.45 A slight modification of the above construction enables us to construct a diffeomorphism F such that (Mm x s 1 ,F) is not only minimal but also weakly mixing (i.e., the direct product .(Mm x s 1 ,F) x (Mm x s 1 ,F) has a dense orbit). It can also be shown that if Mill is a smooth compact connected manifold without boundary and the group sl acts on Mm in a smooth way and without fixed points (i.e., Mm is the total space of a principal smooth fibre bundle with sl as the structure group), then a minimal cascade with phase space Mm can be defined. In particular, there exists a minimal cascade on each sphere s2n+l (Katok [1]). One unresolved conjecture is that it is impossible to define a minimal flow on the three-dimensional sphere. (See the survey paper by Smale [1]). 2•11.46 Let s 1 be the one-dimensional sphere {z[z eCC,[zl = 1}, I the segment [o, 1] and W = s 1 x I. Identifying every two points of Wof the form (z,O) and (z- 1 ,1) (zeS 1 ), we obtain a non-orientable surface which is called the Klein bottle (Hu Szetsen [1]). It will be shown in the Subsection 3•18.16 that the Klein bottle supports a distal minimal cascade, and the Cartesian product of the Klein bottle and sl is minimal under some distal flow. 2•11.47 We present (without proof) some more results concerning topological properties of minimal sets. (a): Jones (see Gottschalk and Hedlund [2], p. 151) constructed a minimal cascade whose phase space is a onedimensional planar continuum that is locally connected at some but not all points. (b): Hedlund [1] has shown that for each positive integer p ~ 2 the space X of all unit tangent vectors of a com-

MINIMAL TRANSFORMATION GROUPS

106

pact orientable surface of genus pis a minimal set under the horocycle flow (X,JR,n). Moreover, P(X)

= Q(X) = XxX.

(c): Chu [3] has proved that neither the universal curve of Sierpinski nor the universal curve of Menger (Aleksandrov [1]) can be the phase space for a minimal flow. Gottschalk [5] has shown that there does not exist a minimal cascade on the universal curve of Sierpinski. However, the universal curve of Menger admits a minimal cascade (Anderson [ 1]) .

REMARKS AND BIBLIOGRAPHICAL NOTES Theorems 2•11.1,3,5 are taken from the book by Gottschalk and Hedlund [2]. Theorem 2•11.3 is a generalization of a theorem due to Hilmy [1]. Theorem 2•11.5 was proved by Markov [1] in the case T = JR. An interesting generalization of Theorem 2·11.5 was found by Chu [3]. Example 2•11.6 is due to Floyd [1] and J. Auslander [1]. The concept of a homogeneous minimal set was introduced by Birkhoff [2]. In that paper (under the numbers 2 and 3), the following problem is posed: to prove that each minimal set is homogeneous in the sense that 'the stream lines are topologically indistinguishable from one another'. But it is clear that by embedding the cascade of Example 2•11.6 in a flow we get a topologically non-homogeneous minimal flow. Nemytskii [2] posed the question, whether or not a minimal flow on a manifold is homogeneous in the sense of Birkhoff. Example 2•11.40, constructed by Bronstein and Kholodenko [1], answers negatively this question. Theorem 2•11.10 is due to Gottschalk [2]. Related results were obtained by Zidkov [1] and Grabar' [1]. The propositions 2•11.1323 follow from results of J. Auslander [4], J. Auslander and Hahn [1], L. Auslander and Hahn [1], Bronstein [1]. The notion of harmonizability was introduced by Markov [3]. Lemma 2•11.23 was proved by Ellis [9] (see also Eisenberg [2]). The material in Subsections 2•11.24-26 is taken from the paper by Kahn and Knapp [1]. The results in Subsections 2·11.28-30 are due to Ellis [8]. The fundamental group and the first cohomology group of a minimal set were considered by Chu and Geraghty [1]. The material presented in Subsections 2•11.23-43 is taken from the paper by Bronstein and Kholodenko [1]. Example 2•11.44 is a particular case of a more general construction due to Anosov and Katok [1].

CHAPTER III

EXTENSIONS OF MINIMAL TRANSFORMATION GROUPS

§(3•12): THE BASIC THEORY OF EXTENSIONS We shall introduce in this section the notion of an extension of a transformation group and we shall study some basic properties of extensions. We shall consider relations between the most important classes of extensions and we shall also investigate the problem of the openness of homomorphisms of minimal sets.

3•12.1

Let T be a fixed topological group, X and Y topological spaces, (X,T,n) and (Y,T,p) transformation groups and~: X +Ya homomorphism of (X,T,n) onto (Y,T,p). In this case we say that (X,T,n) is an EXTENSION OF (Y,T,p) BY THE HOMOMORPHISM ~ and that (Y,T,p) is a FACTOR OF THE TRANSFORMATION GROUP (X,T, n). Often the mapping ~;(X,T,n) + (Y,T,p) itself is called an extension. For each point y e Y the subspace Xy = {x / x e X ,x~ = y} is called the FIBER OF THE EXTENSION~ OVER y. If each fiber Xy (ysY) consists of one point (i.e.,~ is injective and therefore a homeomorphism, whenever Xis compact), then the extension~ is called a TRIVIAL EXTENSION. If the extension is not trivial, then it is called a PROPER EXTENSION. An extension ~:(X,T) + (Y,T) is said to be a MINIMAL EXTENSION whenever (X,T) is a minimal transformation group.

3•12.2

Let ~i:(Xi,T) + (Yi,T) (i = 1,2) be extensions. A pair of homomorphisms ~:(X1,T) + (X2,T), A:(Y1,T) + (Y2,T) is called a MORPHISM OF THE EXTENSION ~1:(X1,T) + (Y1,T) INTO ~2: (X2,T) + (Y2,T) if ~loA = ~0~2, i.e., the diagram

107

108

EXTENSIONS OF MINIMAL

is commutative. Let cp 1 :(X1 ,T) + (Y,T) and cp2:CX2,T) + (Y,T) be extensions of the same transformation group (Y,T). A homomorphism t:(X1,T) + (X2,T) is called a MORPHISM OF THE EXTENSION ..

X

>..) =

(Contd)

EXTENSIONS OF MINIMAL

152

(Contd)

(mA ,m )T( A x A) U (mB ,m )T( A x A);

therefore g e cls 0 A U clsoB.

3•15.13 LEMMA: If¢

f

I AC G, then:

cls 0 A = {glgeG,(\fmeM)(\fae'U[X])(3 heA):(mg;,.,mh;,.)eaT}. Proof: Let m eM. If g eels A, then (mg;,.,mh;,.) e (mA,m)T(;,. x A) Hence there exists a pair (m1,m2) e (mA,m)T with m1A = mg;,., m2A = mA. Since the set (mA,m)T is closed and invariant, we may assume that m2 = m and (m1,m) is an almost periodic point of the transformation group (M x M ,T). Hence there exists an element k e G with m1 = mk. Thus (mg;,. ,m;,.) = (mkA ,m;,.) and (mk ,m) e (mA ,m )T. There exist nets {tn}, tneT and {hn}, hneA, such that mtn + m, mhntn + mk. Thus mktn = mtnk + mk and hence mgtnA = mgAtn = mkAtn = mktnA + mk;,. = mg;,., mhntnA + mk;,. = mg;,., Therefore for any index a eU[X] there exists a t with (mgtnA,mhntnA) ea. Consequently (mg;,. ,mhnA) e aT. Conversely, let g e G be an element such that for every me M and every a e U[X] there is an element he A satisfying the condition (mg;,. ,mh;,.) e aT. By the minimality of the transformation group (M,T) and the uniform integral continuity (Lemma 1•1.5) we can find nets { tn}, tn e T, and { hn}, hn e A, such that mgtnA + mg;,., mhntnA + mg;,., mtn + m. Then mtnA + mA and therefore (mg;,., m;,.) e (mA,m)T(A x A). Thus g eels A, I

3•15.14 LEMMA: Let mo,m1eM be such that (moA,mlA) eP Then the family {Vci(mo,m1)iae'U[X]}, where

P(X,T).

I

Va (mo ,m1) = {g g e G, (mogA ,mo A) e aT, (mogA ,m1A) e aT}, is a base of a-neighbourhoods of the identity of G. Proof: Let a e 'U[ X] be an index. Denote Va = Va (mo ,mo). Then Va(mo ,m1) C Va, Choose i3 e 'U[X] such that 132 C a, i3 = 13-l, We shall show that V13 C Va(mo,m1), Let g e Va, i.e., (mogA,moA) e BT. Since (mogA,moA) is an almost periodic point of the transformation group (Xx X ,T), there exist subsets A C T, K C T such that K is compact, AK= T and (mogA,moA)A C 13. Hence for any t e T we can choose an element kteK with (mog;,.,mo;,.)tkt-lei3. As Xis compact and (moA,mlA) eP(X,T), we have (moA,m1;,.)toK-l C i3 for sonie to e T. Then (mogA ,m1A )tokt -1 e 132 C a, and hence (mogA ,ml A) e aT. Therefore geVaCmo,m1). Now we shall show that if a is open in Xx X, then the set Va is o-open. Let he cls 0 (G,Va), that is:

If g e c,va, then (mog;,.,mo;,.) e Xx x,aT; hence (moh>..,mo>..) e Xx X'aT, and consequently he c-----..va. Thus cls 0 (G'Va) = G'--Va.

TRANSFORMATION GROUPS

153

Finally, let Ube any open a-neighbourhood of the identity e e G. The set F G'----U is a-closed. We shall prove that

=

(moF ,mo )T(}..

x }.. )

11 a: 0 = ¢,

for some a:o e 'U[X]. Suppose the contrary holds. Then for each a: e U[X] there exist elements ta: e T and ga: e F such that (moga:A, mo}..)ta: ea:. Without loss of generality we may suppose that the net {mota:} converges to some point n e M. Then {mota:A} -+ n}.. and {moga:ta:A} -+ n}.. = ne}... This means that e e clsaF = F, contradicting the fact e e U = G'---F. Thus there exists an index a:o e U:[X] with (moF,mo)(}.. x }..)na:oT = ¢, and hence Va: 0 Cmo,m1) C Va: 0 CU, I 3•15.15 LEMMA: The map Lh:(G,a)-+ (G,a) defined by gLh = hg ( g e G) is continuous for every h e G.

Proof: Let¢= AC G, clsaA = A and heG. It will suffice to show that clsa(hA) = hA. Let g e clsa(hA) and me M, By Lemma 3• 15 .13, for the point mh- 1 e M and the index a: e 'U[X] we can find an element a eA such that (mh- 1g}..,mh- 1ha'A) e a:T, i.e., (mh- 1g}.., ma'A)ea:T. By Lemma3•15.13,h- 1geclsaA = A. Hence gehA,I 3•15.16 Let us introduce another topology on the group G. If ¢ = A C G, then we define (taking Lemma 3 • 15. 10 into account): els A

{g [g e G, (mg ,m) e (mA ,m )T ,me M} {g I g e G, 3 mo e M: (m 0g ,mo) e (moA ,mo )T} {g[g e G, 3mo eM:(mog,mo) e l\(M)(A

x

e)} c clsaA,

If A=¢, then we set els A=¢. 3•15.17 LEMMA: The map A~ cls 0 A (AC G) is a closure operator for some topology , on G with , :::> o. If¢ AC G, then

+

cls 0 A = {g[geG,(\fmeM)(\fa:e'U[M])( 3heA):(mg,m)ea:T}.

Let, further, mo,m1eMand (mo,m1)eP(M,T). tem { Wa:(mo ,m1) [ a: e 'U[M]}, where

(15.4)

Then the sys-

Wa:Cmo,m1) = {g[g e G,(mog,mo) e a:T,(mog,m1) e a:T},

is a base for the ,-neighbourhoods of the identity e e G. Moreover, for each heG the map Lh:(G,,)-+ (G,,) is continuous. Proof: The operator A~ cls,A is a particular case of the operator A~ clsaA, when X =Mand 'A:M-+ Mis the identity map. Hence this lemma is a corollary of Lemmas 3•15.12-15. I

EXTENSIONS OF MINIMAL

154

3•15.18 LEMMA: (G,,) is compact and is a T1 space. Proof: Let {Aa} be a filter of ,-closed subsets Aa CG and let mo e M. Then {moAa} is a filter of closed subsets of the compact space M; therefore nmoAa ¢. Let m1 e nmoAa; then

t

(m1 ,mo )T c n (moAa ,mo )T = n (moAa ,mo )T. a a Let (m 2 ,mo) be an almost periodic point of the transformation group (M x M ,T) belonging to the set (m1,mo)T CRµ. In view of the regularity of the extension µ:M + Y, there exists an element go e G with m2 mogo. Then (mogo ,mo) e

na (moAa ,mo )T;

hence goe ncls,Aa = nAa. Thus the space (G,,) is compact. Let A = {h} and m eM. If (mg,m) e (mh,m)T, then there exists a net { tn}, tn e T such that mtn + m and mhtn + mg. But mhtn = mtnh + mh, hence mh = mg (meM), i.e., h = g. Thus clsa{h} {h} (h e G).

I

ll(M)(e xA- 1 ) (¢

3 • 15 .19 LEMMA: l\(M)(A x e) Proof: Let meM, geA and n

mg.

t

AC G).

Then (mg,m) = (n,ng-1) e

l\(M)(e xA-1), and hence ll(M)(A x e) C t(M)(e xA-1). the reverse inclusion is similar. I

The proof of

3•15.20 LEMMA: The maps Rh:G + G (he G), where gRh = gh (g e G), and the map g 1+ g-1 (g e G) are continuous in the ,-topology on G.

(Hence they are ,-homeomorphisms).

Proof: Let cls,A = A and he G. Let us prove that clsT(Ah) = Ah. Let g eels (Ah), i.e., (mg,m) e (mAh,m)T. Then (mgh- 1 ,m) = (mg,m)o

(h- 1 ,e) e (mAh,m)(h- 1 ,e)T = (mA,m)T. Thus, gh-1 e clsTA = A; hence It follows that the map Rh:(G,T) + (G,T) is continuous. Let clsTA = A. We shall prove that clsT(A-1) = A- 1 . If g e

geAh.

clsT(l).-1), then (mg,m) e ll(M)(A-1 x e) = l\(M)(e xA) by Lemma 3•15. 19. Put n = mg; then (n,ng-1) e /:,(M)(e xA), and hence (ng-1,n) e i:,(M)(A x e). Thus g-1 e clsTA = A, and therefore g eA-i. I 3•15.21 Consider the commutative diagram

M~A

Xl

z µ

cp

~y

where µ:(M,T) + (Y,T) is a regular extension. Let G(Z) denote the set of all automorphisms g:(M,T) + (M,T) which preserve the

TRANSFORMATION GROUPS

155

fibers of the homomorphism w:M

G(Z)

+

{gfgeG,mgw =

Z. =

In other words, mw,meM}

{gfgeG, 3moeM:mogw = mow}.

3•15.22 LEMMA: G(Z) is a T-closed subgroup of the group

G

=

G(Y) and, moreover, G(X)

C

G(Z).

Proof: Clearly G(Z) is a subgroup of G and G(Z) :::) G(X). Let heclsTG(Z). Then (mh,m)e(mG(Z),m)T CRw (meM), and hence mhw = mw. Thus he G(Z) by the definition of G(Z). I 3•15.23 Let K be a T-closed subgroup of G satisfying the condition K J G(X). The question arises whether there does exist a transformation group (Z,T) satisfying the commutative diagram (15.5) and the condition G(Z) = K. In general, the answer seems to be in the negative. We shall discuss this question under the additional hypothesis that~ is distal. 3•15.24 LEMMA: Let ~:(X,T)

+ (Y,T) be a distal extension and let K be a T-closed subgroup of G = G(X,Y) such that K J G(X). Then

S(K) = {(mg,m)[meM(X,Y),geK}, is a closed invariant equivalence relation and, moreover, R(>-1) C S(K) C R(µi). Proof: Clearly S(K) is an invariant equivalence relation. Let (m1 ,m2) e SUIT = !:,,(M) (K x e). Since S(K) C R( µ1), there exists an automorphism he G with m1 = m2h (Lemma 3•15.6). Thus (m2h,m2) e Li(M)(Kxe), hence heclsTK = K. This means that (m1,m2) = (m2h, m2) e S(K). Thus the relation S(K) is closed. It follows from K J G(X) and Lemma 3•15.6 that S(K) JR(µ1). I Observe that M(X,Y)/S(K) coincides with the quotient space M(X,Y)/K of K-orbits {mK[meM(X,T)}. 3•15.25 THEOREM: Let ~:(X,T) + (Y,T) be a distal minimal extension, let M = M(X,Y) and G = G(X,Y). Let [X,Y] denote the set of all transformation groups (Z,T) satisfying a commutative diagram of the form (15.3). Let H = G(X) and [H,G] = {KIK is a T-closed subgroup of G,K J H}. Define a map¢: [X,Y] + [H,G] by Z¢ = G(Z) and a map f:[H,G] + [X,Y] by Kf = (M(X,Y)/K,T). Then ¢of is the identity transformation of [X,Y] and fa¢ is the identity transformation of [H,G]. Thus ¢, f establish a Galois connection between [X ,Y] and [H,G].

Proof: Let (Z,T) e [X,Y] and K = G(Z). By Lemma 3•15.22, Ke _ [H,G]. Lemma 3•15.23 asserts that M(X,Y)/K = Z1e [X,Y]. Consider the commutative diagram (15.5), where M= M(X,Y). Since~ is a distal extension, µ :M + Y is a homogeneous extension (Lemma 3• 15.6). Therefore if m1w = m2w, then there exists an automorphism

156

EXTENSIONS OF MINIMAL

g eA(M,T) such that m1 = m2g. Then g e G(Z), according to the definition of the group G(Z). Thus Rw = {(.mg,m)[meM,geK = G(Z)}; hence Z = M(X,Y)/Rw = M(X,Y)/K = Z1. This means that ~o~: [X,Y] + [X,Y] is the identity map. Let Ke [H,G]. By Lemma3•15.24,S(K) = {(mg,m)[meM(X,Y),geK} is a closed invariant equivalence relation and M(X,Y)/S(K) = M(X,Y)/K. Hence it follows that G(M(X,Y)/K) J K. On the other hand G(M(X ,Y)/K) C cls,K = K. Thus G(M(X ,Y)/K) = K. This proves that ~o~ is the identity map of [H,G]. I 3•15.26 In the remainder of this section we shall introduce and study a certain subgroup H(G) of the group G, and then apply the results obtained in the investigation of extensions ~:(X,T) + (Y,T) satisfying the condition Q(R~5) = R~9. Finally, we shall prove a structure theorem which generalizes Theorem 3• 14. 36 on the structure of almost distal extensions. The next lemma is a generalization of Lemma 1•4.11 to the case of T1 spaces. 3•15.27 LEMMA: Let Ebe a compact T1 space provided with a semigroup structure in such a way that for each x e E, the map Rx:E + E, defined by yRx = yx, is continuous and closed. Then there exists an idempotent u e E.

Proof: Let A be the system of all non-empty closed subsets S CE with s2 C S. Note that Ee A. Order A by inclusion. It is easy to verify that Zorn's Lemma is applicable to the system A, and therefore A contains a minimal element Me.A. Let u e M; then Mu C MRu is a non-empty closed subset of M. Since MuMu C MMMu C Mu, we have Mu eA. Hence Mu = M, by the minimality of M. Thus there exists an element x e M with u = xu. The set W = {y[yeM,yu = u} = uRu-l is non-empty and closed in M. Moreover, w2 C W. Thus W = M, according to the definition of M. Hence u 2 = uu = u,I 3•15.28 COROLLARY: Let G be a compact Ti space provided with a group structure in such a way that all the maps Rx:G + G (x e G) are continuous. If M is a non-empty closed subset of G and M2 C M, then M is a subgroup of G.

P1°oof: Since G is a group, the maps Rx (x e G) are homeomorphisms, hence they are closed. Let x e M. The set E = Mx satisfies all the hypotheses of Lemma 3•15.27. Therefore Mx contains an idempotent. But the only idempotent in the group G is the identity e. Hence e e Mx and therefore x-1 e M. Thus M is a subgroup of the group G. I 3•15.29 LEMMA: Let G be a group provided with two topologies

, and (1) :

0, 0

satisfying the conditions: C ,;

(2): (G,-r) is a compact T1 space;

157

TRANSFORMATION GROUPS

(3): The maps Ra:(G,,)-+ (G,,) and La:(G,,)-+ (G,,) are

aontinuous for aU a

e G;

(4): The map La: (G,o) -+ (G,o) is aontinuous for every a e G.

Let No and N, denote neighbourhood bases at the identity e e G in the topologies o and , respeatively, and let H =

n [clsoVI VeNo],

Then Lis a subgroup of G and H

=

L.

Proof: Let x ,Y e L. We shall prove that xy e L. Let N e No, and R eN,. Then x e cls,N. By (3), Rx is a ,-neighbourhood of x; hence RxnN + ¢. Thus rxeN for some element reR. In view of (4) we can choose S eN0 so that rxS C N. Hence, by (3), rx(cls,S) C cls,N;

+

therefore rxy e cls,N. Thus R::cy n cls,N ¢. But R::cy is a ,-neighbourhood of xy by (3). Therefore Rxy n N + ¢. Since R eN, and N e N0 are arbitrary, we get xy e L. By (2) and (3), the group G and the set M = L satisfy all the hypotheses of Corollary 3•15. 28; hence Lis a subgroup of G. Now we shall prove that L = H. Since o C ,, we have cls,NC cls 0 N (Ne.N'0 ); therefore LC H. Let hell and N,ReJ,/0 • By hypothesis ( 4) , hN is a a-neighbourhood of h; hence hN n R ¢. Consequently h(cls,N) n cls,R ¢ by condition (3). Hence it follows that

+

+

{h(cls,N)

n cls,RIR eNo}

is a filter of ,-closed subsets. Taking condition (2) into account, we conclude that h(cls,N) ()L ¢. Since N is here an arbitrary a-neighbourhood of the identity e e G, we conclude from conditions (2) and (3) that hL()L ¢. Since L is a subgroup of G, we get he L, and hence H C L. Therefore H = L. I

+

+

3•15.30 Let ;\

M ----+ X

~l·

(15.6)

y

be a commutative regular diagram, let G be the group of all automorphisms g: (M ,T) -+ (M ,T) preserving the fibers of the map µ :M -+ Y and let o and, be the topologies on G defined in Subsections 3•15,11,12 and Subsections 3•15.16,17 respectively. Let xoeX, YO= xoqi. The set {mlmeM,mµ = Yo} is compact and invariant under the action of the group G on M; hence it contains a G-minimal set Mo, If mo is a specified point of Mo, then moG = Mo.

EXTENSIONS OF MINIMAL

158

3•15.31 LEMMA: Let geG and let V be a neighbourhood of the point mog in M. Denote {h[heG,mohe V} by V(G). Then int,cls,V(G)

+ ¢.

Proof: Since moG is a G-minimal set, moG C VG. There exists a finite family F CG with moG C VF. Therefore G = V(G)F and, moreover, G = cls,V(G)•F. Sets of the form cls,V(G)•,Q, (,Q,eF) are ,-closed and mutually ,-homeomorphic by Lemma 3·15.19. This and the equality G = cls,V(G)•F imply the required assertion because F is finite. I

3•15.32 LEMMA: Let ECG and let cls,int,E = G. moG,

Then moE =

Proof: Let g e G and 1 et V be an arbitrary neighbourhood of mog in M. By Lemma 3•15.31,

Thus

+

+

and, moreover, cls,V(G) n int,E ¢. Hence V(G) n int,E ¢, and therefore V(G) nE ¢. It follows from the very definition of V(G) that moEnE ¢. Since V is an arbitrary neighbourhood of mog, we have mog e moE. Thus moG C moE C moG, whence moE = moG. I

+ +

3•15.33 LEMMA: Leto and, be the topologies on G defined in Subsections 3•15,11,12 and Subsections 3•15,17. Define Hand Las in Lemma 3•15.29 and suppose that H = G and Ve N0 • Then moV = moG, Proof: By Lemma 3 · 15. 29, L = H = G. Denote int, V by E. Since Ee int 0 V C int, V = E, E is a ,-neighbourhood of the identity e e G, and since L = G, we get G = cls,E = cls,int,V. By Lemma 3·15.32, inoE = moG. Thus moV = moG. I

3•15.34 LEMMA: Let H = G and suppose that there exist indices a:i e 'U[ X] ( i = 1, 2, . . . ) such that

If moG is a minimal set of the transformation group (M,G) and (mo A ,m11-) e P n Rq>, then the set {m[memoG,(mA,moA) eP11Rq>,(m1-,m11-) ePnRq>}, is residual in moG, Proof: Put Vi = {g [ g e G, (mogA ,mo A) e a:iT, (mogA ,mo A) e a:iT}.

TRANSFORMATION GROUPS

159

By Lemma3·15.14,VieJ.J'0 • Thus moVi = m G (i = 1,2, ... ) (Lemma 3•15.33). It is clear that

Set

Without loss of generality we may suppose that the ai are open in XxX. Then each set Ni is open in moG, Since moViC Ni, we have Ni= moG. Thus the sets Ni (i = 1,2, ... ) are open and dense in moG. Therefore the set E = n [Ni = 1, 2,.. . ] is residual in moG. If m e E, then

Ii

(m\,mj\) e

n

aiTnRCJl = PnRCJl,

(j

1, 2) .

i=l Note that mo\CJl

= m1\CJl = m\CJ)

whenever me EC moG,

I

REMARK: If the space Xis metrizable, then the requirement imposed on PnRCJl in the preceding theorem is satisfied. 3·15.35 LEMMA: RCJl5

= {(mogt\,mot\)lteT,geG}

(moeM).

Proof: Lt:;t (x,y) e RCJl:1. Since the diagram (15. 6) is regular, there exist me M and g 0 e G such that m\ = y and mg 0 \ = x. Therefore (x,y) e (mg 0 ,m)T(\ x \) = (m 0 g 0 ,m 0 )T(\ x \)

C { (mogt\ ,mot\)

It e T ,g e G}.

e T ,g e G},

(mo e M).

Thus RCJlS C { (mogt\ ,m 0 t\) t

The reverse inclusion follows from the definition of G.

I

3•15.36 LEMMA: I f Q(RCJlS) = RCJlS~ then H = L = G. H.

Proof: In view of Lemma 3•15.29, it suffices to show that G C Let go e G and mo e M. Then we have by the preceding lemma,

It e T ,g e G}) Q( { Cmogt\ ,m 0 t\) It e T ,g e G}) n {(mogt\ ,mot/\) It e T ,g e G} n aT.

(mogo ,mo) e RCJl~ C Q(RCJlS) = Q( { (mogt\ ,mot\)

=

aeU[X]

It follows from Lemma 3•15.14 that the family {Valae'U[X]}, where

EXTENSIONS OF MINIMAL

160

Vix = {g \ g e G, (mogA ,moA) e aT} is a base for the a-neighbourhoods of the identity e e G. The equalities (15. 7) imply that go e {g\g e G,(mogA,moA) e (moVa,mo)T(A x A)} for each index ae'U[X]. G CH,

I

= cls

Hence goe il[cls 0 Va,\aeU[X]]

0

Va,

= H.

Thus

3•15,37 Let I be a minimal right ideal of the semigroup E(X,T), let xo e X, and let "J,.:(I,T) + (X,T) be the homomorphism defined by /;A = x 0 i; (!; e I) and µ = "J,.ocp, As was shown in Subsection 3•15,3, the diagram I - - -A+ X

~l· y

is regular, In the sequel, when using 15.33-36 we shall be assuming that M = The set Io = {I;\ I; e I ,xoi;cp = xoi;} is the semigroup I; hence there exists an Go= Iou is a subgroup of the group Iu

the results of Lemmas 3•

I. a closed subsemigroup of idempotent u e I 0 . Thus with identity u.

3•15.38 LEMMA: The group G of all automorphisms g:(I,T) + (I,T) that preserve the fibers of µ:I+ Y coincides with the group {Ls:I + I\s e Go}. F'roof: Clearly, LseG (seGo), If l;,neio and (1;,n) is an almost periodic point of the transformation group (Ix I ,T), then there exists an idempotent v e Io with l;v = I;, nv = n. Set q (l;v)(nv)-1, where (nv)-1 denotes the inverse element in the group Iov with identity v. Sets= qu. Then I;= qn = qun = sn in accordance with Lemma 1•4.2, and, moreover, s eiovu = Iou = G. The map Ls is an autormorphism sending n to ~.I 3 • 15. 39 LEMMA: If p e Go and p = pw for some idemf!otent w e I, then p e Gow,

F'roof: If p e Go C Io, then p = pw e I 0w = I 0uw = G0w, because uw = w by Lemma 1•4,2.I

3•15.40 LEMMA: There exists an idempotent we Go such that Gow = wG is a minimal set under the action of G = {Ls\ s e Go} on I. F'roof: The set Go is closed in I and invariant under the action of the group G. Hence there exists a point p e Go such that Gap= pG is a minimal set of the transformation group (I,G). Choose an idempotent we I so that pw = p. By the ~recedinf ;lemma there is an element g 0 e Go with p = gow. Then go- p = go- gow = uw = w, and hence we go- 1 Go = go-lGo = Go.

Since Gap

= pG

is a

TRANSFORMATION GROUPS

161

minimal set and w = go- 1p e Gop, we get Gow minimal set of (I,G).I

= Gop.

Thus Gow is a

3•15.41 LEMMA: Let X be a compact space, (X,T) a minimal transformation group, RC Xx X a both cfosed and open invariant equivalence relation and let qi:X + X/R = Y denote the canonical projection. Suppose that (1); R = Ifj; (2): Q•'•(R) = R; (3):

30.i eU[X] :PnR

= (\

[aiTnRli

= 1,2,.,.

];

and suppose that at least one of the following conditions is satisfied: (4): There is a point aeX such that the set {xlxeX,(x,a) e

P()R} is finite or countable;

(5): P(\R is an equivalence relation.

Then R = l:i. Proof: According to Theorem 3•13.15, Q••(R) = Q(R), hence it follows from (1) and (2) by Lemma 3•15.36 that H = L = G. Suppose first that (4) is satisfied. Consider the commutative diagram in Subsection 3•15.37, where it is assumed that M = I is a specified minimal right ideal of E = E(X,T) and the map \:I+ X is defined by s\ = as Cs e I). Choose an idempotent u e I with au = a. Denote Io= {slseI,asqi = aqi} by Io. Then ueio and Go= Iou is a group with identity u. By Lemma 3•15.40 there exists an idempotent we Go such that Gow wG is a minimal set under the action of the group G {L 8 iseGo} on I. Then (a,aw)ePnR. Without loss of generality we may suppose that the indices ai are open in Xx X. Put

=

=

Ni

{mlm e wG,(m\,u\) e aiT,(m\,w\) e aiT} = {mlm e

Gow,(am,a) e aiT,(am,aw) e aiT},

Wi = {x Ix e aGow, (x ,a) e aiT, (x ,aw) e aiT}. It can be seen from the proof of Lemma 3•15.34 (by setting m1 = u, = w) that the sets Ni (i = 1,2, ... ) are open in wG and Ni = wG. Clearly Wi = Ni\; hence

mo

wi = aG 0w,

(i = 1,2, ... ),

Ii

Since the sets Wi are open in aGow, the set E1 = n [Wi = 1, 2, ... ] is residual in aGow. By (3), E1 {xlxeaGow,(x,a)ePnR (x,aw)ePnR}. Because of (4), the set E1 is finite or countable. Set

=

E2

{x Ix e E1, {x} is open in aGow} .

EXTENSIONS OF MINIMAL

162

Let us show that E2 = aGow. Suppose that, to the contrary, E1 '-.....E2 is a finite or countable residual subset of the non-empty open set aGow'--E2. This contradicts the definition of E2. Thus E2 = aGow. Since aGow is dense in aGow and each point x e E2 is open in aGow, it follows that E2 C aGow. Let x e E2. Then x = agw for some geGo. Since xeE2C E1, we get (agw,aw)ePnR; but (gw,w) is an almost periodic point of Ix I, and hence agw = aw. Consequently E2 = {aw}, and thus

aGow

=

E2

=

{aw}.

(15.8)

Let (a,x) e R. Then x = ai; for some i; e Io. Since E;w e Iow = = Gow, it follows from {15.8) that xw = ai;w = aw. Hence (x ,a) e P n R. As X is a minimal set, we get Rj = 6. Therefore the hypothesis (1) implies R = 6. This completes the proof in the first case. Now suppose that (5) is satisfied. Take any point a e X and define M,A,u,Go,w as above. It will suffice to show that aGow = {aw}. Define:

Iouw

E

{m Im e wG, (mA ,uA) e PnR} - {mlmeGow,(am,a) ePnR}.

By Lemma 3·15.34 the set E is residual in Gow. Let g e Go. Since 1+ gi; (E; e M = I) is a homeomorphism of Gow onto itself, we have gE()E 'f ¢. Hence there exists an element s e E with gs e E. Thus (as,a) ePnR, (ags,a) ePnR. Hence (ags,as) ePnR, because PnR is an equivalence relation. Consequently (ag ,a) e PnR, but this is possible only when ag = g. In fact, g e Go = I 0u; hence g = gu, whence (g,u) is an almost periodic point of (IxI,T). Thus ag = a for each g e G0 , and therefore agw = aw (g e Go), i. e,, aG 0 w = aw .1 i;

3•15.42 THEOREM: Let X be a compact metrix space, let (X,T) be a minimal transformation group and let ~:(X,T) 7 (Y,T) be an extension. Suppose that at least one of the following conditions is satisfied: (a) : There is a point a e X such that the set {x Ix e X, (x ,a)

e P()R~} is finite or countable; (b): PnR~ is an equivalence relation (here P denotes the

proximal relation in (X,T)). · Then there exist a minimal transformation group Ul,T) with a compact metrizable phase space and homomorphisms p:(W,T) 7 (X,T) and q:(W,T) 7 (Y,T) such that (1): p ~s a proximal extension;

(2): q ~s a PE-extension; (3): po~ = q.

In particular, if there is at least one R -distal point, then pis almost automorphic and q is a V-extension.

TRANSFORMATION GROUPS

163

Proof: The major part of the proof consists in showing the following proposition. Let X be a compact metric space and let ~:(X,T) (Y,T) be a minimal extension. Then there exist corrrpact metric spaces X* and Y*, minimal transformation groups (X*,T), (Y*,T) and homomorphisms n,~,~* such that PROPOSITION:

+

(i): The diagram

x ..--~n__ x,·,

~

j

j~,·,

y -~E;,"--_ y,•,

is commutative; (2):

~

and n are proximal extensions;

n to each fiber of the extension ~*:(X~,T) + (Y*,T) is injective; ( 4) : ~,., is an open map and R~,.,j = R~>',,

(3): The restriction of

To prove this proposition, let us consider a regular diagram of the form (15.1). Define H = G(X)

{gigeG

= G(Y),mgA

= mA,meM}.

By Lemma 3•15.22 G(X) is a T-closed subgroup of G. Define B = (mH,m)T. Clearly Bis a closed invariant relation on M. Consider now the transformation group (2M,T) induced by (M,T). If Ao e 2M and A1,A2 belong to the orbit closure of Ao in 2M, then A1 UA2 C Ao implies that the points A1 and A2 are proximal in (2M,T). In fact, let

where E(Ao) denotes the orbit-closure of Ao in (2M,T). Note that Ao e.A. If N1::) N2::) ···::)Na::) Na+l::) ••• (Nae.A) and N = nNa, a then N = lim{Na} in 2M. Since NaeE(Ao), N = lim{Na}eE(Ao). Moreover, Na C Ao implies NC Ao. Hence it follows by Zorn's lemma that .A has a minimal element, say L. As LeE(Ao), there exists a net {tn}, tneT, such that {Aotn} +Lin 2M. Without loss of generality we may suppose that {A1tn} + S1 and {A2tn} + S1 in 2M. As A1 U A2 C Ao, it follows that S1 U S2C LC Ao. It is also clear that S1,S2eE(Ao). Hence S1,S2eA, and S1 U S2 C L; therefore S1 = S2 = L by the minimality of L. Thus {A1tn} + L, and {A2tn} + L. Hence A1 and A2 are proximal. Fix some point m0 e M and, as usual, set

moB

= {mlm

eM,(mo,m)

eB}.

164

EXTENSIONS OF MINIMAL

It follows easily from the above considerations that the orbit closure of (mo,moB) in (Mx 2M,T) contains a unique minimal set, say Mi. Let Z denote the image of Mi under the projection M x zM + 2M, We get the commutative diagram

y

P1

+-----

z

where q1 and Pl are proximal extensions. Let us show that moB = (moh)B = (moB)h (he H). Indeed, if m emohB, then (moh,m) eB, hence (mo,m) = (moh,m)(h- 1 ,e) e 6.(M)o (Hx e)(h-1,e) = b.(M)(Hx e) = B, by Lemma 3•15.10. The proof of the remaining inclusion is similar. Next we prove that if (mo,A)eM1, then (moh,A)eM1 for each he H. Let he H, then there exists a net Un}, tn e T such that {motn} + moh. Without loss of generality we may assume that {moBtn} + A1 e zM. Then (moh,A1) e l:(mo ,moB), the orbit closure of (mo,moB)eMx2M. As (mo,A)eM1C l.:(mo,moB), it follows that AC moB. Similarly, A1 C mohB = moB. Since A,A1 e l.:(moB), the points A and A1 are proximal. Hence there exists a net {,n}, 'n e T, with {mo,n} + mo, {A,n} + A, and {A1,n} + A. Thus

{(moh,A1),n}

=

{(mo,nh,A1,n)}

+

(moh,A);

and therefore (moh,A) e l.:(moh,A1) C l.:(mo,moB). Clearly (moh,A) is an almost periodic point. Since M1 is the only minimal set contained in l: (mo ,moB), we have (moh ,A) e M1. We claim that if geH and the points (mo,A1), (mog,A2)el.:(mo, moB) are such that ((mo,A1),(mog,A2)) is an almost periodic point of (M x 2M x M x 2M,T), then A1 = A2. Indeed, A1 C moB, A2 C mogB = moB and A1,A2el.:(moB), hence it follows from the above considerations that A1 and A2 are proximal. Thus (mo,A1), (mo,A2) are also proximal. By hypothesis, ((mo,A1),(mog,A2)) is an almost periodic point, and hence ((mo,A1),(mo,A2)) is also almost periodic. This is possible only when A1 = A2, As a consequence of what has just been proved, we get that Pl is a proximal extension. The homomorphism A:(M,T) + (X,T) induces homomorphisms A1:(2M, T) + (2X,T) and A2 = Ax Al:M x 2M + Xx 2X. Set Y1 = ZAl and X1 = M1A2, We get the commutative diagram on the next page. As Pl is proximal, the extension pis also proximal. It is easy to see that q is also proximal. It follows directly from the construction that the restriction of q to each fiber of ~1 is injective. If~ is open and R~S = R~, then p and q are isomorphisms. If either R~S f R~ or~ is not open, then pis not a homeomorphism, as is seen from the construction itself. In this case

TRANSFORMATION GROUPS

z

--------------+ Al

165

Y1

we repeat the above considerations, taking ~1 instead of~. Proceeding by transfinite induction, we finally obtain minimal sets x·l< and y,·, with the required properties. (As a matter of fact, X* = X1, and Y* = Y1, but the proof of this will be omitted here). Thus the proof of the proposition is complete. The remainder of the proof is very similar to the proof of Theorem 3•14.36, however, we must use Lemma 3•15.41 instead of Lemma 3•14.31. The details are left to the reader. I 3•15.43 REMARK: If Xis a compact metrizable space and ~:(X,T) + (Y,T) is a minimal extension such that there is at least one R~-distal point, then it is not hard to deduce from Theorem 3•15.43 that the set of all R~-distal points is residual. Thus if Xis a metrizable space and the set of all R~-distal points is non-empty, then it is residual.

REMARKS AND BIBLIOGRAPHICAL NOTES The material presented in this section is essentially due to Ellis [13 ,14] (see also Ellis, Glasner and Shapiro * [1], Glasner *[4]), but our approach differs essentially from the original approach suggested by Ellis. We are not in a position to compare these two approaches here, since in this book we have not introduced the concepts from the algebraic theory of minimal sets that were developed by Ellis. We only note that our presentation is shorter and simpler than the one by Ellis. However, this remark is subjective and probably reflects only the preferences of the author. It should be noted that the algebraic method made possible different proofs of the results in Subsections 3•13.7-9 (see Ellis [14]) and to prove some other important statements (see for example, Ellis and Keynes [1,2]). The notion of the Galois group (in the case of a distal minimal extension) was introduced by Bronstein [1?,20*].

EXTENSIONS OF MINIMAL

166

§(3•16): GROUP EXTENSIONS In this section we shall introduce and study group extensions. The totality of group extensions with a given structure group will be characterized in terms of homological algebra. A more complete description will be obtained under some additional conditions by using covering transformation groups. Let X be a topological space and T and G be topological groups. We shall say that (X,T,11,G,o) is a GROUP EXTENSION if (X,T,11) and (X,G,o) are topological transformation groups and, moreover, for each point xe,X the set xG = {xogJge,G} is compact and 3•16.1

xG11t

=

x11tG,

(x e, X, t e, T).

(16.1)

It is evident from (16 .1) that the partition {xG Ix e, X} is invariant under (X,T,11). If the set U is open in X, then UG is also open in X. Thus {xGlxe,X} is a T-invariant star-open partition. By Subsection 1•3.2, the transformation group (X,T,11) induces a topological transformation group (X/G,T,p) on the quotient space X/G = {xG Ix e X}. Suppose in addition that the space X/G is Hausdorff. This holds, in particular, when G is a compact group. Indeed, it follows from Lemma 1•1.3 that in this case the partition {xGlx e,X} is star-closed. The transformation group (X,T,11) is an extension of the transformation group (X/G,T,p) by the canonical projection ~:X + X/G. This justifies the above terminology. Let (X,T,11) be an extension of (Y,T,p). We shall for brevity say that (X,T,11) is a group extension of (Y,T,p), whenever there exist a topological group G and a map o :Xx G + X such that (X ,T, 11,G,o) is a group extension with (X/G,T,p) = (Y,T,p). It follows from (16.1) that for any given x e,X, t e,T and g r;;G, there exists an element g' e G satisfying the condition (16.2) A group extension (X,T,11,G,o) is called a FREE GROUP EXTENSION if the group G acts freely on X, i.e., the equality xg = xog = x holds for a point x e X iff g is the identity of G. Let (X ,T, 11, G,o) be a group extension. If for any two elements t e T and g e G there exists an element g' e G such that (16. 2) holds for every point x e X, then the group extension is called a NORMAL GROUP EXTENSION.

Let G be a compact group and let AutG denote the group of all automorphisms of the group G with composition as the group operation. Provide AutG with the topology of uniform convergence, i.e., with the topology of a subspace of C(G,G). Then AutG is a topological group. If (X,T,11,G,o) is a free normal group extension and the group G is compact, then (16.2) determines a map h:T + AutG, namely,

TRANSFORMATION GROUPS

167

for each element ts T there is a corresponding automorphism ht such that ght = g'. We can easily verify that his a continuous _homomorphism of the group Tinto the group AutG. The homomorphism h:T + AutG is said to be INDUCED by the free normai group extension (X,T,n,G,a) and the group G will be called the STRUCTURE GROUP of this extension. A group extension (X,T,n,G,a) is called a CENTRAL GROUP EXTENSION if (16.3) for all x s X, t s T, g s G. Let (X,T,n,G,a) be a free group extension with a compact structure group. This extension is said to be a TAME EXTENSION if (16.2) implies that (16.4) for all ks G, i.e., for any fixed g and t the element g' in (16. 2) depends only on xcp e X/G, thus being constant on each fibre (xcp)cp-1 C X. Thus, if (X,T,n,G,a) is a tame extension, then we can define a map h:T x Y + AutG by ( x s X, t ET),

(16.5)

where y = xcp. If Y is compact, then h :T x Y + AutG is continuous (the proof is left to the reader). Each central group extension (X,T,n,G,a) is homogeneous because each map ag (g e G) is an automorphism of the transformation group (X,T,n). Hence if Xis a compact space, (Y,T) is a minimal transformation group and (X,T,n,G,a) is a central group extension, then cp:(X,T) + (Y,T) is a distal extension. 3•16.2 THEOREM: Let X be a compact space, (X,T,n) a minimai transformation group, Ga compact topoZogicai group and iet (X,T,n,G,a) be a centrai group extension. Then the canonicai projection cp:(X,T) + (X/G,T) is an equicontinuous extension.

Proof: Suppose the contrary is true. Then for any index as U[X] we can choose points aa e X and ha EX and an element ta e T so that (16 .6) where

ao

is a certain fixed index.

Choose elements ga s G (a e U[X]) such that ha Without loss of generality we may suppose that

=

aaaga - aaga.

{ga}+geG.

(16.7)

EXTENSIONS OF MINIMAL

168

Then it follows from (16.6) that {aaga} = {ba} + a, and hence ag = a. Since (X,T) is a minimal transformation group and the actions of T and G commute, xg = x for al 1 x e X. Further, ba ta = aagata = aatwa· Therefore {bata} + cg and (16.6,7) imply (c,cg) 4 ao, contradicting the equality xg = x (x e X) . I 3•16.3

Note that not every equicontinuous extension is a central group extension. However, it will be shown in subsection 3•17 that the investigation of minimal equicontinuous extensions can be, in some sense, reduced to the study of group extensions.

Two group extensions (X,T,n,G,a) and (X1,T,n1,G,01) are said to be CONGRUENT GROUP EXTENSIONS, if there exists a homeomorphism~ of X onto Xi such that 3•16.4

(x e X, t e T ,g e G) .

It is easily seen that congruent normal extensions induce the same homomorphisms h:T + AutG. By Ext[(Y,T,p),G,h] we shall denote the family of all classes of free normal group extensions (X,T,n,G,a) such that (X/G,T,p) = (Y,T,p) and the induced homomorphism coincides with h:T + AutG. Our next aim is to describe Ext[(Y,T,p),G,h].

of (X,T,n,G,a) is defined to be a map u:Y + X such that uo~ is the identity transformation of Y. The image of a point ye Y under the map u:Y + X will be denoted by u(y) (thus infringing upon our convention of writing the function sign on the right). A function f:Y x T + G, (f:Y x T x T + G) is called a two-dimensional (respectively three-dimensional) COCHAIN OF THE TRANSFORMATION GROUP (Y,T) over the group G if the following equalities are satisfied 3•16.5

A SECTION

f(y ,e) = e,

(f(y,e,s) = f(y,s,e) = e),

3•16.6 Let ye Y, t eT.

(ye Y ,s

e T).

As, (16. 8)

there exists a uniquely defined element f(y,t) e G with

u(y)nt

=

u(ypt)f(y,t),

(ye Y, t

e

T).

(16.9)

Then

u(y)nts = u(yptS)f(y,ts),

(y e Y; t , s e T) •

On the other hand,

u(y)nts

= [u(y)ntJnS = [u(ypt)f(y,t)Jns.

(16 .10)

TRANSFORMATION GROUPS

169

Taking (16.2,9) into account, we get

u(y)nts

u(ypt)nS[f(y,t)]hS =

u(yptS)f(ypt,s)[f(y,t)]hS.

(16.ll)

Since G acts freely on X, comparing (16.10) with (16.11) we obtain

f(y,ts) = f(ypt,s)[f(y,t)]h 8

.

(16 .12)

A function f: Y x T + G which satisfies (16 .12) will be called a TWO-DIMENSIONAL COCYCLE. Let u':Y + X be some other section. Since u'(y)~ = u(y)~ there exists a unique element \ = \(y) e G with u'(y) = u(y)oJ.(y) _ u(y)\(y). Let f':YxT + G be the cocycle corresponding to the section u':Y + X, that is, u'(y)nt = u'(ypt)f'(y,t). As u'(ypt) u(ypt)\(ypt), we have that (16.13) On the other hand, the condition u'(y)

u'(y)nt

u(y)\(y)nt

=

=

u(y)\(y) implies:

u(y)nt[\(y)]ht

u(ypt)f(y,t)[\(y)]ht.

(16.14)

Comparing (16.13) with (16.14) we obtain (16.15) It is easy to see that (16.15) determines an equivalence relation on the set of all cocycles, namely, f' ~ f iff (16.15) is satisfied. The set of all classes of equivalent cocycles will be denoted by C2[(Y,T,p),G,h]. Thus, given a class of congruent extensions, i.e., an element of the set Ext[(Y,T,p),G,h], we have defined a class of equivalent cocycles, i.e., an element of c2[(Y,T,p),G,h]. In the sequel this map will be denoted by w. 3•16.7

Since (x~)~-1

=

xG (xeX) and the group G acts freely on

X, each point x e X can be written uniquely in the form

x = u(x~ )gx, where gx e G.

Then

xnt = u(x~)gxnt = u(x~)nt(gxht) = u(xnt~)f(x~,t)(gxht). Therefore g

xn

t = f(x~, t) (gxht).

Let \J' :X

+

(Y x G) denote the map

defined by x\J' = (x~,gx) (xeX). Then xnt\J' = (x~pt,f(x~,t)(gxht)). Suppose now that f is a two-dimensional cocycle, i.e., f:YxT + G satisfies condition (16.12). Define a map µ:(YxG) xT + (YxG)

EXTENSIONS OF MINIMAL

170

by

(y,g)µt = (ypt,f(y,t)•(ght)), a map a : ( Y x G) x G (y,g)aa

+

(Y

-+

( 16 . 16)

x G) by

= (y,g)a

and a map n: G x G

(y e Y, g e G, t e T),

(y,ga),

(y e Y, g e G, a e G) ,

G by (a,g)n = ag,

(a,g e G).

If the group G is compact and the function f:Y x T -+ G is continuous, then (YxG,T,µ,G,a) is a free normal group extension which will be called the SKEW PRODUCT of the transfoY'lrtation groups (Y,T,p) and (G,G,n) by means of the cocycle f. The induced homomorphism coincides with the given homomorphism h:T + AutG. If the section u:Y + (YxG) is defined by u(y) = (y,e), where e is the identity of G, then the corresponding two-dimensional cocycle is just f:Y x T + G. Concerning the case when f is not continuous, we do not know whether or not there exists a free group extension corresponding to the cocycle f.

3•16.8

Now we shall consider the particular case when the group The group operation in G will be written additively. Given a function >..:Y + G, let o>..:YxT + G denote the function defined by

G is commutative.

Co>..)Cy,t)

= -

>..(ypt) + [>..Cy)]ht.

(16.17)

Condition (16.12) can be rewritten in the form f(y,ts) = f(ypt,s) + [f(y,t)]h 8

,

(16.18)

and condition (16.15) in the form f'(y,t) = f(y,t) + o>..(y,t).

(16.19)

Define a group operation on the set of all two-dimensional cocycles as follows: (f1 + f2)(y,t) = f1(y,t) + f2(y,t).

The totality of two-dimensional cocycles endowed with this operation forms a commutative group which we shall denote by z2

z2 [(Y,T,p),G,h].

A function o>..: Y x T + G of the form (16 .17) wil 1 be called a COBOUNDARY. It is evident that each coboundary is a cocycle. The set 32 = B2 [(Y,T,p),G,h] of all coboundaries is a subgroup

TRANSFORMATION GROUPS

171

of the group z2, Denote z2/B2 by H2. The commutative group H2 = H2 [(Y,T,p),G,h] is called the SECOND COHOMOLOGY GROUP of the transformation group (Y,T,p) over the group G with the homomorphism h:T + AutG. Thus, if the group G is commutative, then the set c2 can be provided with a structure of a commutative group H2. The group H2 [(Y,T,p), G,h] may be regarded as the second cohomology group of some complex of right T-modules (MacLane [1]). Let (X,T,n,G,a) be a normal group extension of the transformation group (X/G,T,p) = (Y,T,p). This extension is said to be AN EXTENSION LOCALLY TRIVIAL AT A POINT ye y if there exist a section u:Y + X and a neighbourhood U of yin Y such that the restriction ulu of the map u:Y·+ X to U is a homeomorphism. An extension is called a LOCALLY TRIVIAL EXTENSION if it is locally trivial at each point ye Y. An extension (X,T,1r.,G,a) is said to be a TRIVIAL EXTENSION if there exists a continuous (and, consequently, a homeomorphic) section u:Y + X. 3•16.9

3•16.10 LEMMA: Let (Y,T,p) be a minimal transformation group. If the extension (X,T,n,G,a) is locally trivial at some point ye Y, then it is locally trivial. Proof: Let z e Y. By hypothesis there exist a section u:Y + X and a neighbourhood U of y e Y such that the restriction u Iu is a homeomorphism. Choose an element t e T and a neighbourhood V of z e Y so that Vp t-l C U. The restriction to V of the section u': Y + X defined u'(y) = u(ypt- 1 )nt is a homeomorphism. Thus the extension is locally trivial at every point z e Y, I 3•16.11 LEMMA: If (X,T,n,G,a) is a free normal group extension which is locally trivial at some point ye Y = X/G and the spaces Y and Gare compact, then there exist a neighbourhood U of y and a homeomorphism 1/Ju:U x G + Ucp-1 such that (z,g)l/Jucp = z (zeU,geG) (cp denotes, as usual, the canonical projection).

Proof: By hypothesis, there exist a section u:Y + X and a neighbourhood U of ye Y such that the restriction u Iu is a homeomorphism. Let x e Ucp-1. As the group G acts transitively and freely on xcpcp-1 there exists a unique element gx e G with x = u (xcp )gx. Define a map Au: ucp- 1 + U x G by xAu = (xcp ,gx), then AU is an injective map. The inverse map 1/Ju = Au-1 is defined by (z,g)l/Ju = u(z)g (z eU,geG) and therefore it is continuous. Without loss of generality we may suppose that the section u: Y + X is a homeomorphism onto the closure D of U. Since Dx G is a compact set, it follows from the preceding considerations that 1/Ju is a homeomorphism. If z eU, geG, then (z,g)l/Jucp = u(z)gcp =

z.l

3•16.12 Let Exti[(Y,T,p),G,h] denote the set of all classes of congruent locally trivial free normal group extensions

EXTENSIONS OF MINIMAL

172

(X,T,n,G,a) which satisfy the condition (X/G,T,p) = (Y,T,p) and induce the given homomorphism h:T + AutG. Let the group G be compact. Let (X,T,n,G,a) belong to some class in Exti[(Y,T,p),G,h] and let Yoe Y. There exist a section u:Y + X and a neighbourhood V of Yo .such that u\v is a homeomorphism. If f:Y x T + G is the cocycle corresponding to the section u:Y + X, i.e.· u(y)nt then the function

=

u(ypt)f(y,t),

f: Y x T V*

=

+

(ye Y, t

e T),

G is continuous on the set

{(y,t)\yeV,yptev}.

Let Zi[(Y,T,p),G,h,yo] denote the set of all those cocycles f:YxT +Gin Z2 [(Y,T,p)jG,h] for which there exists a neighbourhood V of Yo such that f V* is continuous. Let Cf[(Y,T,p),G,h,yo] denote the set of all classes of equivalent (in the sense of (16.15)) cocycles in Zi[(Y,T,p),G,h,yo]. If (Y,T,p) is a minimal transformation group, then the set Ci [(Y,T,p),G,h,yo] does not depend on Yo eY. The proof is similar to that of Lemma 3•16.10. Let w1 be the natural map which assigns to each class of congruent extensions in Exti[(Y,T,p),G,h] an element of Cf[(Y,T,p),

G,h,y 0 ], namely, the class of cocycles equivalent to the cocycle f, which corresponds to that section u:Y + X whose restriction u \u to some neighbourhood U of Yo e Y is continuous. 3•16.13 THEOREM: If Y is a aompaat spaae and (Y,T,p) is a minimal transformation group, when wi is a bijective map of Extt[(Y,T,p),G,h] onto Cf[(Y,T,p),G,h,yo]. Proof: Let f:Y x T + G be a cocycle and suppose that there exists a neighbourhood U of Yo e Y such that the restriction of f to U•'< = {(y,t)\yeU,ypteu} is continuous. We shall construct an extension (X,T,n,G,a) which belongs to some class in Ext 1 [(Y,T,p),G,h] and which has the given cocycle f (for a suitable section). Let X denote the Cartesian product of the sets Y and G, X = Y x G. We shall provide X with a topology which will be defined by assigning a neighbourhood base ~(y,g) to each point (y,g) e y

X

G.

Since (Y,T,p) is a minimal transformation group, there exists an element t e T with yr 1 = ypt-l e U. Let V = V(y) be any neighbourhood of y satisfying the condition vt-1 CU and let W(e) be an arbitrary neighbourhood of the identity e in the topological group G. Then ~(y,g) is defined to be the family of all subsets of Y x G of the form

TRANSFORMATION GROUPS

173

I

A(V,W,t) = {(z,f(zt -1 ,t)•[f(y,t -1 )]ht.bg) zeV(y),beW(e)}. Let us show that the family ~(y,g) in fact does not depend on the choice of the element t e T with yt-1 e U. Let s e T and ys-1 e U. Choose a neighbourhood Wo(e) so that Wo2 C W. Further choose W1(e) so that W1C Wo.and [f(y,s

-1

)]h 8 •Wo :> W1•[f(y,s

-1

)]h 8

•

As the restriction of f: Y x T + G to u1, is continuous, there exists a neighbourhood V1 V1(y) C V such that v1s-lc U and

=

[f(zs

-1

,st

-1

t -1 -1 t )]h e [f(ys ,st ]h •W1

for all z e V1(y). We show that A(V1,W1,s) C A(V,W,t). Indeed, let (z,b)eA(V1,W1,s). Then zeV1C V and b = f(zs- 1 ,s)•[f(y, s- 1 )]h 8 ·b1g, where b1 e W1 C Wo. After some reductions we obtain: b = f(zs

e f(zt

C f(zt = f(zt

-1

-1 -1 -1

,(st

-1

)t)•[f(y,s

,t)•[f(ys ,t)·[f(ys ,t)•{f(ys

-1 -1 -1

,st ,st ,st

-1 -1 -1

-1

)]h 8 ·big

t -1 )]h •W1•[f(y,s )]h 8 •big t

)]h ·[f(y,s )•[f(y,s

-1

-1

)]h 8 •Wobig

)]h 8

t-1

t }h •Wobig.

The expression in curly brackets coincides with f(y,t- 1 ). deed f(y,t

-1

)

=

f(y,s

-1

(st

-1

))

=

f(ys

-1

,st

-1

)• [f(y,s

-1

)]h 8

t-1

In-

•

Hence

b e f( z t

-1

, t) • [f( y , t -1 ) ] h t • Wg,

and consequently (z ,b) e A( V ,W, t). Thus the family CB(y ,g) does not depend on the choice of t e T with yrl e U. It is not difficult to verify that the totality of all families (]3(y ,g) for all ly ,g) e Y x G determines a topology on X = Y x G such that CB(y ,g) is a neighbourhood basis at (y ,g). Let X be the topological space obtained by providing the set YxG with the above topology. Define an action of Ton X as follows:

EXTENSIONS OF MINIMAL

174 (y,g)nt =

(ypt,f(y,t)(ght)),

(t e T).

An elementary but tedious computation shows that the map n: Xx T -+ X is continuous. We do not present here the details but only note that, as it follows from the very definition of the topology on X, it suffices to verify the continuity only on U* x G, where u,•: = {(y,t)lye,U,ypte,U}. The latter follows easily from the continuity of ht and flu*· Thus (X,T,n) is a topological transformation group. Let us define an action of G on X. If (y,g) e,X and a e,G, then we set (y,g)oa = (y,ga). Clearly, (X,G,o) is a topological transformation group and moreover G acts freely on X. Define a map ~:X-+ Y by x~ = y whenever x = (y,g). It is clear that~ is continuous and open and that the set x~~-1 = {(y,g)lgs G} is compact. It follows directly from the definition of the topology on X that the topology of the subspace U x G C X coincides with the topology of the Cartesian product of UC Y and G. In fact, if ye, U and V(y) is a neighbourhood with V(y) CU then, in the capacity of an element tsT with yt- 1 e,V, we can take t = e, where e is the identity of T. Then A(V,W,g) = {(z,h)ize,V(y),he,W(e)g}.

Now it follows easily that the space Xis Hausdorff. Since Y is compact and each fibre x~~-l is compact, the space Xis also compact by the continuity of ~:X-+ Y. If ysY, gsG, asG, tsT, then (y,ga)nt =

(ypt,f(y,t)•[(ga)ht])

Thus (X,T,n,G,o) is a normal group extension and, moreover, the induced homomorphism coincides with the given one, h:T-+ AutG. If we take the section u:Y-+ X defined by u(y) = (y,e), where e is the identity of G, then the cocycle f:Y x T-+ G corresponding to this section coincides with the given function f. In fact, u(y)nt =

(y,e)nt

=

(ypt,f(y,t))

=

(ypt,e) 0 f(y,t)

=

u(ypt) 0 f(y,t).

The extension (X,T,n,G,o) is locally trivial by Lemma 3•16.10, since the aforementioned section u:Y-+ Xis continuous on the set

u.

Thus

(X,T,n,G,o)

is the required extension.

I

3•16.14 In the remainder of this section all topological spaces are assumed to be connected locally pathwise connected and semi-locally simply connected (unless specified otherwise) (Hu [1]) .

TRANSFORMATION GROUPS

175

3•16.15 Let (X,T,n,K,0) be a free normal locally trivial extension of (Y,T,p). Then the map ~:X + Y is a locally trivial fibre bundle and since Y is compact and Hausdorff, ~:X + Y satisfies the covering path axiom (Hu [1]). Let xoeX, Yo= xo~- Let A= E(X,xo) and B = E(Y,yo) be the universal covering spaces of X and Y respectively. Further, let D1 = n1(Y,yo), D2 = n1(K) and D = n1(X,xo), Let us denote by G the universal covering group of the group K. It is known that D2 is a discrete (consequently, a central) normal subgroup of G and K = G/D2, The group D1 acts in a natural way on B = E(Y,yo) (from the right). Let [b] eB, i.e., [b] is a class of equivalent paths in [Y;yo ,Y], and let d1 e D1, i.e., d1 = [o:], where o: is a loop at the point YO· Define a map 01:BxD1 + B by ([b],d1)01 = [o:- 1b]. Clearly, (B,D1,01) is a transformation group. We define the transformation group (A,D,o) in a similar way. The transformation group (X,K,0) induces the universal covering transformation group (E(X,xo),G,a) as follows. Let xeX, I [0,1] and let a:I + X be a path satisfying a(O) = xo, a(l) = x. If g e G, then g = [T], where T(O) = e (e is the identity of the group K). The path s:I + X, defined by s(i) = a(l)0T(i) Ci eI) has x as its initial point. Define a map 0:A xG + A by ([a],g)a = [as]. It is easy to verify that this map is well defined and that (A,G,a) is a topological transformation group. Let us show that the actions of G and Don A just defined commute. In fact, let [a] eA, g = [T] e G and d = [o:] e D. Then [a] ag = [as] where s:I +Xis already defined. Further, [as]od [o:- 1as], On the other hand, ([a]od)ag = [o:-la]crg = [o:-las]. I 3•16.16 Let a:I + X and s:I + X be the previously defined paths. Define a path "A:I + X by "A(i) = xo0T(i) (i e I). Let T(l) = k. Let us denote by ak the path ak:I + X, defined by ak ( i) = a ( i) 0k (i e I) . 3•16.17 LEMMA: [as]

=

["Aak],

Proof: We need to show that the paths as and "Aak are homotopic.

Set { x 0 rr,(2i),

hj(i)

0 ,,; 2i ,,; j ,,; 1,

a(2i - j)0T(j),

0

~

j

a( 1)0T(2i-l),

0

~

j < 2i - 1 ., 1.


Rµ,

Then

n

.l(Jl3

holds iff

(Jl2 .l(Jl3,

Proof: Necessity follows from Lemma 3•19.2. Let (Jl 2 l(Jl 3 , We shall prove that (Jlll(Jl3. Let ;\:X-+ X2 denote the principal group extension adjoint to the equicontinuous extension µ:X1-+ x 2 and let G be its structure group. Then (X2,T) = (X/G,T) and there

EXTENSIONS OF MINIMAL

224

is a closed subgroup K of G such that (X1,T) = (X/K,T). By Theorem 3•17.2, (X1,T)"' ((XxM)/G,T), where M = G/K == {hKiheG} and (X xM,T,G) is an M-associated group extension. By hypothesis, (X2·Z,T) = ((X/G)•Z,T) is a minimal transformation group. Let G act trivially on Z, i.e., zg = z, (zeZ,geG). Then there exist natural isomorphisms between ((X/G)•Z,T) and ((X•Z)/G,T) as well as between (((XxM)/G)•Z,T) and (((X•Z) xM)/ G,T). Consider the commutative diagram 111

((X•Z) xM)/G" ((XxM)/G)•Z _ __,. (X/G)·Z" (X•Z)/G

where ((x,m)G,z)111 = (xG,z). Apply Lemma 3•19.12 to the M-associated group extension ((X•Z) xM,T,G). By hypothesis, the transformation group ((X•Z)/G,T) "' ((X/G)·Z,T) = (X2·Z,T) is minimal. In order to prove that cp 1 lcp 3 it will suffice to show that the transformation group

is minimal.

We shall first prove the inclusion (19. 6)

Let ((a1,z1),Ca2,z 2 )) eR(µ 2 ), where a 1 ,a 2 e (XxM)/G and z 1 ,z 2 eZ. This means that (a1,z1)111 = (a2,z2)112, i.e., (a111,z1) = (a211,z2), hence (a1,a2) eR(µ), z2 = z1 == z. By condition (3), (a1,a2) eQ•', (cp1). Applying Lemma 3•19.16 to the extensions '• (qi) ( )1

X

X

)1).

Further, it follows easily from Q(qi)(µ x µ) = Q(iµ) C Q>'•(ij;),

by applying the inductive process of constructing the relation Q>'•(qi) that Q>'• ( qi ) ( )1

X )1 )

C Q>': ( iµ ) .

I

The next theorem is a generalization of Lemma 3•19.20. 3•19.23 THEOREM: Consider the commutative diagram

of minimal transformation groups and homomorphisms, where (1): A

is an open map;

(2): µ

is an F-extension;

(3): Q>'•( qi) ::> Rµ-

Under these conditions, the extensions joint iff iµ and\ are disjoint.

qi

and

A

are dis-

Proof: We assume that iµ lA and prove that qi lA. By hypothesis (2), µ can be decomposed into a transfinite sequence of equicontinuous extensions qig+ 1 (0 ~a< B): qig+l

(Y,T) - (Yo,T)

+ +

(Y1,T) +

+

... +

(Ya,T) - - (Ya+1,T)

+

(YB,T) - (X ,T).

Since iµ lA, we have that (Yo •Z ,T) is a minimal transformation group. Suppose that (Yy·Z,T) are minimal for all ordinals smaller than some ordinal S. If Bis a limit ordinal, then (Ys·Z,T) is minimal by Lemma 2•9.16. Let B =a+ 1. Consider the commutative diagram

EXTENSIONS OF MINIMAL

226

z

where I;, n and r,; are canonical.

Let us prove the inclusion

R(cpg+ 1 ) C Q>':(s). By Lemma 3•19.22, Q>':(cp)(I; x I;)

= Q>':(r,;),

therefore

By hypothesis, cpg+ 1 is an equicontinuous-extension. Hence it follows from Lemma 3•19.20 and the condition nlA that slA, i.e., the transformation group (Ya+1·Z,T) is minimal. Thus it is shown by induction that (Ys•Z,T) is minimal for all S, S ~ B. Letting S = B we get the required result. I 3•19.24 REMARK: If the space Xis metrizable or if the group T is a-compact, then condition (2) in Theorem 3•19.23 can be replaced by requiring the distality ofµ (Theorem 3•14.24). 3•19.25 THEOREM: Let cp:(X,T) + (W,T) a:nd w:(Y,T) + (W,T) be minimal extensions. Suppose that cp is open a:nd wis an Fextension. Then cp a:nd w are disjoint iff cpo lwo, where (J)O: (X/Q*(cp),T) + (W,T) a:nd wo:(Y/Q*(w),T) + (W,T) are the canonical homomoY7phisms.

Proof: Let (Xo,T) = (X/Q>':(cp),T), (Yo,T) = (Y/Q>':(w),T) and cpol By Corollary 3•19.21, cp.lfo. Apply Theorem 3•19.23 to the following commutative diagram:

WO·

(Y,T)

µ

(Yo,T)

(X,T)

~ct/ It is clear that µ is an F-extension. Since R(µ) = Q>':(w), condition (3) of Theorem 3•19.23 is satisfied. By hypothesis, (J) is an open map. Since cplfo, it follows from Theorem 3•19.23 that

cplw.1 3•19.26 COROLLARY: Let (X,T) and (Y,T) be minimal transformation groups and let (Y,T) belong to the class F. Then

TRANSFORMATION GROUPS (X ,T)

227

l(Y ,T) iff (X/Q•'•(X ,T) ,T) l(Y/Q•'•(Y ,T) ,T).

Proof: Apply Theorem 3•19.25 with W assumed to be a single point. I 3•19.27 LEMMA: If cp:(X,T) 7 (Y,T) is a proximal extension and (Y,T) is minimal, then X contains exactly one minimal set.

Proof: Let Mi and M2 be minimal subsets of X. Since MH = Mn = Y, there exist points m1 e Mi and m2 E M2 such that micp = m2cp. Byhypothesis, the points m1 and m2 are proximal. Therefore m1~ = m2~ for some element~ from the enveloping semigroup E(X,T). Hence it follows that M1nM2 t !tl, and consequently Mi= M2,I 3•19.28 LEMMA: Consider the commutative diagram:

(X,T)

µ

(Y,T)

(Z,T)

~(t/ ~here all transformation groups are minimal andµ is a proximal extension. If 1jJ lA and the set of almost periodic points is dense in (X·Z ,cp ,A) = x~z, then cp lA. Proof: Define a homomorphism s:X•Z 7 Y•Z by (x,z)s = (xµ,z) Clearly, the extensions is proximal. If Y•Z is a minimal set, then X•Z contains a unique minimal set M (Lemma 3•19.27); hence M = X•Z, because the set of almost periodic points is dense in X•Z. Thus, if ijJlA, then cplA. I ((x,z) eX·Z).

3·19.29 COROLLARY: If (X,T), (Y,T) and (Z,T) are minimal transformation groups, µ:(X,T) 7 (Y,T) is a proximal extension and the set of almost periodic points is dense in Xx Z, then (Y,T) l(Z,T) implies (X,T) l(Z,T). 3•19.30 We shall say that a minimal transformation group (X,T) belongs to the class 'X if there exist a point xo EX and an idempotent u e E(X ,T) such that uE is a minimal right ideal of the semigroup E(X,T) and

x 0 utu

=

x 0 ut,

(t ET) .

If the group Tis commutative, then each minimal transformation group (X ,T) belongs to the class 1:.. If the point x 0 EX is distal from all other points x EX, then (X ,T) e°K:. Indeed, in this case xou = xo and (xot )u = xot for all t e T and all idempotents u e E(X ,T). Hence

EXTENSIONS OF MINIMAL

228

xoutu

= (xot)u = xot = xout,

(teT,ueE,u2 = u).

3 • 19. 31 LEMMA: If (X ,T) eX and (Y ,T) is a minimal. transformation group, then the set of al.most periodic points of (X ,T) x (Y ,T) is dense in Xx Y.

Proof: Let p: (Xx Y) + X be the canonical projection and let f&:E(X x Y ,T) + E(X ,T) be the homomorphism of the enveloping semigroups induced by p. By Lemma 1•4.20, there exist a minimal right ideal K of E(X x Y ,T) and an idempotent v e K such that v>& = u. Let q:X x Y + Y denote the projection map onto the second coordinate, let f,:E(X x Y,T) + E(Y,T) be the semigroup homomorphism induced by q: (Xx Y ,T) + (Y ,T) and let w = vf,. We specify some point Yo e Y and proceed to prove that (xout ,YOWT) is almost periodic for all t,T eT. In fact, T-lvT is an idempotent belonging to the minimal right ideal T-lK of E(X x Y ,T) and

but this just means that (xout,yoWT) is an almost periodic point. Clearly, the set of these points for al 1 t, T e T is dense in Xx

Y.I

3•19.32 COROLLARY: Let (X,T), (Y,T) and (Z,T) be minimal. transformation groups and Z.et µ:(X,T) + (Y,T) be a proximal. extension. Asswne moreover that either (X ,T) e:t or (Z,T)e:t. Then (Y,T)J_(z,T) irrrpl.ies (X,T)J_(z,T).

Proof: This follows from Lemma 3•19.31 and Corollary 3•19.29.I 3•19.33 THEOREM: Let (j):(X,T) + (W,T) and ~:(Y,T) + (W,T) be minimal. extensions such that (j) is open and~ is a PE-extension. If the set of al.most periodic points is dense in X•Y, then

where (j)Q and ~o are the same as in Theorem 3•19.25. Proof: As~ is a PE-extension, there exists a projective system {(Ya,,T)lo ~a~>&} such that (Yo,T) = (Y/Q;';(~),T),(Y>&,T) = (Y,T), ~ = (j)g and each extension (j)g+l:(Ya+1,T) + (Ya,,T) is either equicontinuous or proximal (Lemma 3•18.20). Let (j)Ql~o, Then (j)l~o by Corollary 3•19.21. Suppose that (j)Ql~'Y' where ~y:CYy,T) + (W,T) is the natural map, for all ordinals y smaller than some ordinals. If Sis a limit ordinal, then ~sl(j), since (Ys·X,T) is equal to the projective limit of the system {(Yy•X,T)lr < S} of minimal transformation groups. Let S =a+ 1 and let f, denote the natural map of

TRANSFORMATION GROUPS

Y onto Ya+l·

229

Then by Lemma 3•19.22,

R(Ya+l

-+

Yo) :::> R( cpg+ 1 ).

Suppose first that cpg+l is an equicontinuous extension. Theorem 3•19.23 to the diagram

(19. 7) Apply

Condition (3) holds in view of (19.7). The other conditions are also satisfied. Hence 1/Ja+1lcr. Now consider the case when cpg+l is a proximal extension. By hypothesis, the set of almost periodic points is dense in X•Y; hence the same is valid for X·Ya+l· Thus 1/Ja+l.lcp by Lemma 3•19. 28. We have proved by induction that 1/Ja.lcr for all ordinals a, a ( B. Letting a = B, we get wlcr, 1 3•19.34 COROLLARY: Let (X,T) and (Y,T) be minimal transformation groups. Assume that (Y ,T) belongs to the class PE and the set of almost periodic points is dense in (Xx Y ,T) (in particular, (X ,T) e:R: or (Y ,T) e1:). Then

(X,T)l(Y,T) iff (X/Q>'•(X,T),T)l(Y/Q>'' O. Let Vp(I,E) denote the family of all pairs of functions cp,lji e 00 •

C(JR,JRn) such that

where I· I denotes the Euclidean norm in JR.TI. It is easy to verify that the totality of sets of the form Vp(I,E) forms a base for some uniformity in C(JR,JRn), which is weaker than the uniformity discussed in Subsection 4•20.5. Denote by Lp(JR,JRTI) the completion of the uniform space obtained in this manner. It is not difficult to show, by using Lemma 4•20.3, that the transformation group (C(JR,JRTI),JR,o) can be extended by continuity to Lp(JR,JRD). In fact, let I and K be two segments of JR and E > 0. If (cp,lji) e Vp(I + K,E), then (h,ijJT) e Vp(I,E) (T eK).

REMARKS AND BIBLIOGRAPHICAL NOTES The shift dynamical system defined in the space of continuous functions cp:JR +JR was first studied by Bebutov [1]. More general shift transformation groups were considered by Gottschalk and Hedlund [2], Sell [4] and Scerbakov [6]. Lemma 4•20.4 is similar to the well known Kamke Lemma (see, for example, Hartman [1]).

EXTENSIONS

238

§(4•21): EXTENSIONS ASSOCIATED WITH CERTAIN CLASSES OF EQUATIONS In this section we shall show that extensions of dynamical systems can be associated in a natural way with several classes of differential, integral, difference, and algebraic equations. In order to give the first and the simplest examples demonstrating the usefulness of considering relations between equations and the corresponding extensions, we prove the existence of recurrent or Poisson stable solutions of differential equations with recurrent or Poisson stable, respectively, right sides. 4•21.1

Let W be an open subset of the space JR.TI. sider differential equations of the form dx

dt

= f(x,t),

We shall con-

, (21.1)

where the right side f is assumed to be defined and continuous for all x e W and t eJR. Consider the set C(W xJR,JRD) of all such functions f and determine the flow ( C( W xJR,JRD ),JR,o), described in Subsection 4•20.6. For brevity denote (C(WxJR,JRD),JR,o) by (Y,JR~o). The family of all pairs (f,qi), where feC(WxJR,JRD), and qi e C(JR,W) is such that qi:JR -+ W is a solution of Equation (21.1), is denoted by X. It is easy to see that Xis an invariant subset of the Cartesian product of the transformation groups (C(WxJR,JRn),JR,o) and (C(JR,W),JR,o). Therefore we have defined a dynamical system (X,JR,o), for which the canonical projection p:X-+ Y, (qi,f)p = f, is a homomorphism. The fibre of the map p: X -+ Y over the point f e Y coincides, in principle, with the set of all solutions qi:JR-+ W of Equation (21.1). 4•21.2

Let Yo be the subset of Y consisting of all functions f e C(W xJR,JRn) such that for each point a e W there exists

a unique solution qi:JR-+ W of Equation (21.1) with the initial condition qi(O) = a. Clearly, Yo is invariant under shifts. Let x0 = Yo x W. Define a map 11: :X0 xJR -+ x0 by (f ,a, t )11: = Cft,C+J(f,a,t)) (feYo,aeW,teJR), where qi(f,a,•) is the solution of Equation (21.1) with qi(f,a,O) = a. It follows from the basic properties of solutions of differential equations and from Lemma 4•20.4 that qi:Y 0 x WxJR-+ Wis a continuous map. Therefore 11: is also continuous. Clearly (f,a,0)11: = (f,a) (feYo,aeW). Lets, teJR. Since the solutions of Equation (21.1) with feYo are uniquely determined by the initial conditions, we have the equality qi(f,a,t+s) = qi(ft,C+J(f,a,t),s),

which implies that the homomorphism axiom is satisfied. Thus (Xo,JR,11:) is a flow, and the canonical projection p:Xo-+ Yo determines an extension p:(Xo,JR,11:)-+ (Yo,R,o). We note (although it is evident) that the map pis open in this case.

AND EQUATIONS

239

Let Xi= {(f,cp)l(f,cp)eX,feYol. It is easy to see that the dynamical systems (Xo,lR,n) and (X1,lR,o) are isomorphic. The required isomorphism of x1 onto x0 is of the form (f,cp)

f+

(f,cp(O)).

The considerations in Subsection 3•21.1 are valid for equations of the form (4•21.1) defined in a Banach space, i.e., in the case when lRn is replaced by an arbitrary Banach space. As to Subsection 4•21.2, it is necessary to assume in addition the continuity of the map cp :Yo x W x]R + W. The latter condition holds if the function f satisfies, for example, the Lipschitz condition: for ever1, B > 0 there exists a number L(B) > 0 such that llf(x,t) - f(y,t)II ~ L(B)jlx - YII whenever llxll ~ B, IIYII ~ B, It ~ B. 4•21.3

4•21.4

Now we return to Equation (21.1) in Euclidean n-space JRD. Let us specify some solution cp :lR+ JRD 9f this equation and construct an extension of transformation groups as follows. Let H(f) denote the closure of the family of shifts {fT j T elR} in the topology of uniform convergence on compact subsets of JRD x]R, In other words, H(f) is the closure of the orbit of the point fin the shift dynamical system (C(JRD xJR,JRD),JR,o). Hence we get a flow (H(f),lR,a). Similarly, let H(f,cp) denote the orbit closure of the point (f,cp) in the Cartesian product of flows (C(JRD xJR,JRD), JR, a) and ( C(JR,JRD ),JR, a). Let p:H(f,cp) + H(f) be the natural projection. We shall say that the extension p: (H(f,cp),lR,a) + (H(f),lR,a) is associated with the solution cp:lR +]RD of Equation (21.1). It is not difficult to prove that if (g, 1/!) e H(f, cp), then 1/J:lR + JRD is a solution of the equation dx dt=g(x,t).

In fact, if (g,1/J) eH(f,cp), then there exists a sequence {TD} such that {fT } +gin C(JRD x]R,JRD) and {cp, } + 1/J in C(lR,JRn). The set

n · D K, consisting of all points cpT (t)(te [-T,T],n = 1,2, ... ) and D

1/J(t)(te [-T,T]), is a compact subset ofJRD for every T > o. it follows from the inequality

Hence

+ Jlg(cpT (s),s) - g(i/J(s),s)II D

that {fT (cpT (•),·)} + g(i/J(•),•) in the space C(JR,JRD). D

D

EXTENSIONS

240

In the case of an equation of the form (21.1) defined in a Banach space, a similar statement holds under the additional hypothesis that the set {cp(s)\s eR} is compact in E (in this case the solution cp:JR. +Eis called compact). 4•21.5

Fix a number h > 0. Denote by~ the space of all continuous maps cp:[-h,O] +Rn with the norm:

II cp \I

=

sup

Icp ( 11) I,

-h~11~0

where I ·I denotes the Euclidean norm in Rn. If 1)!:[-h,A) +Rn is a continuous map, then ijJt'(o ,,: t < A) denotes the element of the space~ defined by: 1)Jt(11) = ijJ(t + 11) Let F:JR.+ x ~ + JR.D be a continuous map.

An equation of the form

x'(t) = F(t,xt),

(21.2)

where x'(t) is the right derivative of x(t), is called a FUNCTIONA continuous function x: [-h,A) +JR.n is said to be a solution of this equation with the initial condition cp e ~, i f the function x = x(t) satisfies Equation (21.2) for all t e [O,A) and x(11) = cp(11), i f -h ~ 11 ~ 0. Let Y = C(JR.+ x ¢,Rn) be the space of all continuous maps F:JR.+ x ~ +Rn provided with the uniformity corresponding to the uniform convergence on compact subsets of JR.+ x ~. Then we have a semigroup shift dynamical system (C(JR.+ x ~,JR.n),JR.+ ,a). Let X be the family of all pairs of the form (F ,cp), where Fe c(JR.+ x ~,Rn) and cp: [-h, + Rn is a solution of Equation (21. 2). We can define a shift transformation semigroup (X,JR.+,a) so that the canonical projection p:X + Y is a homomorphism.

AL-DIFFERENTIAL EQUATION.

00 )

4•21.6

Consider the NON-LINEAR INTEGRAL VOLTERRA EQUATION x(t) =

f(t) + f:a(t,s)g(x(s),s)ds

(t ~ 0),

(21. 3)

where f:JR.+ + JR.TI, g:JR.n xJR.+ + JR.TI and a:JR.+ xJR.+ + AfD are given continuous functions (Afil denotes the space of all real (n x n)-matrices) and x = x(t) is the unknown function. Let C = c(lR+,JR.D), A = c(JR.+ x]R_+ ,Afil) and let G = c(JR.n XJR.+,JR.TI) be the spaces of continuous maps provided with the uniformity of uniform convergence on compact subsets. Let Go denote the subspace of G consisting of the functions g e G satisfying the following Lipschitz condition: for every compact subset KC Rn and every segment I= [O,T],O < T < there exists a number L = L(g,K,I) such that 00

241

AND EQUATIONS

lg(x,s) - g(y,s)I


O. It is also easy to verify that the solution depends continuously on f1:oC, acA, gcGo and tc [o,h] (moreover, the continuity with respect to f,a,g is uniform on [O,h]). Let x0 denote the subspace of the uniform space C x A x G0 consisting of all triples (f,a,g) such that the solution of (21.3) is defined on the hal £-line ID.+. Let (f ,a ,g) c Xo. Then we can associate with an equation of the form (21.3) an extension p:(X 0,ID.+,1l)-+ CY 0 ,1R+,o) as follows.

(21. 4)

Define

((f,a,g) cXo,T :> O), (21.S) where (STf)(t)

=

f(t + T) + jTa(t + T,s)g(x(s),s)ds,

(21. 6)

0

aT(t,s) gT(x,s)

=

=

a(t + T,S + T),

g(x,s + T),

( t ,s, T clli.+ ;x clli.n),

(21. 7) (21. 8)

and x(s) = x(f,a,g,s) is the solution of Equation (21.3). Let Yo be the image of x 0 under the canonical projection p:C x Ax G0 -+ Ax G0 . It is clear that the equalities (21. 7) and (21. 8) determine a transformation semigroup (Yo,JR+,o). Let us show that (21.S) determines a transformation semigroup (Xo,ID.+,n). The identity axiom is evidently satisfied. The homomorphism axiom follows from the equalities

f(t+q+T2)+

JTl0 a(t+T1+,2,s)g(x(s),s)ds

EXTENSIONS

242

To prove the continuity axiom, we shall make use of Lemma 4· 20.4. The continuity of the map (21.6) and consequently of the map rr at T em+ is evident. Let x(s) = x(f ,a ,g ,s), T > 0 and let K be the closure of the spherical neighbourhood with radius one of the set {x(s)!o'< s < 2T}. If the point (f1',a''',g''')eXo is sufficiently near (f,a,g), then x1 0 and a segment I2 CIB. such that l[f(x,t) - f(x,t +

T)I!


(1) is essentially the same as the proof of Lemma 4•22.10. Evidently (1) and (3) are equivalent. I Now we shall present some simple conditions that ensure the existence of synchronous and uniformly synchronous solutions of the equation (22.3). 4•22.13 LEMMA: If cp is a compact solution of (22.2) and H(cp) does not contain other compact solutions of this equation (in particular, if there is no solution w distinct from cp and satisfying the condition w(JR) C cp(JR)), then cp is a synchronous solution.

Proof: The fibre of the map p:Hk(f,cp) 7 Hk(f) over the point f consists of pairs (f,w) such that f = limf, and w = limcp, for n

n

some sequence {,nln = 1,2, ... }. Therefore w is a compact solution of the equation l22. 3). The hypothesis of the lemma implies that the map pis one-to-one at the point (f,cp). Hence cp is synchronous by Lemma 4·22.10. I 4•22.14 COROLLARY: If cp is the unique compact solution of (22.3), then it is synchronous. 4•22.15 LEMMA: Let cp be a compact solution of (22.3), K= cp(JR) and let f:KxJR 7 Ebe Lagrange stable. If for every function g e Hk(f) the equation

dx dt - g(x,t)

(22.4)

EXTENSIONS

250

has no more than one solution 1/; with 1/; e H(cp) (in particular if this equation has no more than one compact solution), then cp is a uniformly synchronous solution.' Proof: It follows from the hypotheses of the lemma that p :Hk Hk(f) is a bijective map. To conclude the proof we refer to Lemma 4•22.12. I (f,q,)-+

4•22.16 We shall say that the compact solutions of the equation (22.3) are POSITIVELY SEPARATED SOLUTIONS if for each compact solution cp and every solution 1/; satisfying the conditions 1/; e H( cp), 1/; t cp, there exists a number !l > O, such that II cp(t) - ij;(t) II ?> !l for all t ;;, 0. In the case of non-homogeneous linear differential equations the above property is equivalent to the following condition on positive separation from zero of the non-trivial compact solutions of the corresponding homogeneous equation: for every such solution cp :IR-+ E there exists a number !l > O such that II (J)(t) II ; ,. for all t ;;, o.

!l

4 • 22. 17 THEOREM: Let (J) :JR -+ E be a compact solution of Equation (22.3), K = (J)(lR) and let the function feC(KxJR,E) be recurrent in the sense of Birkhoff. If cp is the only compact solution of the equation (22.3) and every equation of the form (22.4) with geY = Hk(f) satisfies the condition on separation of compact solutions in the positive direction, then cp is a uniformly synchronous recurrent (in the sense of Birkhoff) solution.

Proof: By hypothesis, Y is a compact minimal set. According to Lemma 4 21.10, the space X = Hk(f,cp) is also compact. Therefore X contains some minimal set M. Let us prove that X = M. Since Mp = Y, there is a pair (f ,ij;) e M. Clearly ij; is a compact solution of the equation (22.3), and hence ij; = cp by hypothesis. Thus X =Mand the map p:X-+ Y is injective at the point (f,(J)) Hence it follows from Lemma 3•12.7 that pis a proximal extension. On the other hand, the positive separation of compact solutions implies that the extension p:(X,1R+)-+ (Y,1R+) is distal. By Lemma 4•22.4 the extension p:(X,1R)-+(Y,JR) is also distal. This is possible only when p:X-+ Y is a bijective map. By Lemma4·22.12, (J) is a uniformly synchronous solution. I 4•22.18 COROLLARY: Let cp:JR-+ Ebe a compact solution of the equation (22.3), K = (J)(lR) and let the function f:KxJR-+ E be almost periodic in t (uniformly with respect to x e K). Suppose further that (J) is the only compact solution of (22.3) and every equation of the form (22.4) with geHk(f) satisfies the condition on positive separation of compact solutions. Then cp is a uniformly synchronous almost periodic solution. 4•22.19

Consider the Linear non-homogeneous differential equation

251

AND EQUATIONS

x ' = A ( t )x + f ( t

),

(22.5)

where A e C(JR., L(E, E)), (L(E, E) is the normed ring of continuous linear maps of E into E, and f e C(JR.., E). It is useful to consider at the same time the corresponding homogeneous equation X' = A(

t

(22.6)

)x.

For each point xo e E there exists a unique solution cp :lR + E of the equation (22.5) with the initial c0ndition cp(O) = xo, and this solution depends continuously on (A,f,x 0 ). Therefore we can construct an extension of dynamical systems associated with Equation (22.5) in the way it has been done in Subsections 4•21.2. Let Y denote the closure of the family { (AT ,fT) T eJR} in the cartesian product C(JR.,L(E ,E)) x C(JR.,E). Set X = Y x E. The maping (B,g,x)rcT

(TeJR, (B,g)eY,

xeE),

where cp(t;B,g,x) is the solution of the equation

x' = B(t)x + g(t), with the initial condition cp(O;B,g,x) = x determines a dynamical system (X,]R,rc). The projection p:X + Y, (B,g,x) + (B,g), is a homomorphism. 4• 22. 20 THEOREM: If (22. 5) has a compact solution cp: JR+ E and (22.6) does not have non-trivial compact solutions, then cp is a synchronous solution

Proof: This follows from Corollary 4•22.14, since cp is the only compact solution of the equation (22.5). I 4•22.21 THEOREM: If the functions AeC(JR,L(E,E)) and feC (JR.,E) are Lagrange stable and Equation (22.5) has a compact solution cp: JR-+E, whilst each equation of the form x' = B(t )x,

(22.8)

with Be H(A) has no non-trivial compact solutions, then cp is a uniformly synchronous solution. Proof: This follows directly from Lemma 4•22.15. 4•22.22

I

Let V(t) be the Cauchy operator of the Equation (22.6) and let U(t,T) = U(t)u-l(T) be the evolution operator. Following Daletskii and Krein [1], we shall say the equation (22.6) satisfies the condition of [REGULAR] EXPONENTIAL DICHOTOMY on the real line JR an brief, Equation (22.6) is e-DICHOTOMIC on JR) if for some (and hence for every) number t 0 eJR the space E can be decomposed into a direct sum of closed linear subspaces

EXTENSIONS

252

(22.9) so that (a) if x 1 (t) = U(t,t 0 )x 1 ° is a solution of (22.6) with x1° x1Cto)eE1(to), then

for some numbers N1

O, V1

>

>

O;

(b) if x2(t) = U(t, o)x2°,tx2(to)

for some N2

>

O, V2

>

x2°eE2(to), then

O;

(c) if EJ O and t > O there bers o > O and L > O such that if for some :\, I; eJR

sup

IIA ( t

-t-

:\)

-

A(t

-t-

i;)

sup

11

f( t

-t-

;\)

-

f( t

-t-

i;) II
0 there is a number o > 0 such that if llx1 - x2 < o, x1,x2eK and geH(f), then the solutions cp1(t), cp2(t) of

II

the Equation (23.2) with the initial conditions cpi(O) = Xi (i = 1,2) satisfy the inequality II cp1(t) - cp2(t) II < E: for all t ;;, 0, Let X = H(f) x]Rn. Define a dynamical system (X ,;IR) as was described in Subsection 4•21.2. Then the natural homomorphism p:(X,;IR+) + (H(f),;IR+) is an equicontinuous extension. 4·23.6 THEOREM: Consider Equation (23.1) in the case when n = 2. Suppose that this equation is uniformly positively stable and its right side f e c(!Rn xJR,;IR) is almost periodic. If this equation has at least one solution which is bounded on JR+, then it has an almost periodic solution cp(t) (which is not necessarily synchronous).

Proof: If the hypotheses of the theorem are satisfied, then all solutions are bounded fort;;, 0 and the fibres of the extension are homemorphic to the planelR.2. The required result follows now from Theorem 4•23.4(2) and Subsection 4•23.5.I 4·23.7 THEOREM: Consider the Equation (23.1) with n = 3. Suppose that this equation is uniformly positively stable and that the right side f(x,t) is almost periodic int (uniformly on x varying in every bounded set). If all the solutions of (23.1) are bounded on the entire axis, then there is at least one almost periodic solutoin of this equation. Proof: This follows directly from Theorem 4•23.4(3).1 4•23.8 THEOREM: Let Ebe a Banach space, Wan open subset of E, f: Wx]R + E a continuous mapping, cp :JR + W a compact solution of the equation

dx dt = f(x,t),

( x e W,

t eJR.),

(23. 3)

and K = cp(!R). Assume further that the function f(x,t) is almost periodic in t elR (uniformly relative to x e K), H( cp) contains finitely many solutions of (23.3) and each equation of the form dx

dt

=

g(x,t),

(x

e W, t eJR.),

(23.4)

where g e HK(f), satisfies the condition on postive separation of compact solutions 1jJ eH(cp). Then cp is an almost periodic solution. Proof: Apply Theorem 3•17.12 to the extension p:HK(f,cp) + HK(f). By hypothesis, HK(f) is an almost periodic minimal set, the fibre over the point f consists of a finite number of points and the extension pis distal in the positive direction. By

265

AND EQUATIONS

Lemma4•22.4,p is distal in both directions. It follows that HK(f,~) is a compact minimal set. By Theorem 3•17.12,HK(f,~) is almost periodic, and hence the function~ is also almost periodic. I 4•23.9 COROLLARY: Let equation

~:JR+

Ebe a compact solution of the

x' = A ( t )x + f ( t \

(23.5)

where A e C(JR,L(E ,E)) and f e C(JR,E) are almost periodic functions. Assume further that each equation of the form x' = B(t)x~ uJhere Be H(A), satisfies the condition on positive separation from zero of non-trivial compact solutions and H(~) contains only finitely many solutions of Equation (23.5). Then~ is an almost periodic solution. 4•23.10 Let X and Y be metric spaces, (X,m.) and (Y,lR) be flows, and let p: (X ,m.) + (Y ,lR) be an extension. We shall say that a set ACX is UNIFORMLY STABLE IN THE POSITIVE DIRECTION RELATIVE TO THE EXTENSION p if for each numbers> 0 there exists a o = o(s) > o such that p(at,xt) < s for all t ~ O whenever aE A, xeX, p(a,x) < 6 and ap = xp (where p denotes the metric in X). 4•23.11 LEMMA: If the set A is uniformly stable in the posititive direction relative to the extension p and p:X + Y is open at all points of the set (Ap)p-1, then A has the above stability property.

Proof: Givens> o, choose a number o = o(s) according to the definition of uniform stability of the set A in the positive direction relative to p. Let us prove that if a e A, x e X, ap = xp and p(a,x) < (l/3)o(s/3), then p(at,xt) < s for all t ~ 0. Suppose that to the contrary there exist ao e A, xo e X and to > O such that aop = x 0 p, p(a 0 ,x 0 ) < (1/3)o(s/3) and p(a 0t 0 ,x 0t 0 ) ~ s. Choose a number 61 so that p(xoto,x1to) < s/3 whenever p(xo,x1) < 61. Since the map p is open at the point xo e (Ap )p- 1 , there exists a number 62 > 0 such that for every point a' in the 62-neighbourhood of ao there is a point x' e X with x' p = a' p, p (xo ,x' ) < 01, Suppose that moreover 62 is small enough so that p(aoto, a'to) < s/3, whenever p(ao,a') < 62, Suppose further that 02 < o1 < (l/3)o(s/3). Since a 0 eA, there is a point a1eA belonging to the 62-neighbourhood of ao, Then p(aoto,a1to) < s/3 and there exists a point x1 e X for which a1p = a1p and p(x1,xo) < 61. Thus p(x1to,xoto) < s/3. Since p(x1,a1) < p(x1,xo) + p(xo,ao) + p(ao,a1) < o(s/3), it follows that p(x1to,a1to) < s/3, by the choice of o(s/3). Hence p(x 0t 0 ,a 0t 0 )

~

p(x 0t 0 ,x 1 t 0 ) + p(x 1 t 0 ,a 1 t 0 ) + p(a 1 t 0 ,a 0t 0 )


0 such that for every£> O there is a number L(c) > 0 with the property that a e A, z e X, ap = zp and p(a ,z) < o0 imply p(at ,zt) < £ for all t ~ L(c). 4•23.13 LEMMA: Let ACX. If the extension pis open at all points Xe (Ap)p-1 and the set A is uniformly positively attracting relative to the extension p, then so is A.

Proof: The proof is similar to that of Lemma 4•23.11 and therefore it will be omitted.I 4•23.14 An extension p:(X,JR)

+ (Y,JR) is said to be POSITIVELY (NEGATIVELY) LOCALLY PROXIMAL IN THE FIBRE x0 = {x \ x e X, xp = yo} OVER A POINT yo e Y if for every point x GXo there exists

a number ox> 0 such that p(x,x') < ox and x'p = xp imply that the points x and x' are positively (negatively) proximal, i.e., inf p(xt,x't) = 0 (inf p(xt,x't) = 0). t>O t O according to the local proximality condition. Select a finite subcovering of the set Mo from the covering {S(x,ox) \x eMo}. Clearly each subset of the form S(x,ox) nMo consists of the single point x. Hence Mo is a finite fibre and since p:(M,JR) + (Y,JR) is a distal minimal extension, all the fibres consist of the same number of points. Moreover, the map p:M + Y is open (Corollary 3•12.25). Consequently pis a local homeomorphism.I 4•23.16 THEOREM: Let p:(X,JR) + (Y.JR) be an extension, (Y,JR) an almost periodic minimal transformation gr:Z!:P_and xo e X a point such that xo:JR+ is compact in X (i.e., xo:JR+ is compact) and uniformly stable in f!:!:§_positive direction relative to the extension p. If p:(xoJR\JR) + (Y,JR) is locaUy proximal in a certain direction at least in one fibre and p:X + Y is open___gJ_aU points x e (xopJR+)p-1, then there exists a point zo exoJR+ such that zo:JR is an almost periodic minimal set.

Proof: The compactness Gx 0JR+ such that z0JR is the set xo:JR+ is uniformly tive to the extension p. (zoJR,JR) + (Y,JR) is distal

z0

of xoJR+ ensures the existence of a point a compact minimal set. By Lemma 4•23.11, stable in the positive direction relaThis implies that the extension p: in the negative direction and thus, by

267

AND EQUATIONS

Lemma 4•22.4, it is distal in both directions. By Theorem 4•23.15, the fibres of this extension are finite. By Theorem 4•17.12, the transformation group (zcjlR,:m.) is almost periodic.I 4•23.17 A

POINT x eX is called UNIFORMLY ASYMPTOTICALLY STABLE RELATIVE TO THE EXTENSION p: (X,:m.) -+ ( Y,:m.) if the set xJR+

is uniformly stable in the positive direction relative top and uniformly attracting relative to p. A point x e Xis called a POSITIVELY ASYMPTOTICALLY ALMOST PERIODIC POINT, if there exist an almost periodic minimal set M C X and a point a e M such that lim p(xt,at)

= O.

t-++oo

4•23.18 THEOREM: Let p:(X,:m.)-+ (Y,:m.) be an extension, where (Y,:m.) is an almost periodic minimal transformation group. Let xo e X be a uniformly asyl'fff!J.2!_icaUy stab le point re iative t~ p, such that th~ set xo:m.+ is compact+and the map p: X-+ Y ~s open at all po~nts of the set (xop:m. )p-1, Then the point xo is positively asymptotically almost periodic.

Proof: By Lemmas 4•23.11 and 4•23.13, the set xo:m.+ is uniformly stable in the positive direction and uniformly attracting relative to p. Therefore the extension p:(x 0:m.+,:m.) -+ (Y,:m.) is locally positively proximal in every fibre. By Theorem 4•23.16, there exists an almost periodic minimal set MC xo:m.+, By Lemma 4•22.29, there is a point a e M such that ap = x 0 p and the points a and x 0 are proximal. Let£ be an arbitrary positive number. Choose oo and L(E) as required by the condition that xo:m.+ is uniformly attracting relative top. Since the points a and xo are proximal, there is a number to e:R with p(xoto,ato) < oo, Then p(xot,at) < £ for all t ;i, to+ L(d. This means that x 0 is a positively asymptotically almost periodic point.I 4•23.19 Consider the equation dx dt = f(x,t)

(23.6)

in a Banach space E, where f is assumed to be continuous on Wx:m. (Wis an open subset of E). Suppose that for each equation of the form dx dt = g(x,t),

(23. 7)

where gaH(f) eC(Wx:m.,E), and for each point z eW there exists a unique solution ~::m.-+ W with the initial condition ~(O) = z and that this solution depends continuously on the initial point and on the right side g e H(f). Define the dynamical system (H(f) x W,JR) as was done in Subsection 4•21.3, Let p:H(f) x W-+ H(f) be the canonical projection. Clearly pis open.

EXTENSIONS

268

Recall the following definitions. A solution cp:]R+ +Wis said to be uniformly etable in the positive direction, if for every £ > 0 there exists a o > 0 such that if t 0 ~ 0 and a solution 1/J:lR+ + W of (23.6) satisfies the condition l\1/J(to) - cp(to)\\ < 8, then lli/J(t) - cp(t)II 0 such that for every£> Owe can choose an L > 0 so that if lli/J(to) - cp(to)\\ < 80 for some to eJR+, then \11/J(t) - cp(t)\I O there exists a number 8 > 0 such that if a function F:W x]R+ + E is continuous and satisfies the condition \IF (xlt)II < 8(xeW, t?: O) and yoeW is any point with llcp(t 0 ) Yo I< 8, then for each solution y(t) of the equation

y'

=

f(y,t) + F(y,t),

with the initial condition y(to) = Yo the inequality llcp(t) y(t)II 0 there exist numbers 8 > O and£> 0 such that if geHk(f) and llg(x,t) f(x,t + T) II< 8 for some T ?: O and all x eK, ltl ~ £, where iµ(t) is a solution of (23.7) with 111/J(O) - cp(T)I\ < 8, then \\iµ(t) cp(t + T)\\ 0 such that for every solution ;>,.::JR-+ Ko of Equation (23.7) with 11;>,.(o) - 1/J(O)II < oo there is a sequence {tk}, {tk}-+ for which 111/J(tk) A ( tk ) 11 -+ 0 • Then there exists an almost periodic solution ~::JR-+ Ko of Equation (23.6). 00

Proof: By hypothesis (1), (H(f),:JR) is an almost periodic minimal transformation group. It follows from Condition (2) that the closure of the positive semiorbit of the point (f,~(O)) e H(f) x W is compact and uniformly stable in the positive direction relative to the extension p. Condition (3) implies that the extension is positively locally proximal in the fibre over the point f. Thus by Theorem 4•23.16, H(~) contains an almost periodic solution ~::JR-+ Ko of Equation (23.6).1 4•23.21 THEOREM: Let f(x,t) be almost periodic int, uniformly in x e K for every compact set KC W. Suppose that ~::JR+ -+

Wis a compact uniformly asymptotically stable solution of Equation (23.6). Then the function~ is positively asymptotically almost periodic. Proof: This is an immediate consequence of Theorem 4•23.18.I 4•23.22 The following examples illustrate Theorems 4•23.20,21. Let {wn} be a sequence of positive numbers satisfying 00

the condition

l

nwn


0). Then each solution \:T-+ a: t6T of (23.9) is almost periodic.

Proof: Since the discriminant is separated from zero, the extension p:H(\,a)-+ H(a) is distal. Clearly, all the fibres of this extension are finite. To conclude the proof we refer to Theorem 3•17.12. I Now we shall give an example which shows that in Theorem 4•23.25 the hypothesis infJD(t)J > O cannot be replaced by the condition D( t) + 0 (t 6 T) . Moreover, we shall ~roduce a Bohr almost ~eriodic function f:JR-+ a: such that Jf(t) I > O (t..,, R), inf Jf(t)j = O, but the solutions of the equation \2 - f(t) =Oare not Levitan almost periodic. We precede this with a proof of the following lemma. 4•23.26 LEMMA: Let f:JR-+ a: be a Bohr almost periodic function with Jf(t) J > 0 (t 6:IR), infJf(t) J = O. Suppose that each function g e H(f) vanishes at no more than one point. Then Levitan almost periodicity of the function u(t) = /f(t) (t .;JR) is Bohr almost periodicity.

Proof: Here /f(t) means one of the two continuous solutions of the equation u2 - f(t) = o. Suppose that u(t) is a Levitan almost periodic solution. This means that there exist a compact commutative group G, a continuous homomorphism a:JR-+ G onto a dense subgroup G1 = {a( t) Jt 6:IR} of G and a continuous function v:G 1 -+ a: such that v(a(t)) = u(t). Moreover we may suppose that there exists a continuous function F:G-+ a: such that F(a(t)) = f(t) (t t:IR). Leto= Jf(o)J/2 > o. Choose a sufficiently small neighbourhood U of the identity e = a(O) of the group G such that

o (g t U) ; (2) If ~:U-+ a: is a continuous branch of the function IF(g), (1)

J

F (g )

J

>

then J~(gi) + Hg2)J ~ o for all g1,g2 eU.

Since u(t) = v(a(t)) (t 6:IR) and the function v:G1-+ a: is continuous, there exists a neighbourhood U1 C U of the point e e G, such that Ju(O) - u(t)J = Jv(e) - v(a(t))! < o/2 whenever a(t) e u1 . It follows from Condition (2) that the function v(a(t)), when restricted to U1, agrees with one of the branches of IF(g). To be precise, let:

v(a(t))

=

~(a(t)),

This means that the function u(t) = v(a(t)) can be extended by continuity onto U1,

272

EXTENSIONS

Let us show that such an extension is possible on the whole space G. Let g e G. We shall prove that if fo(tnn -+ g, then {v(aCtn))} is also convergent. Suppose that, to the contrary, there exist two subnets {t~} and {t;} such that lim v(a(t~))

+ lim v(a(t~)). n-+oo

The function F(ga(t)) belongs to H(f), and therefore it vanishes at no more than one point. Hence we may suppose that F(ga(t)) + 0 fort> 0. There exists a number,> 0 such that ga(,) eU 1 . Since u(t) = v(a(t)) = (a(t)) for a(t) eU1 and limfo(T + tn)} lim{a(tn)·a(T)} = ga(T), we have lim{v(a(T + tn))} = lim{v(a(T + t~))} n-+oo

= lim{v(a(T

+ t~))}

= (ga(T)).

n-+oo

+

Since F(ga(t)) 0 fort> 0, v(a(T + t~)) there exists a number y > 0 such that:

+ v(a(T

+ t;)) and

\v(a(T + t~)) - v(a(, + t;))\ ~ y for all sufficiently large subscripts n. But this leads to a contradiction. Thus the function u(t) = v(a(t)) can be extended by continuity onto G. Hence u(t) is a Bohr almost periodic function. I 4•23.27 We shall construct now a Bohr almost periodic function f:JR-+ ~ which satisfies the hypotheses of Lemma 4•23.26 and which has the property that the function u(t) = lf(t) is not Levitan almost periodic. Let G be the two-dimensional torus, a:JR -+ G a dense one-parameter subgroup and (G,JR,o) the flow defined by

(g,t)o

=

ga(t),

(t sJR,g e G) •

Let g1, g2 be two points of the torus with distinct orbits. Each continuous function F:G-+ ~ can be considered in a natural way as a vector field on G. Define a function F:G-+ ~ so that: (1) F(g1) = F(g 2 ) = o; F(g) O, if g g1, g g 2 ; (2) the degree of the singular point gl of the vector field Fis equal to 1 and the degree of g 2 is equal to (-1). It is easy to see that such functions exist. Let go be a point of G outside the orbits of the points gl,,~2 in the dynamical system (G,JR,o). The function u(t) = IF(goa(t ) is bounded and uniformly continuous. Let X = H(F(g 0a(t),u(t)), i.e., Xis the orbit closure of the pair of functions F(g 0a(t)) and u(t) in the corresponding shift transformation group. There

+

+

+

AND EQUATIONS

273

exists a homomorphism p of the flow (X,:JR) onto the flow (G,:JR,o) which sends the point (F(goa(t)),u(t)) to the point g 0 . The fibre Xo over the point go consists of no more than two points (which correspond to different branches of the function IF(g 0 a(t))). It follows from the choice of the point g 0 and from property (1) of the function F that if x0 contains two points, then they are distal. Therefore Xis a minimal set. We shall prove that the fibre Xo consists in fact of two points. Suppose the contrary holds. Then p:X + G is an almost automor-· phic extension; hence (F(g 0 a(t)),u(t)) ·is Levitan almost periodic and we deduce from Lemma 4•23.26 that u(t) is Bohr almost periodic. Hence it follows that Xis an almost periodic minimal set, each fibre consists of a single point, and the function /F(g 0 a(t)) can be continuously extended to the whole torus G. But this is impossible because the increase of the argument of the function F after a turn around the point g 1 along a small curve is equal to 2rr. This contradiction shows that the fibre x 0 consists of two distal points, and therefore (F(g 0 a(t)),-u(t)) sX. Since Xis a minimal set, we conclude that (k(t),9.(t)) e.X implies (k(t) ,-9.(t)) e.X. Identifying such pairs of points we obtain a minimal set y,•,. The canonical map s:X + Y* is a twofold covering. The projection on the first coordinate determines a homomorphism r:(Y*,:JR) + (G,:JR,o). Clearly, r is an almost automorphic extension. Since the function u(t) is not Bohr almost automorphic, the map r is not injective. Thus the extension p:(X,:JR) + (G,:JR) is decomposed into a twofold coverings and an almost automorphic extension r. The last statement is in accordance with the theorem on the structure of extensions with a finite fibre (Theorem 3•13.18 and Remark 3•13.21).

REMARKS AND BIBLIOGRAPHICAL NOTES Subsections 4•23.1-9 contain an exposition of results due to Zikov [2] (with some generalizations and improvements). In connection with Theorems 4•23.6,7 it is natural to suggest the following conjecture: if an equation dx

dt

=

f(x,t),

(xs JR.TI, t e, JR)

with right side almost periodic int satisfies the conditions on uniform positive stability and positive dissipativity, then it has at least one almost periodic solution. As was shown by Fink and Frederickson [1], dissipation alone does not imply the existence of almost periodic solutions. At the same time it is well known that in the periodic case, dissipativity ensures the existence of a periodic solution. It follows from the dissipation condition that x 0w is a retract of some ball of:JRn. By Theorem 4•23.2, in order to prove the above conjecture it will suffice to prove the following statement: each compact connected topolog-

274

EXTENSIONS

ical group G of homeomorphisms of a compact contractible subset It is easy to verify that this statement is true for commutative groups, but in the general case the answer is unknown to the author. Conner and Montgomery [1] have shown that the group S0(3) can act onJRl2 without fixed points. This shows that if the assumption on compactness of A is dropped, the statement becomes false. Theorem 4•23.8 was proved by Amerio [1,2]. The material presented in 4•23.10-18 is taken from the paper by Bronstein and Cernii [1]. The relationship between various types of stability and the existence of almost periodic solutions of differential equations was studied by many authors (Deysach and Sell [1], Fink [1,2], Fink and Seifert [1], Miller [1], Seifert [1-6], Yoshizawa [1-3] and others). Their proofs are based either on considerations similar to the one at the end of Subsection 4•23.19 or on the fact that uniform asymptotic stability implies (under some additional conditions) total stability (Barbasin [3], Kato and Yoshizawa [1]). The proofs of Theorems 4•23.20,21 were found independently by Zikov. Relations between almost periodic differential equations and almost periodic flows were investigated by Cartwright [1-3]. Examples presented in Subsection 4•23.22 are due to Bohr [2]. Asymptotically almost periodic functions are introduced by Fr§chet [1]. Theorem 4•23.24 was proved by Bronstein [19]. Theorem 4•23.25 is the main result of the papers by Walters [1] and Bohr and Flanders [1]. The proof of this theorem based on the theory of distal extensions is due to Zikov [2]. So are Lemma 4•23.26 and Example 4•23.27.

A of the space JRD has a fixed point.

§(4•24): LINEAR EXTENSIONS OF DYNAMICAL SYSTEMS AND LINEAR DIFFERENTIAL EQUATIONS We shall introduce here and study linear extensions of semigroup dynamical systems. We shall extend to such extensions the well known minimax method due to Favard. Several general theorems on the existence of proximal and almost periodic extensions will be proved. These lead to some corollaries on synchronous solutions of linear differential equations. We investigate conditions under which the primitive of a recurrent (in particular, almost periodic) function is isochronous with the given function. Results due to Bohl, Bohr and Kadets on this subject will be presented. 4·24.1

Let X be a topological space, Ea linear topological space over the field of real or complex numbers and 0: X x E -+ X a continuous action of the additive group E on X. Henceforth it is assumed that the action 0 is free, i.e., xaa x for a t O and al 1 x e, X. Denote R = {(x,xaa)lxe,X,ae,E}. If x1,x2 e,R, then there is a uniquely defined element n (x 1 ,x2) e, E satisfying the condition

+

AND EQUATIONS

275

The function n:R +Eis called the shift map. Let Y = X/R = {xElx eX} be the space of E-orbits, i.e., the quotient space of X with respect to the equivalence relation R, and let p:X + X/R denote the canonical projection. Then (X,p,Y) is called an EEXTENSION. If the shift map n:R +Eis continuous, then the Eextension is called a PRINCIPAL E-EXTENSION. Each fibre Xy = {x Ix e X ,xp = y} (y e Y) is in this case homeomorphic to the space E. Indeed, if xo is any fixed point of the fibre Xy, then the map x 1+ n(xo ,x) (x e Xy) is a homeomorphism. Let (X,p,Y) be a principal E-extension, Sa topological semigroup with identity e and (X,S,11) a transformation semigroup satisfying the condition (x e X ,s e S) .

(24.1)

Since p:X + Y is an open map, (X,S,11) induces a transformation semigroup (Y,S,p). For xeX, seS and aeE there exists by (24.1) an element F;,(x,s,a) eE such that (24.2) Indeed, f;,(x,s,a) = n(x11 8 ,xaa11 8 ). It is easily seen that F;,:XxSx +Eis a continuous map. For each element b e E we have by (24. 2) that

E

(24. 3) On the other hand,

Since the action a is free, equating (24.3) and (24.4) yields the identity

f;,(x,s,a + b)

=

F;,(x,s,a) + f;,(xaa,s,b),

(xeX,seS;a,beE). Consider the special case when

F;,(x,s,a) = f;,(xab,s,a), Then we can define a function r;, 1 : Y

l;,i(y,s,a)

=

f;,(x,s,a),

(x eX,s eS;a,b eE). x S xE

+ E:

(24.5)

EXTENSIONS

276

It follows from the continuity of~ and the openness of the map p :X -+ Y that ~1 is also continuous. For y e Y and s e S define a map A(y,s):E-+ Eby

A(y,s)a

=

~1(y,s,a),

(a e E).

(24 ;5) implies the following identity:

A(y,s)(a + b)

=

A(y,s)a + A(y,s)b, (24.6)

(a, b e E; y e Y, s e S) .

If Eis a linear space over JR., then it follows from the continuity of A(y,s):E-+ E and from (24.6) that A(y,s):E-+ Eis a linear map (ye Y,s e S). If E is considered over the field of complex numbers, we also have to require that

A ( y , s ) ( ia)

=

iA ( y , s ) a ,

(y e Y, s e S, a e E) ,

where i is the imaginery identity. linear map and

Then A(y,s) is a continuous

(yeY,xeXy,seS,aeE), in view of (24.2).

(24.7)

Hence it follows that

(ye Y;s1,s2

e S).

(24. 8)

If Sis a group, then also the following identities hold:

A(y,e)

=

I,

[A(y,s)]

-1

=

A(yps,s

-1

),

(y e Y, s e S) , ( 2 4 . 9)

where e is the identity of the group Sand I is the identity map of E. Let L(E,E) denote the set of all continuous linear maps of E into E. If there is a map A: Y x S -+ L(E ,E) satisfying the identity (24.8), then p:(X,S,n)-+ (Y,S,p) is called an E-LINEAR EXTENSION. Since the map (y,s,a) f+ A(y,s)a is continuous and satisfies (24.8), the map (y,a)n1 8

=

(yp 8 ,A(y,s)a),

(yeY,seS,aeE)

(24.10)

determines a transformation semigroup (YxE,S,n1). The extension q:(YxE,S,n1)-+ (Y,S,p), where q:YxE-+ Y denotes the canonical projection, is E-linear and it is called the HOMOGENEOUS LINEAR EXTENSION CORRESPONDING TO THEE-LINEAR EXTENSION p:(X,S,n)-+ (Y,S,p). This agrees with the terminology adopted in the theory of linear differential equations. Yet, it would be more precise

277

AND EQUATIONS

to use the term affine for linear extensions and the term linear for homogeneous linear extensions. We shall now consider an important particular case of Eextensions. Let (Y,S,p) be a given dynamical system, X = YxE and let p:X --r Y be the canonical projection. Suppose that there is defined an E-linear extension p:(X,S,n) --r (Y,S,p) with the natural action of the additive group E on X = Y x E: 4•24.2

(y,a)ab = (y,a + b),

(ye Y;a,b e E).

The equality (y,O)nS

=

(ypS,f(y,s)),

determines a continuous map f: Y identity holds:

x S --r E

(ye Y ,s e S)

(24 .11)

such that the following

(24.12)

It follows from (24.11) and (24.12) that (y,a)n 8 = (yp 8 ,f(y,s) + A(y,s)a), (y e Y; a e E, s e S) .

(24.13)

Conversely, let f:YxS --r E and A:YxS --r L(E,E) be functions satisfying the equalities (24 .12) and (24. 8) . If the maps (y ,s) f+ f(y ,s) and (y ,s ,a) f+ A(y ,s )a (y e Y ,s e S ,a e E) are continuous, then the formula (24.13) determines a transformation semigroup (YxE,S,n) such that q:(YxE,S,n) --r (Y,S,p) is an E-linear extension. The latter will be called a SKEW PRODUCT EXTENSION. 4•24.3 LEMMA: Let Ebe a quasi-corrrplete linear topological space, p:(X,S,n) --r (Y,S,p) an E-linear extension and let xo e X. Suppose that there exists a non-errrpty compact set KC Y such that Kp- 1 nxoS is corrrpact. Let F(y) denote the closed convex hull of the set xoS n Xy in the affine space XyThen the set F = LJF(y) is corrrpact and F(y)SnXy C F(y) yeK ( y e Kn x 0 Sp). Proof: Let y e K and let U be an arbitrary neighbourhood of the set F(y). Let us prove that there exists a neighbourhood V of y such that F(z) c U (z e VnK). Suppose that, to the contrary, then there exists a net {ya} such that {ya} C K, {ya} --r y, but

for some fixed neighbourhood Vo of F(y).

Choose arbitrary points

EXTENSIONS

278

Xa ex 0snxYa C x 0Sr1Kp- 1 .

Without loss of generality we may as-

sume that the net {xa} converges to some poii:i:_t x,·, e xoS n Xy. Let N(x) = {n(x,z)[zexoS,xp = zp} (xexoSr'iKp-1). The set N(x) is compact because it is homeomorphic to the compact set x 0 SnXXR' Since the space E is quasi-complete the closed convex hull coN(x) of N(x) in Eis compact (Bourbaki [1]). Further, it follows from the definition of the set F(y) that (x,coN(x))o

=

F(xp),

For the neighbourhood u0 of F(y) we can find a convex neighbourhood W of the zero element of the space E and a neighbourhood Vo of the point x''' such that (z,coN(x''') + W + W)oc Uo, Hence for sufficiently large a: Cxa ,coN(x''') + W + W)o C u0 . Since {xa}

-+

x''' and the set x 0 Sr'iKp-1 is compact, we have N( Xa) C Hence it follows that

N(x*) + W for all sufficiently large a. coNCxa) C coN(x''') + W. Therefore

coNCxa) C coN(x''') + W + W. Thus there exists an index a 0 such that

for all a> a 0 . But this contradicts (24.14). Thus for every point y e K and every neighbourhood U of F(y) there exists a neighbourhood V of y such that F(z) C U for all z "vnK. Since the set K is compact and each fibre F(y) of the continuous map p:F-+ K is compact, the set Fis also compact. Since the set xoS is invariant and the semigroup S acts affinely on the fibres X , we have F(y )ITS C F(yp 8 ) (y e xoSp n K). It follows from the above considerations that F(y)snxy C F(y) for all y 6KnxoSp. I 4•24.4 COROLLARY: Let Ebe a quasi-complete linear topological space, and let p:(X,S,n)-+ (Y,S,p) be an E-linear extension. Let xo e X be such that the set xoS is compact. Then the set F = U [F(y) [ y e xopS] is compact and invariant. Proof: This follows directly from Lemma 4•24.3, if we take K to be equal to xopS. I

4•24.5 THEOREM: Let Ebe a quasi-complete linear topological space, p:(X,S,n)-+ (Y,S,p) an E-linear extension, xo eX and Yo= xoP· Suppqse {urther.that for some neighbourhood Vo of Yo, the set Vop- nxoS i-s compact. Then there exists a

AND EQUATIONS

279

point z e.F(yQ)_= co(x 0snxy 0 ~ sue~ that the extension p: (zS,S,n) + (yoS,S,p) is prox~mal ~n the fibre over the point YO· Proof: Apply Lemma 4•24.3 with K equal to the set (Vop- 1 nx 0 S)p. Then the set F = U [F(y) IY e (Vop-lnxoS)p] is compact. Let .A. denote the family of all non-empty convex compact subsets DC F(y 0 ) which satisfy the condition

By Lemma 4 • 24. 3, F(yo) e..A. It is easy to verify that the set inductive. Hence, by Zorn's Lemma, there exists a minimal element Do e.A. We shall prove that the extension p: (Dos ,S. n) + (yoS ,S. p) is proximal in the fibre over YO· Let x1,x2 e.Dosnxy 0 • Then x1x2 e

A is

Do, Let us prove that the points x1 and x2 are S-proximal. Suppose the contrary holds. Putz= ½Cx1 + x2). Let w be an arbitrary point in zS()XYO C D0 • Then there exists a net {sa} such

that w = lim{zcr 8 a}. Hence {zo 8 ap} + y 0 • Since the set Fis compact, x1,x2 e.Do C F(yo) and {x 1sap} = {x 2sap} = fasap} + Yo, we have x1sa e.F, x2sa e.F for all sufficiently large a. Therefore we may assume without loss of generality that {x1sa} + z1, {x2sa} + z2, where z1z2 are some points from Dosnxy 0 C Do. Since the semigroup S acts affinely on the fibres, w = ½Cz1 + z2). We note that z1 z2 because x1 and x 2 are supposed to be distal. Thus we have shown that none of the points w e zS n Xy 0 is extremal for

+

the set Do, It is ~nown (cf. Dunford and Schwartz [1]) that if Q is a compact subset of E and the closed convex hull coQ is compact, then the extremal points of coQ are contained in Q. Therefore the compact set co(zS()Xy 0 ) C D0 does not contain extremal points of the set Do. In fact, if xis an extremal point of Do and xe.co(zS()Xy 0 ), then xis an extremal point of co(zsnxy 0 ), and hence x e. zS () Xy 0 , which is impossible, as was shown above. But Do has extremal points (see Dunford and Schwartz [1], p.476); hence co(zS()Xy 0 ) is a proper subset of D0 . Since the set F is compact and z e.F(y 0 ), Kp- 1 nw is a closed subset of F. Hence we may apply Lemma 4 • 24. 3 to the point z. We obtain QS () XYO C Q, where Q

co(zS n Xy 0 ). Thus Q e..A, Q C Do and Q contradicts the minimality of D0 .I =

+ Do.

But this

4•24.6 COROLLARY: Suppose that aU the conditions of the preceding theorem are satisfied. Then either there exists a point z e.F(y 0 ) such that the extension p:zS + y 0S is almost automorphic or there exists a point ao e. E, ao o, such that for every neighbourhood V = V(O) we can choose an element s e. S with A(yci,s)ao e. V(O).

+

EXTENSIONS

280

Proof: If z 1 ,z 2 12zsnxy 0 and z1 There exists a net Therefore

{s 0 ) ,

t

z2, then set ao

Sae S, such that lim{z1sa}

= =

n(z1,z2), lim{z2sa}.

In the case of a skew product extension Theorem 4•24.5 can be reformulated as follows. 4·24.7 THEOREM: Let Ebe a quasi-complete linear topological space, (Y,S,p) a dynamical system, X = YxE, p:(X,S,1l)-+ ( Y, S, p) an E- linear skew product extension, Yo e Y and a 0 e E. Suppose there exists a neighbourhood V0 of the point Yo such that the set Vo ny 0 S is compact. If, moreover, there exists a compact set KC E with

{f(Yo,s) + A(yo,s)aols 12S,yos e Vo} c K, then there is a point b e coK such that the extension p:((yo,b)S,S,1l)-+ (yoS,S,p)

is proximal in the fibre over YO· If this fibre contains more than one point, then {A(yo,s)cols 12S}nV(O) t 0 for some element co12E, cot O, and every neighbourhood V(O),I Now we shall assume that Eis a uniformly convex Banach space and E 0 is the linear topological space obtained from the linear system Eby endowing it with the weak topology o = o(E,E') (Dunford and Schwartz [1]). Let X = (X,T) be a topological space and p:((X,T),S,1l)-+ (Y,S,p) an E-linear extension. Suppose further that we have a topology To on X such that p: ((X,T 0 ),S,1l)-+ (Y,S,p) is an E0 -linear extension. This situation occurs, for example, when we are considering E-linear skew product extensions p:(YxE,S,1l)-+ (Y,S,p). In this case (X,T) = YxE and (X ,T 0 ) = Y x E0 • Under the above circumstances the following theorem holds. 4•24.8

4 • 24. 9 THEOREM: Let a e X and b neighbourhood Vo of a, the set

=

ap.

Suppose that for some (24.15)

is bounded. Then there is a point c contained in the closed convex hull F of the set aS 0 nx1 (where as 0 denotes the closure of aS in the space (X,T 0 )) such that inf lln(x1IT 8 ,x2IT 8 seS

for every x1 ,x2 e csa n F.

)

II

= 0

281

AND EQUATIONS

Proof: Since the map p:(X,Ta) + Y is open, the function n: (R ,Tax Ta) + Ea is continuous and the norm in Ea is lower semicontinuous, we conclude that the set (24.16)

{n(a1,z) \a1 e Vo,z e asa ,(a1,z) e R}

is bounded. Recall that the fibre (Xb ,Ta) is homeomorphic to Ea. It follows from the boundedness of the set (24.16) that the set asa ('\ Xb is bounded (and weakly closed) , hence asa n Xb is weakly compact. By the Krein-Smulian theorem {see: Dunford and Schwartz [1]) the closed convex hull F of the set asa n Xb is weakly compact . Let x e X. Put M(x)

sup{\!n(a1,z)II la1 e Vo,z exS,(a1,z) eR} = sup { 11 n ( a 1 , z )

11

Ia 1 e Vo , z e xsa , ( a 1 , z ) e R} ,

( 2 4 . 1 7)

M=infM(x).

(24.18)

xeF

Since a eF and the set (24.16) is bounded, we have M < We shall prove that there exists a point c e F with M( c) = M. Indeed, let {xn}, Cxn e F), be a minimizing sequence, i.e., 00 •

(24.19) Since the set Fis weakly compact, there exists a subsequence of {xn} which is weakly convergent to a point c e F. We prove that M(c) = M. Suppose the contrary holds. Then there exists a number 80 >Osuch that M(c) > M + 89. We can choose an element s'''eS and a point a'''eVo so that iln(a''',cs''')II > M + 8 0 . Since the map p :xa + Y is open, the function n: (R ,Tax Ta) + Ea is continuous and the norm is lower semi-continuous on Ea, we can choose an arbitrarily large subscript n and a point an e Vo in such a way that (an ,xns''') e R and II n(an ,xns''') II > M + 80 > M + 1/n. But this contradicts (24.19). Thus we have proved the existence of a point c e F for which

Let x e csa n F. Then M( x) ;;, M, by the definition of M. On the other hand, x e csa. Hence xS° C cS 0 , and consequently M(x) ::; M(c) = M. Thus M(x}, = M for all xecs 0 nF. Let x 1 ,x 2 e cS 0 n F and let x 3 = ½Cx 1 + x 2 ). Then x 3 e F by the convexity of F. Hence M(x 3 );;, M, and thus for every positive integer n there exists a point an e Vo and an element Sn e S such that (an,X3Bn) eR and

Since the semigroup S acts affinely on the fibres, we have

EXTENSIONS

282

Therefore

Let Pn = nCan,x1sn), qn = nCan,x2sn). Then IIPnll ,:, M, M, ½IIPn + qnll = lln x,yeE, Jlxll,, M, IIYII ~ M ,::; £. Thus

0 there exists a o > 0 such that if and ½llx + YIJ ;;, M - o, then IJx - YII

(n -+ +co) .

I

4•24.10 THEOREM: Let Ebe a uniformly convex Banach space, (Y,S,p) a dynamical system, Y compact, X = YxE, p:(X,S,ll)-+ (Y,S,p) an E-Unear skew product extension, Yo eY and ao eE.

Suppose that there is a neighbourhood V0 of Yo such that the set {f(Yo,s) + A(yo,s)aoJse,S,yoseVo} is bounded. Then there is a point b e, E such that either the extension p:((yo,b)S,S,ll)-+ (yoS,S,p) is almost automorphic or some point c 0 e, E, c 0 t O, satisfies the condition inf IJA(y 0 ,s)coll = o.

se,S 4•24.11 Let Ebe a Banach space and let q:(YxE,S,1!1)-+ (Y,S,p) be a homogeneous E-linear extension. We shall say that this extension satisfies the condition on SEPARATION FROM ZERO of compact (bounded) non-trivial motions in the fibre over ye, Y if for each element 9., e,E, 9., t 0, the compactness (boundedness) of the set {A(y,s)9.,Js e,S} implies inf JIA(y,sHII > 0.

se,S It is clear how one extends this definition to the case of a linear topological space E. 4•24.12 COROLLARY: Let Ebe a quasi-complete linear topological space, (Y,S,p) a dynamical system, q:(YxE,S,1! 1 )-+ ( Y ,S, p) an E-Unear skew product extension, Yo e, Y and ao e, E. Suppose that the sets YoS and {f(Yo,s) + A(yo,s)aois e,S} are compact. If the non-trivial compact motions of the corresponding homogeneous extension are separated from zero in the fibre over the point Yo, then there is a point b e E such that the extension q:((y 0 ,b)S,S,1!1)-+ (y 0 S,S,p) is injective

283

AND EQUATIONS

at the point (yo,b). If, moreover, the set YoS is minimai and the condition on separation from zero is satisfied for aU points y e T[os, then there is a point be E such that the extension q:((yo,b)S,S,n1) + (y 0 S,S,p) is an isomorphism. Proof: The first assertion is an immediate consequence of Theorem 4•24.7. The second assertion follows from the first one by virtue of Corollary 3•12.8.I 4•24.13 COROLLARY: Let Ebe a uniformiy convex Banach space, (Y,S,p) a dynamicai system with Y compact, q:(YxE,S,n1) + (Y,S,p) an E-Unear skew product extension and Zet Yoe Y and ao eE be such that the set {f(Yo,s) + A(yo,s)aols eS} is bounded. If the non-triviai bounded motions of the corresponding homogeneous extension·are separated from zero in the fibre over Yo, then there exists a point b e E such that the extension q:(yo,b)S 0 + YoS is one-to-one at the point (yo,b). If, moreover, YoS is a minimai set and the condition on separation from zero is satisfied for aU points ye YoS, then q:((yo,h)S 0 ,S,n1) + (yoS,S,p) is an isomorphism.

Proof: This follows directly from Theorem 4•24.9.I 4•24.14 LEMMA: Let Ebe a Banach space, iet q:(YxE,S,n1) + (Y,S,p) be a homo?eneous E-Zinear extension. If the set E1 = {,e,IQ,eE,supllA(b,s),e,ij < is closed for some point seS beY, then q:((b,,e,)S,S,n 1 ) + (Y,S,p) is stabie in the fibre over the point b (Q, e E1). 00 }

Proof: Clearly E1 is a linear system and since E1 is closed, it is a subspace of the Banach space E. By the Banach-Steinhaus theorem (see: Dunford and Schwartz [1] p.52), the family {A(b,s): E1 + Else S} is equicontinuous. I 4•24.15 COROLLARY: Let Ebe a EucZidean space and Zet q: (YxE,S,rr1) + (Y,S,p) be an E-Unear homogeneous extension. If b eY and Q, eE are such that {A(b,s),e,js eS} is bounded, then the extension q:((b,,e,)S,S,rr1) + (Y,S,p) is stabZe in the fibre over the point b. 4•24.16 Now we shall apply the above results to linear differential equations. Let Ebe a Banach space, A eC("JR,L(E,E)) and f e C(JR,E). The differential .equation dx dt =

A(t)x + f(t)

(24.20)

determines an E-linear skew product extension in the following manner. Let Y = H(A,f) and let (Y,JR,p) be the shift dynamical system. The map n:YxExJR + YxE defined by (B,g,x,t)n = (Bt,gt, qi(t,x,B,g))((B,g) eH(A,f),xeE,teJR), where qi(t,x,B,g) denotes the solution of the equation

EXTENSIONS

284

dx

dt

=

(24.21)

B(t)x + g(t),

with the initial condition ~(O,x,B,g) = x, determines a transformation group (.YxE,JR,11). The canonical projection p:.YxE-+ .Y is an E-linear skew product extension. The corresponding homogeneous extension is determined by the homogeneous equations

x' = B(t)x,

(24. 22)

where B e H(A). 4•24.17 THEOREM: Suppose that the functions A and fare Lagrange stable, there exists a compact solution ~:JR -+ E of (24.20) and each non-trivial compact solution iµ of the homogeneous equation corresponding to (24.20) is separated from zero (i.e., inflli/J(t) II > o). Then Equation (24.20) has at teJR

least one synchronous solution. If the rent in the sense of Birkhoff, then the if the separation condition is replaced tion condition (i.e., infllt(t) II > O).

pair (A,f) is recursame is true even by the semiseparaIf, moreover, the

t';30

condition on semiseparation of the compact solutions is satisfied for all equations of the form ( 24. 22) with B e H(A), then (24.20) has a uniformly synchronous solution. Proof: This follows from Corollary 4•24.12 and Lemmas 4•22. 10,4,12.1

4•24.18 A solution iµ:JR-+ E of Equation (24.20) is said to be a WEAKLY SYNCHRONOUS SOLUTION if for every neighbourhood u of iµ e C(JR,E 0 ) (where E0 denotes the linear topological space obtained from the linear system Eby endowing it with the weak topology o(E,E')) we can choose a number o > O and a segment I CJR so that

IIA(t) -

A(t +

T)II


a for some a:;, 0. By Lemma 4•24.23, we may also suppose (by passing to a subsequence, if necessary) that 00

(24.26)

for all i; e Kn-1, where Kn-1 denotes the set of all possible sums of the form 6n-i·tn-l + ... + 61·t 1 , oi is equal to either O or 1

AND EQUATIONS

287

and O•t = O, l•t = t. Let tk = t 11k, (k = 1, 2, ... ) denote any terms of the sequence {tn}. We shall apply the following identity, which is a consequence of (24.24):

m-1 =

I

{F(xntk+ 1 ,tk + ... + ti) - F(x,tk + ... + ti)}.

(24.27)

k=l Taking (24.27) and (24.26) into account, we conclude that m

II I

k=l

Thus the series

I F(x,tn)

(24. 28)

F(x,tk)il~L+l.

is divergent, but the equality (24.28)

n

holds for every subsequence {F(x,tk)} and every positive integer m. It follows from a theorem due to Pelczynski [1,2] that Econtains a subspace isomorphic to~, but this contradicts the hypothesis. Define an almost periodic function ~::JR+~ by: ~(t)

= {(1/n)cos(t/n)[n = 1,2, ... }.

The primitive

F(t) =

I:

~(t)dt

=

{sin(t/n)ln

1,2, ... }

of this function is bounded, but it is not even Poisson stable. Hence Fis not synchronous with ~-1 4•24.25 COROLLARY: If the space E does not contain a subspace isomorphic to~, ~::JR+ Eis a uniformly continuous function and the set ~(:JR) is compact, then the boundedness of the primitive (24.23) implies that the primitive and the function~ are both isochronous.

Proof: Let {tn} be a sequence such that t+tn { f o ~C~)d~}

ft o~C~)d~

+.

(24.29)

in the space C(:JR,E). We shall prove that {~(t + tn)} + ~(t) in C(:JR,E). Suppose the contrary holds. Since the family of all shifts of the function~ in the space C(JR,E) is compact, there

EXTENSIONS

288

Let t 0 and t be arbitrary elements of the group T with (xnt,

xn to) e c5.

It follows from (24. 30) that

xn to+k for some element keK.

e

xoS

Since (xnt,xnto) ec5 c y, it follows from

the choice of y that (xnt+k,xnto+k) es.

xn t +k

(24. 33)

e

Hence

xn to+k S C xoS 2 C xoa.

(24.34)

It is easily seen from (24.24) that F(x,t) - F(x,to) = {F(x,t + k) - F(x,to + k)} + {F(xntO,k) - F(xnt,k)}.

(24.35)

Since the oscillation of p,•, in xoa is less than c/2, we get from (24.33) and (24.34) that IJF(x,t + k) - F(x,to + k) II - p,',(xnto+k)IJ < s/2.

(24.36)

But (xnt,xnto) ec5, keK and (24.32) implies IIF(xnt 0 ,k) - F(xnt,k)II
'< :xT + E is continuous at some point x 0 e X = xT. Then F>'< is unifoI'ITlly continuous on xT, and hence i t can be extended by continuity onto X. Proof: Let c be an arbitrary positive number and let the index

a e U[X] be such that the oscillation of the function F>'< :xT + E in the neighbourhood x 0 a is smaller than c/2. Let S e U[X] be an index with s2 c a, S = s- 1 . Since the transformation group (X,T,rr)

is minimal, there exists a compact subset KC T such that X C xoSC - K)

= (x 0 S, - K)rr.

(24. 30)

Using the property of uniform integral continuity (see Subsection 1•1.5), we choose an index yeU[X] so that (24.31)

(t e K).

cS C

By Lemma 4•24.23 we can choose an index y and

cS e'U[X]

such that

(t e K).

(24. 32)

EXTENSIONS

290

(ye X, t

e

T).

The function f:X-+ C(T,E) is also continuous in view of the uniform integral continuity property. Since (X,T,n) is a minimal transformation group and fx(t) = F*(xnt), the function F*(xnt) = Fx(t) is recurrent in the sense of Birkhoff and f is a homomorphism of (X,T,n) onto the minimal transformation group (X*,T,a), where X* is the orbit closure in (C(T,E),T,a) of the point Fx:T-+ E.

I

4•24.29 COROLLARY: Let Ebe a Banach space which does not contain a subspace isomorrphic to~ and let ~:JR-+ Ebe a recurrent function in the sense of Birkhoff (in particular, a Bohr almost periodic function) whose primitive (24.23) is bounded. Then this primitive is a recurrent (respectively, almost periodic) function uniformly isochronous with~Proof: Since the function ~ e C(JR,E) is recurrent, the orbit closure X of the point~ in (C(JR,E),JR,o) is a compact minimal set. Let ~:JR-+ E denote the primitive (24.23) and let X* denote the orbit closure of~ in (C(JR,E),JR,o). It can be seen from the proof of Theorem 4•24.28 that there exists a continuous homomorphism f:X-+ X*. It remains to prove that the map f is bijective. Let I/Ji, 1/!2 e X and"f( 1/Ji) = f( 1/!2). Since X is a minimal seh there exist sequences {tn(l)} and {tn(2)} such that {~(t + tn(lJ)}-+ 1/Ji(t) (i = 1,2) in the space C(JR,E). Then lim f n-+oo

tn(i)+t O

~(s)ds

=

lim [

ftn(i)

~(s)dt +

ft

o~(s + tn(i))ds]

0

n-+oo

where c(l) and c(2) are elements of the space E. from the equality f(1/J1) = f(1/J2) that 1/!1 = 1/!2-I

Hence it follows

4•24.30 THEOREM: Let (X,T,n) be a minimal transformation group with a compact metrizable phase space X, Ea Banach space, F:XxT-+ Ea cocycle over (X,T,n) and let xsX be such that the set {F(x,t) Jt e T} is weakly relatively compact. Then the point Fx e C(T,E), where Fx(t) = F(x,t), is uniformly synchronous with x. Proof: Let f be a linear functional defined on E and let q,(x,t) = (f,F(x,t)) (tsT). Clearly ¢:XxT -+JR is a cocycle. Therefore the function ¢x:T-+ JR, ¢x(t) = q,(x,t) (t e T), is uniformly synchronous with the point x. Hence the map ¢*:xT -+JR, where ~}~(xnt) = ~(x,t) (t sT), is uniformly continuous. Since the weak closure of the set {F(x, t) t e T} is weakly compact, the map F*:xT-+ Eis weakly uniformly continuous, and therefore it J

291

AND EQUATIONS

can be extended continuously onto X = xT. Denote this extension by F1*:X ~ E. It follows from a result due to Gel'fand [1] that the function F1 is strongly continuous at some point of X. By Lemma 4•24.27, F 1>'•:X ~Eis a (strongly) continuous function. Hence it follows that the point Fx is uniformly synchronous with

x. 4•24.31 COROLLARY: Let Ebe an arbitrary Banach space 1 and

let