Topological Dynamics

AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XXXVI

TOPOLOGICAL DYNAMICS

BY

WALTER HELBIG GOTTSCHALK ASSOCIATE PROFESSOR OF MATHEMATICS UNIVERSITY OF PENNSYLVANIA

AND

GUSTAV ARNOLD HEDLUND PROFESSOR OF MATHEMATICS YALE UNIVERSITY

PUBLISHED BY THE

AMERICAN MATHEMATICAL SOCIETY 80 WATERMAN STREET, PROVIDEINCE, R. I.

1955

COPYRIGHT, 1955, BY THE

AMERICAN MATHEMATICAL SOCIETY All Rights Reserved No portion of this book may be reproduced without the written permission of the publisher

PREFACE By topological dynamics we mean the study of transformation groups with respect to those topological properties whose prototype occurred in classical dynamics. Thus the word "topological" in the phrase "topological dynamics" has reference to mathematical content and the word "dynamics" in the phrase has primary reference to historical origin. Topological dynamics owes its origin to the classic work of Henri Poincard and G. D. Birkhoff. It was Poincare who first formulated and solved problems of dynamics as problems in topology. Birkhoff contributed fundamental concepts to topological dynamics and was the first to undertake its systematic development.

In the classic sense, a dynamical system is a system of ordinary differential equations with at least sufficient conditions imposed to insure continuity and uniqueness of the solutions. As such, a dynamical system defines a (one-parameter or continuous) flow in a space. A large body of results for flows which are of interest for classical dynamics has been developed, since the time of Poincard,

without reference to the fact that the flows arise from differential equations. The extension of these results from flows to transformation groups has been the work of recent years. These extensions and the concomitant developments are set forth in this book.

Part One contains the general theory. Part Two contains notable examples of flows which have contributed to the general theory of topological dynamics and which in turn have been illuminated by the general theory of topological dynamics.

In addition to the present Colloquium volume, the only books which contain extensive related developments are G. D. Birkhoff [2, Chapter 7], Niemytzki and Stepanoff [1, Chapter 4 of the 1st edition, Chapter 5 of the 2nd edition] and G. T. Whyburn [1, Chapter 12]. The contents of this volume meet but do not significantly overlap a forthcoming book by Montgomery and Zippin. The authors wish to express their appreciation to the American Mathematical Society for the opportunity to publish this work. They also extend thanks to Yale University and the Institute for Advanced Study for financial aid in the preparation of the manuscript. The second named author extends to the American Mathematical Society his thanks for the invitation to give the Colloquium Lectures in which some aspects of the subject were discussed. Some of his work has been supported by the United States Air Force through the Office of Scientific

Research of the Air Research and Development Command. PHILADELPHIA, PENNSYLVANIA

NEw HAVEN, CONNECTICUT

July, 1954 m

CONVENTIONS AND NOTATIONS

Each of the two parts of the book is divided into sections and each section into paragraphs. Cross references are to paragraphs. 4.6 is the sixth paragraph of section 4. In general, a paragraph is either a definition, lemma, theorem or

remark. A "remark" is a statement, the proof of which is left to the reader. These proofs are not always trivial, however.

References to the literature are, in general, given in the last paragraph of each section. Numbers in brackets following an author's name refer to the bibliography at the end of the book. Where there is joint authorship, the number given refers to the article or book as listed under the first named author. An elementary knowledge of set theory, topology, uniform spaces and top-

ological groups is assumed. Such can be gained by reading the appropriate sections of Bourbaki [1, 2, 3]. With a few exceptions to be noted, the notations used are standard and a separate listing seemed unnecessary. Unless the contrary is specifically indicated, groups are taken to be multiplicative. Topological groups are not assumed to be necessarily separated (Haus-

dorff). The additive group of integers will be denoted by J and the additive group of reals by a. Contrary to customary usage, the function or transformation sign is usually placed on the right. That is, if X and Y are sets, f denotes a transformation of X into Y and x E X, then x f denotes the unique element of Y determined by x and f. In connection with uniform spaces, the term index is used to denote an element of the filter defining the uniform structure, thus replacing the term entourage as used by Bourbaki [2]. In keeping with the notation for the value of a function, if X is a uniform space, a is an index of X and x E X, then xa denotes the set of all y E X such that (x, y) E a. Unless the contrary is stated, a uniform space is not necessarily separated.

V

TABLE OF CONTENTS SECTION

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1. TRANSFORMATION GROUPS

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2. ORBr cLosuRE PARTITIONS .

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3. RECURSION .

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4. ALMOST PERIODICITY . .

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5. REGULAR ALMOST PERIODICITY

6. REPLETE SEMIGROUPS . . . 7. RECURRENCE

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8. INCOMPRESSIBILITY

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9. TRANSITIVITY .

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31

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49

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64

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73

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12. SYMBOLIC DYNAMICS . .

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102

10. AsnnproTICrrY

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11. FUNCTION SPACES .

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13. GEODESIC FLOWS OF MANIFOLDS OF CONSTANT NEGATIVE CURVATURE . .

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14. CYLINDER FLOWS AND A PLANAR FLOW . . . BIBLIOGRAPHY

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133

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143

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INDEX .................................. 149

PART I. THE THEORY 1. TRANSFORMATION GROUPS 1.01. DEFINITION. A topological transformation group, or more briefly, a transformation group, is defined to be an ordered triple (X, T, T) consisting of

a topological space X, a topological group T and a mapping T: X X T -' X such that:

(1) (Identity axiom) (x, e)T = x (x E X) where e is the identity element of T. (2) (Homomorphism axiom) ((x, t)T, s)T = (x, ts)T (x E X; t, s (E T). (3) (Continuity axiom) T is continuous. If (X, T, T) is a transformation group, then { X } { T T } is called the phase { space } { group } { projection } of (X, T, 7r).

1.02. DEFINITION. Let X, Y be { topological } uniform) spaces and let (X, T, jr), (Y, S, p) be transformation groups. A { topological } {uniform } isomorphism of (X, T, T) onto (Y, S, p) is defined to be a couple (h, cp) consisting of a { homeomorphism } { unimorphism } h of X

onto Y and a homeomorphic group-isomorphism So of T onto S such that (xh, tp)p = (x, t)Th (x (E X, t E T). The transformation groups (X, T, T) and (Y, S, p) are said to be (topologically} {uniformly} isomorphic (each to or with the other) provided there exists a {topological} {uniform} isomorphism of (X, T, T) onto (Y, S, p). 1.03. DEFINITION.

Let X be a { topological } { uniform } space and let (X, T, r)

be a transformation group. An intrinsic { topological } { uniform) property of (X, T, r) is a property of (X, T, T) definable solely in terms of the { topological }

{uniform} structure of X, the topological structure of T, the group structure of T, and the mapping T.

1.04. REMARK. We propose in this monograph to study certain intrinsic properties of transformation groups. It is clear that intrinsic {topological} { uniform } properties of transformation groups are invariant under { topological } { uniform) isomorphisms. 1.05. NOTATION.

Let (X, T, r) be a transformation group. If x E X and

if t E T, then (x, t)T is denoted more concisely by xt when there is no chance for ambiguity. Then the identity and homomorphism axioms may be restated as follows:

(1) xe = x (x E X). (2) (xt)s = x(t8) (x E X; t, s (E T). 1.06. TERMINOLOGY. The statement "(X, T, T) is a transformation group" may be paraphrased as "T { is } { acts as } a transformation group { of } { on } X 1

TOPOLOGICAL DYNAMICS

2

[1.06]

with respect to 7r". By virtue of 1.05 it often happens that a symbol for the phase projection does not occur in a discussion of a transformation group. In such an event we may speak simply of (X, T) as the transformation group where X is the phase space, T is the phase group and the phase projection is understood. The statement "(X, T) is a transformation group" may be paraphrased as "T { is ) { acts as) a transformation group { of } [on) X". Thus the transformation group (X, T, 7r) may be denoted by (X, T) or even by T provided no ambiguity can occur.

Generally speaking, the statement that the transformation group (X, T, 7r) has a certain property may be paraphrased as either T has the property on X or X has the property under T. If x E X, then the statement that (X, T, Tr) has a certain property at x may be paraphrased as either T has the property at x or x has the property under T. 1.07. STANDING NOTATION. Throughout the remainder of this section (X, T, 7r) denotes a transformation group.

1.08. DEFINITION.

If t E T, then the t-transition of (X, T, 7r), denoted

7r`, is the mapping ir' : X --> X such that x7r` = (x, t)7r = xt (x E X). The transition group of (X, T, 7r) is the set G = [ir` t E T]. The transition projection of (X, T, a) is the mapping A : T -* G such that to = Tr' (t E T). I

If x E X, then the x-motion of (X, T, 7r), denoted 7rs , is the mapping as : T -* X

such that t7rs = (x, t)7r = xt (t E T). The motion space of (X, T, 7r) is the set M = [az I x E X]. The motion projection of (X, T, a) is the mapping u : X -' M

such that xu = a, (x E X). 1.09. DEFINITION.

The transformation group (X, T) is said to be effective

provided that if t E T with t 5;6 e, then xt 54 x for some x E X. 1.10. REMARK. Let {G} J X) {M) J u) be the I transition group) {transition projection) { motion space } { motion projection } of (X, T, 7r). Then

(1) it is the identity mapping of X.

(2) If t, s E T, then 7r` r = a". (3) If t E T, then 7r` is a one-to-one mapping of X onto X and (7r`)-' = a`-' (4) If t E T, then jr' is a homeomorphism of X onto X. (5) G is a group of homeomorphisms of X onto X. (6) A is a group-homomorphism of T onto G. This justifies the name "homomorphism axiom" of 1.01 (2). (7) A is one-to-one if and only if (X, T, 7r) is effective.

(8) If x E X, then 7ri is a continuous mapping of T into X. (9) u is a one-to-one mapping of X onto M. 1.11. REMARK. Let t E T and let (p, : T -* T be defined by r(p, = t-'rt (r E T). Then (7r°, cp,) is a topological isomorphism of (X, T, 7r) onto (X, T, 7r). 1.12. DEFINITION. Let X be a topological space. A topological homeomorphism group of X is a topologized group 4' of homeomorphisms of X onto X

TRANSFORMATION GROUPS

[1.171

3

such that 4) is a topological group and p : X X P -p X is continuous where p is defined by (x, cp) p = xcp (x E X, v E 4>)

1.13. REMARK. The effective topological transformation groups and the topological homeomorphism groups are essentially identical in the following sense:

(1) If (X, T, ir) is an effective topological transformation group, then the transition group G of (X, T, r), topologized so that the transition projection of (X, T, 7) becomes a group-isomorphic homeomorphism of T onto G, is a topological homeomorphism group of X. (2) If 4> is a topological homeomorphism group of a topological space X, then (X, (b, p) is an effective topological transformation group where p : X X 4> --> X is defined by (x, cp) p = xcp (x E X, p (E 4>).

In particular, a notion defined for topological transformation groups is automatically defined for topological homeomorphism groups.

1.14. DEFINITION. A discrete transformation group is a topological transformation group whose phase group is discrete. A discrete homeomorphism group

is a topological homeomorphism group provided with its discrete topology. A homeomorphism group is a group of homeomorphisms. The total homeomorphism

group of a topological space X is the group of all homeomorphisms of X onto X. 1.15. REMARK. It is clear from 1.13 that the effective discrete transformation groups and the discrete homeomorphism groups are to be considered as

identical. Since a homeomorphism group may be considered as a discrete homeo-

morphism group, a notion defined for transformation groups is automatically defined for homeomorphism groups. 1.16. NOTATION.

If A C X and if B C T, then (A X B)-7r = [xt I x E

A&t E B] is denoted more concisely by AB when there is no chance for ambiguity. In particular, we write At in place of A[t] where A C X and t E T; and we write xB in place of [x]B where x E X and B C T. By the homomorphism axiom, xts is unambiguously defined, where x E X and t, s E T; likewise ABC

where A C X and B, C C T; etc. 1.17. LEMMA.

Let X, Y, Z be topological spaces and let c : X X Y - Z be

continuous. If A, B are compact subsets of X, Y and if W is a neighborhood of (A X B),p, then there. exist neighborhoods U, V of A, B such that (U X V),p C W.

PROOF. We write (x, y)cp = xy (x E X, y E Y). Let x E A. We show there exist open neighborhoods U, V of x, B such that UV C W. For each y E B there exist open neighborhoods Uy , Vy of x, y such that U,,VV CTTW. Choose a finite subset F of B for which B C

UVEF

V, . Define

U = n ,,F Vy and V = UYEF Vy For each x E A there exist open neighborhoods U. , V. of x, B such that UUVV C W. Choose a finite subset E of A for which A C U = U.EE U. and V= n.EE V. . Then U, V are neighborhoods of A, B such that UV C W.

4

TOPOLOGICAL DYNAMICS 1.18. LEMMA.

[1.18]

The following statements are valid:

(1) If A C X and if t E T, then _At = At. (2) If A C X and if B C T, then AB C AB and AB = AB = AB. (3) If A, B are compact subsets of X, T, then AB is a compact subset of X.

(4) If A, B are compact subsets of X, T and if W is a neighborhood of AB, then there exist neighborhoods U, V, of A, B such that UV C W. (5) If A is a closed subset of X and if B is a compact subset of T, then AB is a

__

closed subset of X.

(6) If A C X and if B is a compact subset of T, then AB = AB. _PROOF.

A= At.

(1) Since 7r` : X -* X is a homeomorphism onto, At = Aar` _

_

(2) Since a : X X T -* X is continuous, AB = (A X B),r = (A X B)ir C (A X B)ir = AB. The last conclusion follows from AB C AB C AB and AB C AB C AB. (3) AB = (A X B)ir is a continuous image of the compact set A X B. (4) Use 1.17.

(5) Let x E X - AB. Then xB-1 (1 A = 0. By (4) there exists a neighborhood U of x such that UB-1(l A= 0 whence U (l AB = 0 and U C X - AB.

(6) By (2) and (5), AB = AB = AB. 1.19. LEMMA.

Let X, Y be uniform spaces, let p : X - Y be continuous and

let A be a compact subset of X. If ,6 is an index of Y, then there exists an index a of X such that x E A implies xacp C x(p,6. PROOF. Let y be a symmetric index of Y such that y2 C ,6. For each x E A there exists a symmetric index as of X such that xaicp C xrpy. Choose a finite subset E of A for which A C IJxaz . Define a = f yeE ax . Let x E A. There

exists z E E such that x E zaz . Since xcp E za;cp C zjpy, it follows that xacp C za.acp C za2Ap C ztpy C xgy2 C x(p,6. The proof is completed.

1.20. LEMMA. Let X be a uniform space, let A, B be compact subsets of X, T and let a be an index of X. Then: (1) There exists an index ,6 of X and a neighborhood V of e such that x E A

and t E B implies x(3tV C xta and x0Vt C xta. (2) There exists an index 6 of X such that x E A and t E B implies x,Bt C xta and xt,6 C xat. (3) There exists an index ,l3 of X such that x E A implies x(3B C xBa and xB,6 C xaB.

(4) There exists an index 6 of X such that t E B implies A,6t C Ata and At,6 C Aat. PROOF.

Since 7r : X X T -* X is continuous and A X B is a compact subset

of X X T, (1) follows from 1.19. The first part of (2) follows immediately from (1).

Since AB and B-1 are compact, there exists an index ,6 of X such that x E AB and t E B-1 implies x0t C xta. Hence, x E A and t E B implies xt$t-1 C xtt-1a

TRANSFORMATION GROUPS

[1.27]

5

and xto C xat. This proves the second part of (2). Finally (3) and (4) are easy consequences of (2). 1.21. LEMMA. Let X be a compact uniform space, let a be an index of X and let K be a compact subset of T. Then there exists an index # of X such that:

(1) x E X and k E K implies x,6k C xka. (2) x E X and k E K implies xka C xak. (3) (x, y) E l3 and k E K implies (xk, yk) E a. (4) (x, y) E a' and k E K implies (xk, yk) E ,B'. PRooF.

Use 1.20 (2).

1.22. DEFINITION.

Let A C X and let S C T. The set A is said to be invariant

under S or S-invariant provided that AS C A. When S = T, the qualifying phrase "under T" and the prefix "T-" may be omitted. 1.23. REMARK. The following statements are valid:

(1) If A C X, then the following statements are pairwise equivalent: A is T-invariant, that is, AT C A; AT = A; t E T implies At C A; t E T implies At = A; t E T implies At D A. (2) X and 0 are T-invariant. _ (3) If A is a T-invariant subset of X, then A' = X - A, A, int A are T-invariant.

(4) If A and B are T-invariant subsets of X, then A - B is T-invariant. (5) If a is a class of T-invariant subsets of X, then f a and U a are T-invariant. (6) If A C X and if S C T, then A is S-invariant if and only if A' is S-1-invariant. 1.24. REMARK.

Let Y C X, let S be a subgroup of T, let Y be S-invariant

and let p = 7r I Y X S. Then (Y, S, p) is a transformation group. In particular, T acts as a transformation group on every T-invariant subset of X, and every subgroup of T acts as a transformation group on X. 1.25. DEFINITION. Let Y be a T-invariant subset of X. The transformation group (X, T) is said to have a certain property on Y provided that the transformation group (Y, T) has this property. 1.26. DEFINITION. Let x E X and let S C T. The orbit of x under S or the S-orbit of x is defined to be the subset xS of X. The orbit-closure of x under S or the S-orbit-closure of x is defined to be the subset xS of X. An {orbit} {orbit-

closure } under S or an { S-orbit } { S-orbit-closure }

is defined to be a subset A

of X such that A is the { S-orbit I { S-orbit-closure } of some point of X. When S = T, the phrase "under T" and the prefix "T-" may be omitted. 1.27. DEFINITION.

Let X be a set. A partition of X is defined to be a disjoint

class a of nonvacuous subsets of X such that X = Ua.

TOPOLOGICAL DYNAMICS

6

[1.281

1.28. REMARK. The following statements are valid:

(1) If x E X, then the orbit of x under T is the least T-invariant subset of X which contains the point x.

(2) If x E X and if y E xT, then yT = xT. (3) The class of all orbits under T is a partition of X. (4) If x E X, then the orbit-closure of x under T is the least closed T-invariant subset of X which contains the point x.

_

(5) If x E X and if y E xT, then yT C xT. (6) The class of all orbit-closures under T is a covering of X. 1.29. REMARK. The following definitions describe various methods of constructing transformation groups. 1.30. DEFINITION.

Let n be a positive integer. An n-parameter [discrete)

{continuous} flow is defined to be a transformation group whose phase group is 1.4"1 { (R." } . The phrase "one-parameter {discrete) {continuous} flow" is shortened to " { discrete } { continuous } flow". 1.31. REMARK. Let n be a positive integer. An n-parameter discrete flow (X, 6", a) is characterized in an obvious manner by n pairwise commuting homeomorphisms of X onto X, namely ,r" which are

said to generate (X, J", a). In particular, a discrete flow (X, 5, a) is characterized by a single homeomorphism of X onto X, namely 7r1, which is said to generate (X, 6, a). The properties of a discrete flow (X, 6, 7r) are often attributed to its generating homeomorphism a1. 1.32. DEFINITION.

Let S be a subgroup of T and define p = a X X S.

The transformation group (X, S, p) is called the S-restriction of (X, T, a) or a subgroup-restriction of (X, T, ir).

Let Y be a subset of X such that (Y X T)ir = Y and define p = 7r I Y X T. The transformation group (Y, T, p) is called the Y-restriction of (X, T, a) or a subspace-restriction of (X, T, 7r).

Let S be a subgroup of T, let Y be a subset of X such that (Y X S)r = Y and define p = it I Y X S. The transformation group (Y, S, p) is called the (Y, S)-restriction of (X, T, 7r) or a transformation subgroup of (X, T, a). Let S be a topological group, let cp : S -* T be a continuous homomorphism into and let p : X X S ---> X be defined by (x, s) p = (x, s(p) 7r (x E X, s E S). The transformation group (X, S, p) is called the (S, (p)-re8triction of (X, T, 7r). Let S be a topological group, let (p : S -* T be a continuous homomorphism

into, let Y be a subset of X such that (Y X Scp)a = Y and let p : Y X S -> Y be defined by (y, s)p = (y, sSp)ir (y E Y, s (E S). The transformation group (Y, S, p) is called the (Y, S, (p)-restriction of (X, T, 7r). 1.33. REMARK. We consider every partition a of a topological space X to be itself a topological space provided with its partition topology, namely, the greatest topology which makes the projection of X onto a continuous.

TRANSFORMATION GROUPS

[1.42]

1.34. DEFINITION.

7

Let X be a set, let a be a partition of X and let E C X.

The star of E in a or the a-star of E or the saturation of E in a or the a-saturation of E, denoted Ea, is the subset U [A I A E a, A (1 E 01 of X. The set E is

saturated in a or a-saturated in case E = Ea. 1.35. DEFINITION. Let X be a topological space and let a be a partition of X. The partition a is said to be { star-open } { star-closed } provided that the a-star

of every { open) { closed } subset of X is { open } { closed } in X. 1.36. REMARK.

Let X be a topological space and let a be a partition of X.

Then the following statements are pairwise equivalent (1) a is { star-open) { star-closed 1.

(2) If x E X and if U is a neighborhood of { x } { xa 1, then there exists a neighborhood V of { xa } { x and therefore xa } such that { V C Ua Va C U } . (3) The projection of X onto a is { open } { closed 1. 1.37. DEFINITION. Let X be a topological space. A decomposition of X is a partition a of X such that every member of a is compact.

Let X be a compact metrizable space and let a be a decom{ star-open } { star-closed } if and only if xo , x, E X with ]imn.m xn = X. implies

1.38. REMARK.

position of X. Then a is x2 ,

{ xoa C lira inf xna } { 1im sup xna C xoa 1. 1.39. DEFINITION. Let a be a {star-open partition} {star-closed decomposition } of X, let Air` E a (A E a, t (=- T) and let p : a X T -> a be defined by (A T)p = Aa° (A E a, t E T). The transformation group (a, T, p) is called the partition transformation group of a induced by (X, T, a).

1.40. DEFINITION.

Let 4' be a group of homeomorphisms of X onto X such

that v r' = ir`cp (,p E 4), t E T) and let a = [x(b I x E X] whence a is a staropen partition of X such that A7r` E a (A E a, t (E T). The partition transformation group of a induced by (X, T, ,r) is called the 4,-orbit partition transformation group induced by (X, T, 7r). 1.41. DEFINITION. Let 4, be a group of homeomorphisms of X onto X such that cp r` = a`cp ((p E 4), t E T) and let a = [x4, 1 x E X] be a partition of X

whence a is a star-open partition of X such that Air` E a (A E a, t E T). The partition transformation group of a induced by (X, T, 7r) is called the 4-orbit-closure partition transformation group induced by (X, T, 1r). 1.42. DEFINITION. {Let rp be a continuous-open mapping of X onto X1 (Let X be a compact T2-space, let (p be a continuous mapping of X onto X } such that rpa` _ ir`cp (t E T) and let a = [x(p ' I x E X] whence a is a {staropen partition } { star-closed decomposition } of X such that Aa` E a (A E a,

8

TOPOLOGICAL DYNAMICS

[1.42]

t E T). The partition transformation group of a induced by (X, T, ir) is called the So-inverse partition transformation group induced by (X, T, ir). 1.43. DEFINITION. Let S be a topological group, let T be a topological subgroup of S, for x E X and u E S define {A(x, a) _ [(x1r', ur) I r E T]} {A (x, Q) = [(x7r', r -'a) I r E T]}, define the star-open partition a = [A (x, v) x E X, a E S] of X X S and let p : a X S - a be defined by { (A(x, a), s)p = A(x, s -'a) } { (A (x, v), s)p = A(x, as) } (x E X; v, s (E S). The transformation

group (a, S, p) is called the {left} {right} S-extension of (X, T, 2r).

1.44. REMARK. We adopt the notation of 1.43. Consider the transformation

group (X X S, S, I) where 1 : (X X S) X S --' X X S is defined by { ((x, Q), s)rt = (x, 8'q) } { ((x, a), On = (x, as) } (x E X; a, s E S). The partition transformation group of a induced by (X X S, S, o) coincides with the { left } { right } S-extension of (X, T, 7r). 1.45. REMARK. Let S be a topological group and let T be a discrete topological

subgroup of S. Then (X, T, 7r) is isomorphic to a transformation subgroup of the { left } fright } S-extension of (X, T, ir).

c E I) of sets is

1.46. NOTATION. The cartesian product of a family (X,

denoted X,(=I X, . The direct product of a family (G, I a E I) of groups is denoted DX,EI G,

1.47. REMARK. We consider the cartesian product of every family of {topological } { uniform) spaces to be itself a { topological } { uniform } space provided

with its product

{ topology) { uniformity } ,

namely, the least { topology If uni-

formity} which makes all the projections onto the factor spaces [continuous} f uniformly continuous } . 1.48. DEFINITION. Let ((X, , T, , 7r,) c E I) be a family of transformation groups. The {cartesian} {direct} product of ((X, , T, , 7r,) c (E I), denoted { X,EI (X, , T, , 7r,) } {DX,EI (X, , T, , 7r,)), is the transformation group (X, T, ir) I

I

where X = X,EI X, , {T = X, E, T,} {T = DX,EI T,} and it : X X T - > X is defined by (x, t)ar = (x,7r,' c E I) (x = (x, c E I) E X, t = (t, t E I) (E T). I

I

1.49. DEFINITION. Let ((X, , T, 7r,) c E I) be a family of transformation groups. The space product of ((X, , T, 7r,) c (E I), denoted ,X,EI (X, , T, 7r,), is the transformation group (X, T, 7r) where X = X,EI X, and it : X X T -> X is defined by (x, t)7r = (x,ir,' , (E I) (x = (x, e E I) E X, t E T). I

I

J

1.50. REMARK. Both the direct and space products of a family of transformation groups are subgroup-restrictions of the cartesian product of the family.

Let T be a topological group. The left transformation group of T is defined to be the transformation group (T, T, A) where X : T X T -* T is defined by Cr, t)X = t 'r(r, t E T). The right transformation group of T is defined to be the transformation group (T, T, A) where µ : T X T --> T is defined by (r, t)µ = rt (r, t E T). 1.51. DEFINITION.

TRANSFORMATION GROUPS

[1.591

9

The bilateral transformation group of T is defined to be the transformation group (T, T X T, t) where t : T X (T X T) -). T is defined by (r, (t, s)) = t-'-rs

(r, t,sET).

1.52. DEFINITION.

Let X be a uniform space and let x E X. The trans-

formation group (X, T, 7r) is said to be { equicontinuous at x } { equicontinuous } {uniformly equicontinuous} provided that the transition group [T` t E T] is { equicontinuous at x } { equicontinuous } { uniformly equicontinuous 1. I

1.53. REMARK.

Let X be a uniform space. The following statements are

pairwise equivalent: (1) (X, T, 7r) is uniformly equicontinuous.

(2-5) If a is an index of X, then there exists an index a of X such that

{xEXand tETimplies xfltCxta} { x E X and t E T implies xt,8 C xat } { (x, y) E a and t E T implies (xt, yt) E a} { (x, y) E a' and t E T implies (xt, yt) (E #'J. 1.54. REMARK. The {left) fright) transformation group of a topological group is uniformly equicontinuous relative to the { left) { right } uniformity of the phase space. 1.55. NOTATION. Let H be a subgroup of a group G. The left quotient space [xH I x E G] of G by H is denoted G/H. The right quotient space [Hx I x E G] of G by H is denoted G\H.

1.56. DEFINITION. Let S be a subgroup of a topological group T. The left transformation group of T/S induced by T is defined to be the transformation group (T/S, T, A) where A : T/S X T -* T/S is defined by (A, t)X =

t-'A (AEE T/S,tET). The right transformation group of T\S induced by T is defined to be the transformation group (T\S, T, µ) where p : T\S X T -* T\S is defined by (A, t)µ =

At (A E T\S, t (E T). 1.57. REMARK. Let S be a subgroup of a topological group T and let (T, T, 77) be the { left) { right } transformation group of T. Then the { left) { right } trans-

formation group of {T/S} {T\S} induced by T coincides with the partition transformation group of {T/S} {T\S} induced by (T, T, n). 1.58. DEFINITION.

Let rp be a continuous homomorphism of a topological

group T into a topological group S and let p : S X T - S be defined by { (s, t) p = (t-',p)s} { (s, t)p = s(tcp) }

(s E S, t E T). The transformation group (S, T, p)

is called the {left} {right} transformation group of S induced by T under rp.

1.59. REMARK. Let (p be a continuous homomorphism of the topological group T into the topological group S. Then the {left} {right) transformation group of S induced by T under (p coincides with the (T, p)-restriction of the {left) {right} transformation group of S.

TOPOLOGICAL DYNAMICS

10

[1.60]

1.60. REMARK. Let cp be a continuous homomorphism of a topological group T into a topological group S. Then the { left } { right } transformation group of S induced by T under cp is uniformly equicontinuous relative to the {left} {right} uniformity of S.

1.61. REMARK. Let (p be a homomorphism of 9 into a topological group T, let t = 1-p and let 0 be the { left } { right } translation of T induced by { t-' j { t 1. Then the { left) { right } transformation group of T induced by J under p coincides with the discrete flow on T generated by 0. 1.62. DEFINITION. Let T be a topological group, let Y be a uniform space, let - be the class of all { right } { left) uniformly continuous functions on T to Y,

let 4) be provided with its space-index uniformity and let p : 4) X T -* 4) be defined by { ((p, t) p = (t-rgp I r (E T) } { (gyp, t) p = (rt-irp I r E T) } (,p (E 4), t E T).

The uniformly equicontinuous transformation group ('F, T, p) is called the {left} {right} uniform functional transformation group over T to Y. 1.63. DEFINITION.

Let T be a locally compact topological group, let Y

be a uniform space, let - be the class of all continuous functions on T to Y, let 4) be provided with its compact-index uniformity and let p : CF X T -* 4) be defined by { (,p, t) p = (trop 1 r E T) } { (,p, t) p = (rt-lcp I r E T) } (rp E 4), t E T). The transformation group ('F, T, p) is called the {left} {right} functional transformation group over T to Y. 1.64. REMARK. A particular case of 1.63 arises when T is discrete. In such an event a different notation may be used, as indicated by the following statements: (1) CF = YT = X,ET Y, where Y, = Y (r E T). (2) The point-index (= compact-index) uniformity of CF coincides with the product uniformity of X,E T Y, (3) If y = (y, J r E T) E X,E T Y, and if t E T, then { (y, t) p = (y,, I r E T) } {(y, t)p = (y,e-. r E T)}. I

1.65. LEMMA. Let T be a locally compact topological group, let Y be a uniform space, let ('F, T, p) be the {left} {right} functional transformation group over T to Y, let (p E 4) and let' C (D. Then:

(1) The orbit (pT of (p is totally bounded if and only if (p is {left} {right} uniformly continuous and bounded. (2) If Y is complete, then the orbit-closure VT of V is compact if and only if (p is {left} {right} uniformly continuous and bounded.

_

(3) 'T is totally bounded if and only if 'P is {left} {right} uniformly equicon-

_

tinuous and bounded. (4) If Y is complete, then iT is compact if and only if 'P is { left } { right } uniformly equicontinuous and bounded. PROOF.

Use 11.31 and 11.32.

TRANSFORMATION GROUPS

[1.701

11

1.66. DEFINITION. Let X be a uniform space, let each transition r` : X -* X (t E T) be uniformly continuous, let the motion space [,r, : T -* X I x E X] be equicontinuous, let Y be a uniform space, let (F be the class of all uniformly continuous functions on X to Y, let (F be provided with its space-index uniformity and let p : (F X T -* (F be defined by (co, t)p = a`-'ip ((p E 4), t (E T). The uniformly equicontinuous transformation group ((F, T, p) is called the

uniform functional transformation group over (X, T, 7r) to Y. 1.67. REMARK. Let T be a topological group, let (T, T, 71) be the {left} {right} transformation group of T and let Y be a uniform space. Then the uniform functional transformation group over (T, T, ,) to Y coincides with the {left} {right} uniform functional transformation group over T to Y.

1.68. DEFINITION.

Let T be locally compact, let Y be a uniform space,

let (F be the class of all continuous functions on X to Y, let 4) be provided with

its compact-index uniformity and let p : 4) X T - (F be defined by (v, t)p = ,r`-'(o (,p (E (F, t E T). The transformation group (4), T, p) is called the functional transformation group over (X, T, 7r) to Y.

1.69. REMARK. Let T be a locally compact topological group, let (T, T, q) be the { left } { right } transformation group of T and let Y be a uniform space. Then the functional transformation group over (T, T, 77) to Y coincides with the { left } { right) functional transformation group over T to Y. 1.70. NOTES AND REFERENCES.

(1.01) The concept of a transformation group for which the topology of the group plays a role appears to have originated in the latter part of the nineteenth century (cf., e.g., Lie and Engel [1]). A system of n differential equations of the first order defines, under suitable conditions, a transformation group (X, T, ir) for which X is an n-dimensional manifold and T is the additive group of reals. Thus a classical dynamical system with n degrees of freedom defines a transformation group for which the phase space is the 2n-dimensional manifold customarily associated with the term. See also Zippin [1]. (1.35) For a decomposition of a compact metric space, the equivalence of { star-open } { star-closed } with { lower semi-continuous } { upper semi-continuous }

is readily verified (cf. Whyburn [1], Ch. VII).

(1.40) (F-orbit partition transformation groups arise naturally in the study of geodesic flows on manifolds (cf. §13).

2. ORBIT-CLOSURE PARTITIONS 2.01. STANDING NOTATION.

Throughout this section T denotes a topological

group.

2.02. DEFINITION. A subset A of T is said to be { left } { right } syndetic in T provided that { T = AK) { T = KA } for some compact subset K of T.

2.03. REMARK. The following statements are valid. (1) If A C T, then A is {left} {right} syndetic in T if and only if there exists a compact subset K of T such that every {left} {right} translate of K intersects A.

(2) If A C B C T and if A is {left} {right} syndetic in T, then so also is B. (3) If A C T, then A is {left} {right} syndetic in T if and only if A-' is {right} {left} syndetic in T. (4) If A C T and if A is symmetric or invariant (in particular, if A is a subgroup of T or if T is abelian), then A is left syndetic in T if and only if A is right

syndetic in T. In such an event, the equivalent phrases "left syndetic", "right syndetic" are contracted to "syndetic". (5) If A is a syndetic subgroup of T, then the left, right quotient spaces T/A, T\A are compact. (6) If T is locally compact, if A is a subgroup of T and if some one of the left, right quotient spaces T/A, T\A is compact, then A is syndetic in T. (7) If T is discrete and if A is a subgroup of T, then A is syndetic in T if and only if A is of finite index in T. (8) If A is a {left} {right} syndetic subset of T and if U is a compact neighborhood of e, then { A U } { UA } is { left } { right } syndetic relative to the discrete topology of T. 2.04. EXAMPLE.

Let T be the discrete free group on 2 generators a, b and

let { A B } be the set of all words of T which in reduced form do not { end } {begin} with {a'} {b'}. Then: (1) A is left syndetic in T but A is not right syndetic in T. (2) B is right syndetic in T but B is not left syndetic in T.

(3) A U B is both left and right syndetic in T but there is no compact (= finite) subset K of T such that every bilateral translate of K intersects

AUB.

2.05. DEFINITION. Let G be a group. A semigroup in G is defined to be a subset H of G such that HH C H. 2.06. LEMMA. Let S be a left or right syndetic closed semigroup in T. Then S is a subgroup of T.

PROOF. We assume without loss that S is left syndetic. Let s E S and let 12

ORBIT-CLOSURE PARTITIONS

12.101

13

U be a neighborhood of the identity e of T. It is enough to show that s' U (1 S F4- 0. Let V be a neighborhood of e such that VV-' C U and let K be a compact

subset of T such that T = SK. There exists a finite class Y of right translates of V such that K C U 5. Choose k, E K. Now s-'ko = s,k, for some s, E S and some k, E K. Again s 'k, = s2k2 for some s2 E S and some k, E K. This may be continued. Thus there exist sequences ko , k, , in K and s, , s2 , . in S such that 8-'k; = s;+1k:+,(i = 0, 1, ). Select integers m, n(0 < m < n) and Vo E 9 such that km , k E Vo . Now s 'k,,,kn' = (s 'kmkm+,) (km+,km+2) sm+,ssm+2 - - ss E S. Also s 1kmkn' E s 'V0Vo' C s -'U. Hence s ' U (1 S 0 0. The proof is completed. 2.07. STANDING NOTATION. For the remainder of this section (X, T, 7r)

denotes a transformation group. 2.08. DEFINITION. Let x E X and let S C T. The x-envelope of S, denoted S. , is defined to be the subset [t t E T, xt E xS] of T. I

__

Let x E X. Then: (1) If S C T, then S. = xS7rz', S. is closed in T, Sx D S and xSx = xS.

2.09. LEMMA.

(2) If S is an invariant semigroup in T, then S. is a semigroup in T. PROOF.

(1) Obvious.

(2) xSxSx C x$8. C xSS, = xSrS C xSS = xSS C xS whence S.S. C S. 2.10. LEMMA.

.

Let x C X and let S be a syndetic invariant subgroup of T. Then:

(1) Ss is a subgroup of T. (2) If U is a neighborhood of e, then x x(T - SxU). (3) If T is locally compact and if U is a neighborhood of x, then there exists a compact subset M of T such that xM C U and S. C SM-'. PROOF.

(1) Use 2.06 and 2.09.

(2) We first show that if t E T - S= , then x xSSV0 for some neighborhood Vo of t. Let t E T - Sx . Since t-' ($ Sx by (1), xt' xSx and x ($ xSt. There exist neighborhoods W of x and V of e such that V = V-' and WV (1 xSt = 0. It follows that W n xSxtV = 0. Define Vo = W. We may assume that U is open. Let K be a compact subset of T such that

T = SK. Define H = K - S.U. Using (1) we conclude that T = SK C S(H U SxU) C SH U SSx U C SxH U SxU and SxH (l SxU = 0. Hence T - SxU = SxH. By the preceding paragraph to each t E H there corresponds a neighborhood V, of t such that x Er: x$ V,. Since H is compact, there exists a finite subset E of H such that H C U I E E V, . Hence x xSxH = x(T - S. U). (3) We may assume that U is open. Let K be a compact subset of T such

that T = SK. Define H = K n Sx . If t E H, then xt E xS and x E xSc-'. Then t H implies the existence of s, E S such that xs,t' E U and hence the existence of a compact neighborhood V, of s,t-' such that xV, C U. Since H is compact, there is a finite subset E of H for which H C U,EE Vj's, . Define M = U, EE V, . Clearly xM C U. If t E Sx , then t = sk for some s E S and

TOPOLOGICAL DYNAMICS

14

[2.10]

some k E K, k = s-'t E S. and t E S(K (1 Ss) = SH. Thus S. C SH. Since SH C UtEE VT'S = SM-', we have S. C SM-'. The proof is completed. 2.11. DEFINITION. Let A C X and let S C T. The set A is said to be minimal under S or S-minimal provided that A is an orbit-closure under S and A does not contain properly an orbit-closure under S. When S = T, the phrase "under T" and the prefix "T-" may be omitted. We often use the more colorful phrase "minimal orbit-closure" in preference to "minimal set".

2.12. REMARK. Let A C X. Then the following statements are pairwise equivalent: (1) A is a minimal orbit-closure under T.

(2) A 5,z- 0 and xT = A for each x E A. (3) A is nonvacuous closed T-invariant and A is minimal with respect to this property.

(4) A is nonvacuous closed and UT = A for each nonvacuous subset U of A which is open in A.

Let A be a minimal orbit-closure under T. Then: 0. (1) A is open in X if and only if int A (2) If int A 0, then A is a union of components of X. (3) If int A 96 0, and if T is connected, then A is a component of X.

2.13. REMARK.

(4) If int A 0 0, and if X is connected, then A = X. 2.14. LEMMA.

Let n be a positive integer, let X be an n-dimensional manifold

and let A C X. Then dim A = n if and only if int A 0 0. PROOF.

Cf. [Hurewicz-Wallman [1], pp. 44-46].

2.15. THEOREM.

Let n be a positive integer, let X be an n-dimensional manifold

and let A be a minimal orbit-closure under T such that A X X. Then dim A

n - 1. PROOF.

Use 2.13 (4) and 2.14.

2.16. DEFINITION. A Cantor-manifold is defined to be a compact metrizable

space X of positive finite dimension n such that X is not disconnected by a subset of dimension 5 n - 2. 2.17. LEMMA.

Let X be a compact metrizable space of positive finite dimension.

Then there exists a subset C of X such that C is a Cantor-manifold and dim C = dim X. PROOF.

Cf. [Hurewicz-Wallman [1], pp. 94-95].

2.18. THEOREM. Let X be a finite-dimensional compact metrizable space such that X contains more than one point, let X be minimal under T and let T be connected.

Then X is a Cantor-manifold and hence X has the same dimension at every point of X.

ORBIT-CLOSURE PARTITIONS

[2.241

15

PROOF. Let n = dim X. Assume X is not a Cantor-manifold. Then there exist closed proper subsets A, B of X such that X = A U B and dim (A (1 B) < n - 2. By 2.17 there exists C C X such that C is a Cantor-manifold

and dim C = n. Let { E } IF) be the set of all t E T such that { Ct C A) { Ct C B1. Clearly E and F are closed disjoint subsets of T. If t E T, then Ct is an n-dimen-

sional Cantor-manifold and hence Ct C A or Ct C B. Thus T = E U F. It follows that T = E or T = F. If IT = E} {T = F}, then {CT C A} {CT C B} and CT

X. This contradicts the minimality of X. The proof is completed.

2.19. REMARK.

If the hypothesis that T be connected is omitted from 2.18,

the conclusion fails. Cf. [Floyd [1]]. 2.20. REMARK.

Let p be a continuous homomorphism of a topological

group T into a topological group S and let (S, T, p) be the { left } { right } transformation group of S induced by T under cp. Then S is minimal under (S, T, p) if and only if Tcp = S. 2.21. REMARK.

If A and B are minimal orbit-closures under T, then

A (1 B = 0 or A = B. In other words, the class of all minimal orbit-closures under T is disjoint. 2.22. THEOREM.

Let X be compact. Then there exists a minimal orbit-closure

under T. PROOF. Let a be the class of all nonvacuous closed invariant subsets of X. Since X E a, we have a 5,!,- 0. Partially order a by inclusion. By the extremum law there exists a minimal element A of a. By 2.12, A is a minimal orbit-closure under T. The proof is completed.

2.23. REMARK. The following statements are pairwise equivalent: (1) The class of all orbit-closures under T is a partition of X.

(2) If x E X and if y E xT, then yT = xT. (3) Every orbit-closure under T is minimal under T. (4) The class of all minimal orbit-closures under T is a covering of X. (5) The class of all minimal orbit-closures under T is a partition of X. 2.24. THEOREM. Let S be a syndetic invariant subgroup of T. Then the class of all orbit-closures under S is a partition of X if and only if the class of all orbitclosures under T is a partition of X.

PROOF. Assume that the class of all orbit-closures under T is a partition of X. Let x E X and y E xS. It is enough to show that TS = xS. Let K be a compact

subset of T for which T = SK. Since x E yT = ySK-' = ySK-I by 1.18 (6), there exists k E K such that xk E S. It follows that xkS C yS C xS. Since S is a subgroup of T by 2.10 (1) and k E S , we have TS = xSs = xSsk = xSzk = xSk = xSk = xkS C xS. Thus yS = xS. Assume that the class of all orbit-closures under S is a partition of X. Let x E X and y E xT. It is enough to show that x E yT. Let K be a compact

TOPOLOGICAL DYNAMICS

16

[2.24]

subset of T for which T_= SK. Since y E xT = xSK-' = xSK-' , there exists k E K such that yk E xS. Now x E ykS. Hence x E yT. The proof is completed. 2.25. THEOREM. Let X be minimal under T, let S be an invariant subgroup of T, let K be a compact subset of T, let T = SK and let a be the class of all orbit-

closures under S. Then a is a partition of X and

crd a = crd T\S. <_ crd K(x E X). PROOF. By 2.24, a is a partition of X. If x E X and if r, o- E T, then xrS = xrS if and only if rQ ' E Ss . The conclusion follows. 2.26. REMARK. Let X be minimal under T, let x, y E X, let S be a syndetic invariant subgroup of T and for t E T define sit : S -> S by sit = t-'st (s E S). Then there exists t E T such that (at, tp,) in a topological isomorphism of the

transformation group (xS, S, r I xS X S) onto the transformation group (YS, S, 7r YS X S). PROOF.

Let K be a compact subset of T such that T = SK. Since

X = xT = xSK, there exists t E K such that y E St,= xtS whence xSt = yS by 2.24. The conclusion follows. '

2.27. DEFINITION. A subset A of X is said to be totally minimal under T provided that A is a minimal orbit-closure under every syndetic invariant subgroup of T. Let T be discrete. Then every connected minimal orbit-closure

2.28. THEOREM.

under T is totally minimal under T. PROOF. Use 2.25. 2.29. REMARK.

Let a be the class of all orbits under T. Then:

(1) If E C X, then ET = Ea. (2) a is star-open. 2.30. REMARK.

Let the class a of all orbit-closures under T be a partition

of X. Then:

(1) If E C X, then ET C Ed = UaEE xT C ET. (2) If E is an open subset of X, then Ea = ET. (3) a is star-open. Consider the following statements: (I) The class of all orbit-closures under T is a star-closed partition of X. (II) If x E X and if U is a neighborhood of xT, then there exists a neighborhood V of x such that y E V implies yT C U. (III) If x E X and if U is a neighborhood of xT, then there exists a neighborhood V of x such that VT C U. (IV) If x E X and if DI is the neighborhood filter of x, then n vE" VT = xT. 2.31. REMARK.

Then: (1) If X is a T1-space, then I is equivalent to II.

[2.36]

ORBIT-CLOSURE PARTITIONS

17

(2) If X is a uniformizable space and if every orbit-closure under T is compact,

then I is equivalent to III. (3) If X is a compact T2-space, then I is equivalent to IV. 2.32. THEOREM. Let X be compact, let X be minimal under T and let S be a syndetic invariant subgroup of T. Then the class of all orbit-closures under S is a

star-closed decomposition of X. PROOF. Let a be the class of all orbit-closures under S. By 2.24, a is a decomposition of X. Let x E X and let K be a compact subset of T for which T = SK.

Since X = xT = xSK = xSK, it follows that a = [Ck I k E K] where C = xS. Let E be a closed subset of X. Define H = [t I t E T, Ct (1 E 0 0]. By 1.18 (4), T - H is open whence H is closed in T. Define M = H (1 K. Now M is compact and therefore CM = Ed is closed in X. The proof is completed. 2.33. THEOREM. Let X be a locally connected {normal T 1-space } { T2-space } , let T be discrete, let the class of all orbit-closures under T be a star-closed { partition }

{decomposition} of X and let S be a syndetic invariant subgroup of T. Then the class of all orbit-closures under S is a star-closed {partition} {decomposition} of X.

PROOF. By 2.24, the class of all orbit-closures under S is a {partition) {decomposition} of X. Let x E X and let U be a neighborhood of xS. There

exists a finite subset K of T such that T = SK. Since xT = xSK = xKS= U,EK xtS, there exists a finite subset H of T such that e E H, xT = UtEH xtS, and xtS (l xsS = 0 (t, s E H; t 0 s). For each t E H choose an open neighbor-

hood U, of xtS so that U, C U and Ut n U. _ 0 (t, s E Hj_t 5-1 s). There exists a connected neighborhood V of x such that y E V implies yT C UtEH U,

.

It remains to show that y E V implies yS C U. Assume there exists y E V such that yS c U. Then ys E Ut for some s E S and for some t E H with t 0 e. It follows that the connected set Vs intersects U. and U, whence Vs (l bdyU, 0 0. This contradicts VT C UtEg U, . The proof is completed. 2.34. DEFINITION.

Let (X, `U.) be a uniform space and let a be a partition

of X. For a E 'U. let a* = [(A, B) I A, B E a, A C Ba, B C Aa]. The uniformity of a generated by the uniformity base [a* I a E ti.] of a is called the partition uniformity of a. The partition uniformity of a may fail to be separated. 2.35. DEFINITION. Let X be a uniform space and let a be a partition of X. The partition a is said to be star-indexed provided that the following equivalent conditions are satisfied: (1-2) If a is an index of X, then there exists an index

of X such that x E X implies {xa,3 C xaa} {x$a C xaa}.

2.36. REMARK. Let (X, 9t) be a uniform space, let a be a star-indexed partition of X, let `U be the partition uniformity of a and for a E `U. let « _ [(xa, ya) I (x, y) E a]. Then: (1) abase of U. (2) The projection of (X, 'U,) onto (a, U) is uniformly continuous and uniformly open.

18

TOPOLOGICAL DYNAMICS

[2.36]

(3) a is star-open. (4) The topology of a induced by *0 coincides with the partition topology of a. 2.37. REMARK.

Let X be a uniform space and let a be a decomposition of X.

Then:

(1) If a is star-indexed, then a is star-open and star-closed.

(2) If X is compact and if a is star-open and star-closed, then a is starindexed.

Let X be a uniform space, let T be equicontinuous and let

2.38. THEOREM.

a be the class of all orbit-closures under T. Then:

(1) a is a partition of X. (2) If every orbit-closure under T is compact, then a is a star-closed decomposition of X. PROOF. (1) Let x, y E X and let y E xT. It is enough to show that x E yT. Let a be a symmetric index of X. There exists a neighborhood U of y such that

Ut C yta (t E T). Choose t E T such that xt E U. Then x E Ut 1 C yt_1a, yt-'

E xa and yT n xa

0. This shows x E yT.

(2) Let A E a and let a be an index of X. For each x E X there exists a neighborhood U. of x such that Ust C xta (t E T). Define U = UsEA U,, Then U is a neighborhood of A and UT C xTa. The proof is completed. Let X be a uniform space, let T be uniformly equicontinuous

2.39. THEOREM.

and let a be the class of all { orbits } { orbit-closures } under T. Then a is star-indexed. PROOF.

Obvious.

2.40. DEFINITION. Let x E X and let S be the class of all closed syndetic invariant subgroups of T. The trace of x under T or the T-trace of x is defined to be the subset f sE s xS of X. A trace under T or a T-trace is defined to be a subset A of X such that A is the T-trace of some point of X. The phrase "under T" and the prefix "T-" may be omitted when there is no chance for ambiguity. 2.41. REMARK.

Let 8 be the class of all closed syndetic invariant subgroups

of T and for x E X let x* = n. $ xS be the T-trace of x. Then:

(1) If xEX,then xEx*.

(2) If x, y E X, then x* C y* if and only if x E y*. (3) If x, y E X, then x* = y* if and only if x E y* and y E x*. (4) [x* I x E X] is a covering of X. (5) [x* x E X] is a partition of X if and only if x, y E X with x E y* implies x* = y*. 2.42. REMARK.

Let X be minimal under T and let x, y E X. Then the

following statements are equivalent: (1) The T-traces of x and y coincide.

__

(2) If S is a closed syndetic invariant subgroup of T, then xS = yS.

ORBIT-CLOSURE PARTITIONS

[2.441

19

2.43. REMARK. Let X be a compact uniform space, let X be minimal under T and let x, y E X. Then the following statements are equivalent: (1) The T-traces of x and y coincide. (2) If a is an index of X And if S is a closed syndetic invariant subgroup of T,

then there exists s, r E S such that (xs, yr) E a. 2.44. NOTES AND REFERENCES.

(2.02) The term syndetic is from the Greek avvSerucos, meaning to bind together. When T = 6t, a subset of T is syndetic if and only if it is relatively dense.

(2.06) This lemma is proved in Gottschalk [8]. (2.11) The concept of a minimal set is due to G. D. Birkhoff (cf. Birkhoff [1], vol. 1, pp. 654-672).

(2.15) A restricted form of this theorem is proved in Hilmy [1]. (2.18) The proof is essentially that due to A. Markoff for the case of a continuous flow (cf. Markoff [1]).

(2.22) Concerning the extremism law, see Gottschalk [11]. A proof of the existence of a minimal set for the case of a flow in a compact space was given by Birkhoff (see (2.11)). (2.40) The concept of T-trace appears in the thesis of Gottschalk [1].

3. RECURSION 3.01. STANDING NOTATION.

Throughout this section (X, T, 7r) denotes a

transformation group. We shall often use the phrase "the transformation group

T" or simply the symbol "T" to stand for the phrase "the transformation group (X, T, 9r)." 3.02. DEFINITION. Let x E X. The period of T at x or the period of x under T is defined to be the greatest subset P of T such that xP = x. 3.03. REMARK.

Let x E X and let P be the period of T at x. Then

(1) P = x7rs 1.

(2) P is a subgroup of T. (3) If X is a T0-space, then P is closed in T. (4) If t E T, then t-'Pt is the period of T at xt. PROOF.

(3) Let t E P. Since xP C xP = x, we have xt E x and xt C Y.

Now t-1 E P_1 = P-1 = P. As before, xt-1 E x whence x E xt = xt and x C xt.

We conclude that xt = x, xt = x and t E P. 3.04. DEFINITION. The period of T is defined to be the greatest subset P of T such that x E X implies xP = x. 3.05. REMARK. Let P be the period of T. Then: (1) P is an invariant subgroup of T. (2) If P. (x E X) is the period of T at x, then P= ntEx Px (3) T is effective if and only if P = [e]. (4) If X is a To-space, then P is closed in T and the transformation group (X, T/P, p) is effective where p : X X TIP -+ X is defined by (x, tP) p =

xt(xEX,t(=- T). 3.06. DEFINITION.

Let x E X. The transformation group T is said to be

periodic at x and the point x is said to be periodic under T provided that the period of T at x is a syndetic subset of T. The transformation group T is said to be fixed at x and the point x is said to be fixed under T provided that xT = x,

that is, the period of T at x is T. The transformation group T is said to be pointwise periodic provided that T is periodic at every point of X. The transformation group T is said to be periodic provided that the period of T is a syndetic subset of T. 3.07. REMARK. Let x E X and let T be periodic at x. Then: (1) T is pointwise periodic on xT. (2) If T is abelian, then T is periodic on xT. (3) xT is compact. 20

RECURSION

[3.13]

3.08. LEMMA.

Let x E X, let P be the period of x under T and let p : T\P

21

xT

be defined by (Pt)(p = xt (t E T). Then: (1) rp : T\P -' xT is one-to-one continuous onto. (2) The following statements are pairwise equivalent: (I) p : T\P --j xT is homeomorphic. (II) irs : T --* xT is open.

(III) Try : T -. xT is open at e. (3) If xT is a second-category T2-space and if T is separable locally compact, then p : T\P --* xT is homeomorphic.

(3) We may assume that X = xT. We first show that if V is a neighborhood of e, then int (xV) 0 0. Let V be a neighborhood of e. Choose a compact neighborhood W of e such that W C V. There exists a countable subset C of T such that T = WC. Now X = xT = xWC = Ueec xWt and each set xWt (t E C) is closed in X. If int (xW) = 0, then int (xWt) = 0 (t E C) and X is of the first category. Hence int (xW) 0 0. Since xW C xV, we have int (xV) 0 0. PROOF.

It is enough by (2) to show that 1r= : T -* X is open at e. Let U be a neighborhood of e. Choose a neighborhood V of e such that VV-' C U. Now int (xV) 0 0. Hence there exists t E V such that xV is a neighborhood of xt. It follows that

xVt-' is a neighborhood of x, xVV-' is a neighborhood of x and xU = U7r. is a neighborhood of x. The proof is completed. 3.09. THEOREM. Let x E X, let X be a T,,-space and let T be separable locally compact. Then x is periodic under T if and only if xT is compact.

PROOF. Use 3.08 (3).

3.10. REMARK. The following statements are valid:

(1-2) If x E X, then T is periodic at x if and only if there exists a syndetic { subset) { subgroup) A of T such that xA = x. (3-4) T is periodic if and only if there exists a syndetic {subset} {invariant subgroup) A of T such that x E X implies xA = x. 3.11. REMARK. The recursion notions defined below are generalizations of periodicity notions. 3.12. STANDING NOTATION. Let there be distinguished in the phase group T certain subsets which are called admissible. Let a denote the class of all admissible subsets of T.

3.13. DEFINITION.

Let x E X. The transformation group T is said to be

recursive at x and the point x is said to be recursive under T or T-recursive provided that if U is a neighborhood of x, then there exists an admissible subset

A of T such that xA C U. The transformation group T is said to be pointwise recursive provided that T is recursive at every point of X. Let X be a uniform space. The transformation group T is said to be recursive

22

TOPOLOGICAL DYNAMICS

[3.131

provided that if a is an index of X, then there exists an admissible subset A of T such that x E X implies xA C xa. Let x E X. The transformation group T is said to be locally recursive at x and the point x is said to be locally recursive under T or locally T-recursive provided that if U is a neighborhood of x, then there exist a neighborhood V of x and an admissible subset A of T such that VA C U. The transformation group T is said to be locally recursive provided that T is locally recursive at every point of X. Let X be a uniform space. The transformation group T is said to be weakly, recursive provided that if a is an index of X, then there exist an admissible subset A of T and a compact subset K of T such that x (=- X implies the existence

of an admissible subset B of T such that A C BK and xB C xa. Let X be a uniform space. The transformation group T is said to be weakly, recursive provided that if a is an index of X, then there exist an admissible subset A of T and a finite subset K of T such that x E X implies the existence of k, h E K such that xkAh C xa. Let x E X. The transformation group T is said to be locally weakly recursive at x and the point x is said to be locally weakly recursive under T or locally weakly T-recursive provided that if U is a neighborhood of x, then there exist a neighbor-

hood V of x, an admissible subset A of T and a compact subset K of T such that y E V implies the existence of an admissible subset B of T such that

A C BK and yB C U. The transformation group T is said to be locally weakly recursive provided that T is locally weakly recursive at every point of X. Let x E X. The transformation group T is said to be regionally recursive at x and the point x is said to be regionally recursive under T or regionally T-recursive provided that if U is a neighborhood of x, then there exists an admissible subset

A of T such that a E A implies U (l Ua 54 0. The transformation group T is said to be regionally recursive provided that T is regionally recursive at every point of X. 3.14. DEFINITION. In agreement with 1.03 a recursion property of the transformation group T is said to be intrinsic provided that the property

"admissible subset of T" can be characterized solely in terms of the topological and group structures of T.

3.15. DEFINITION. An intrinsic recursion property relative to the discrete topology of T is indicated by placing the { adjective discrete } { adverb discretely }

before the {substantive} {adjectival} phrase which refers to the property.

3.16. REMARK. When an intrinsic recursion property is characterized intrinsically but independently of the topology of T, then this recursion property implies the corresponding discrete recursion property (and indeed the corresponding recursion property relative to every topology of T which makes (X, T, 7r) a transformation group).

RECURSION

[3.221

23

3.17. REMARK. The relative strength of recursion properties is indicated in Table 1. No reverse implication is universally valid. (See Part II.) 3.18. REMARK. The following statements are valid: (1) If T is recursive, then T is weakly, recursive. (2) If T is weakly2 recursive and if tas C Ct (t, s E T), then T is pointwise recursive.

3.19. REMARK. Our intent is that weakly recursive should mean the strongest

property such that, under reasonable hypotheses, the following statement holds: If x (E X and if T is recursive at x, then T is weakly recursive on xT. Sometimes weakly, recursive is indicated and sometimes weakly, recursive. 3.20. REMARK. The following are certain "universal" theorems on recursion properties. TABLE 1 T recursive

T locally recursive

T locally recursive at x

I

1

T weakly recursive

1

T locally weakly recursive

T locally weakly recursive at x

I

1

uniform recursive properties

3.21. THEOREM.

T pointwise recursive

T recursive at x

T regionally recursive

T regionally recursive at x

pointwise recursive properties

Let x E X, let T be

recursive properties

at a point { recursive } { locally recursive) { locally

weakly recursive} {regionally recursive} at x and let tat-' C d (t E T). Then T is { recursive) { locally recursive } { locally weakly recursive } { regionally recursive } at every point of xT.

PROOF. Use 1.11. 3.22. THEOREM.

Let R be the set of all

{ T-recursive } { locally T-recursive }

{ locally weakly T-recursive } { regionally T-recursive } points of X and let tat 1 C

a (t E T). Then R is invariant under T.

TOPOLOGICAL DYNAMICS

24 PROOF.

[3.22]

Use 3.21.

3.23. THEOREM.

Let x E X, let T be

{ recursive } { regionally recursive }

at x

and let p be a continuous mapping of X into X such that 7r`cp = cp7r` (t E T). Then T is {recursive} {regionally recursive) at x(p. PROOF.

Obvious.

3.24. THEOREM. Let x E X, let T be { locally recursive) { locally weakly recursive} at x and let rp be a continuous-open mapping of X onto X such that a`cp = rpr` (t E T). Then T is { locally recursive } { locally weakly recursive } at xcp. PROOF.

Obvious.

3.25. THEOREM. Let x E X, let T be locally recursive at x, let xT be minimal under T, and let t(ts C d (t, s E T). Then T is locally recursive at every point of xT. PROOF. Let y E xT and let U be an open neighborhood of y. Choose t E T such that xt E U whence x E Ut-'. There exist an open neighborhood W of x

and an admissible subset B of T such that WB C Ut-' whence WBt C U. Choose s E T such that ys E W and then choose a neighborhood V of y such that Vs C W. It follows that VsBt C WBt C U. Define A = sBt. Since A is an admissible subset of T such that VA C U, the proof is completed. 3.26. THEOREM. Let R be the set of all regionally T-recursive points of X. Then R is closed in X. PROOF.

Obvious.

3.27. THEOREM. Let x E X, let T be regionally recursive at x and let tat-' C a (t E T). Then T is regionally recursive at every point of xT. PROOF.

Use 3.22 and 3.26.

3.28. THEOREM. Let R be the set of all T-recursive points of X. Then every point of R is regionally T-recursive. PROOF.

Use 3.26.

3.29. DEFINITION. A base of a is defined to be a subclass a3 of a such that A E d implies the existence of B E (R such that B C A.

3.30. THEOREM. For a C X X X and A C T let E(A, a) denote the set of all x E X such that xA X xA C a. Let X be a uniform space, let U be a base of the uniformity of X, let a3 be a base of a such that B E (R implies e E B and let R be the set of all T-recursive points of X. Then R= (lee, UBem E(B, (3). PROOF.

Obvious.

3.31. THEOREM. Let X be metrizable, let P, , P2 , - - - be a sequence of subsets of T, for A C T let A E (t if and only if A (1 P. 56 0 for every positive integer n, and let R be the set of all T-recursive points of X. Then:

RECURSION

[3.341

25

(1) R is a G8 subset of X. (2) If T is regionally recursive, then R is a residual subset of X. PROOF. Let p be a metric in X compatible with the topology of X. For positive integers n and m, let E(n, m) denote the set of all x E X such that p(x, xt) >_ 1/m for all t E P . It is clear that X - R = U,;,;,,=1 E(n, m). For fixed positive integers n and m, the set E(n, m) is closed in X. Thus X - R

is an F. set and R is a G1 set.

Now assume that T is regionally recursive. Let n and m be fixed positive integers. Suppose int E(n, m) 0. Then there exists a nonvacuous open subset U of X such that U C E(n, m) and p(x, y) < 1/m (x, y (E U). Since U (1 Ut 0 0

for some t E P , we can find x E U such that xt E U whence p(x, xt) < 1/m. This contradicts the definition of E(n, m) and therefore int E(n, m) _ 0. Thus X - R is a first category subset of X and the proof is completed. 3.32. REMARK. Let X and Y be topological spaces, let A be a dense subset of X, let a be a closed subset of Y X Y, and let (p and ¢ be continuous maps of X into Y such that (xvp, x') E a for all x E A. Then (xlp, 4) E a for all x E X. 3.33. THEOREM. Let X be a uniform space, let Y be a T-invariant subset of X and let T be { recursive } { weakly2 recursive } on Y. Then T is { recursive } { weakly, recursive } on Y. PROOF.

First reading. Let a be a closed index of X. There exists an admissible

subset A of T such that x E Y and t E A implies (x, xt) E a. By 3.32, x E Y and t E A implies (x, xt) E a. The proof of the first reading is completed. Second reading. Let a be a closed index of X. There exists an admissible subset A of T and a finite subset K of T such that y EY implies the existence of

k, h E K such that ykAh C ya. We show that x E Y implies the existence of k, h E K such that xkAh C xa. Let x E Y. Choose k, h E K such that for every neighborhood U of x there exists y E U (1 Y such that ykAh C ya, that is,

t E A implies (y, ykth) E a. By 3.32, t E A implies (x, xkth) E a. Hence, xkAh C xa. The proof is completed. 3.34. THEOREM. Let X be a uniform space, let Y be a T-invariant subset of X, let T be weakly, recursive on Y and suppose that A, B, K C T with A admissible, K compact and A C BK implies B is admissible. Then T is weakly1 recursive on Y. PROOF. Let a be an index of X. Choose a symmetric index ,3 of X such that N3 C a. There exist an admissible subset A of T and a compact subset

K of T such that y E Y implies the existence of a subset C of T such that A C CK

and yC C y13. We show x E Y implies the existence of asubset B of T such that A C BK and xB C xa. It is enough to show that x EY and a E A implies

the existence of k E K such that xak-' E xa. Let x E Y and let a E A. By 1.20 (4), there exists an index y of X such that y C ,B and xyak-' C xak-',B (k (E K). Choose y E xy n Y. There exists k E K such that yak-' E y,8. It follows that (x, xak-') = (x, y) (y, yak-')(yak-", xak-1) E yii2 C #3 C a and xak-' E xa. The proof is completed.

26

TOPOLOGICAL DYNAMICS

[3.35)

3.35. THEOREM. Let X be a uniform space, let x E X, let T be recursive at x, let T be equicontinuous at x, and let T be abelian. Then T is recursive on xT. PROOF.

Let a be a closed index of X. Choose an index $ of X such that

xflr C xra (r (E T). There exists an admissible subset A of T such that xt E x3 (t E A). Then xTt = xtT E x$r C xra (T E T, t (E A) and (xr, xTt) E a (T E T, t (E A). By 3.32, (y, yt) E a (y E xT, t E A). The proof is completed. 3.36. INHERITANCE THEOREM.

For each subgroup G of T let there be distin-

guished in G certain subsets, called G-admissible, such that: (I) If A, B, C C T such that A C BC, if A is T-admissible, and if C is compact, then B is T-admissible. (II) If G is a closed syndetic subgroup of T and if A C G, then A is G-admissible if and only if A is T-admissible. Let T be locally compact and let S be a closed syndetic invariant subgroup of T. Then:

(1) If x E X, then S is recursive at x if and only if T is recursive at x. (2) S is pointwise recursive if and only if T is pointwise recursive.

PROOF. We first show that if T is recursive at x, then S. is recursive at x Suppose T is recursive at x. Let U be a neighborhood of x. There exist neighbor-

hoods V of x and W of e such that W = W-', W is compact and VW C U. By 2.10(2), we may suppose that V n x(T - SSW) = 0. There exists a T-admissible subset A of T such that xA C V. Clearly A C SAW and xAW C U. Define B = S. (1 AW. Since A C BW, B is a T-admissible subset of T. Also B C S. and xB C U. Thus S. is recursive at x. We next show that if Sz is recursive at x, then S is recursive at x. Suppose Sx is recursive at x. Let U be an open neighborhood of x. By 2.10(3), there exists a compact subset M of T such that xM C U and S. C SM-'. Let V be a neighborhood of x for which VM C U. There exists an Ss admissible subset

A of S. such that xA C V. Now xAM C U. Define B = S (l AM. Since A C BM-', B is a T-admissible subset of T. Also B C S and xB C U. Thus S is recursive at x. It now follows that if T is recursive at x, then S is recursive at x. The converse is obvious. The proof is completed. 3.37. DEFINITION. Let T be a topological group. A subset S of T is said to be replete in T, provided that S contains some bilateral translate of each

compact subset of T. A subset A of T is said to be extensive in T provided that A intersects every replete semigroup in T. 3.38. DEFINITION. If in 3.13 the term admissible set is replaced by {left syndetic set) { syndetic invariant subgroup) I translated syndetic invariant sub-

group) { extensive set), then the term recursive is replaced by { almost periodic } { regularly almost periodic } { isochronous } { recurrent 1. By { weakly almost periodic }

{ weakly isochronous } we shall mean { weakly, almost periodic } { weakly, isochronous } .

RECURSION

13.431

27

3.39. REMARK. The choice of "left" rather than "right" in the first reading of 3.38 is dictated by the notation adopted in 1.01 and 3.05, namely,

7r:XXT-4X, xt= (x,t)ir

(xEX,tET)

whence the homomorphism axiom takes the form (xt)s = x(ts)

(x E X; t, s E T).

If the adopted notation were

7r:TXX-*X, tx=ir(t,x)

(tET,xEX)

with the homomorphism axiom written as

s(tx) = (st)x

(t, s E T; x (E X),

then we would choose "right" rather than "left". In the latter notation the order of terms in the image of a point under a mapping and in a mapping product

would be the reverse of the order in the former notation. An isomorphism between the two notations is established by the group inversion of T. 3.40. REMARK. The recursion properties of 3.38 are intrinsic. They include all of the recursion properties that we shall study. 3.41. REMARK. Let admissible and recursive refer to one of the specializations

of 3.38. If 3 and 8 are topologies of the group T such that 3 C 8, if A C T and if A is admissible relative to 8, then A is admissible relative to 3. Hence the strongest recursion properties are those relative to the discrete topology of T. 3.42. DEFINITION. space Y.

Let p be a function on a topological group T to a uniform

The function cp is said to be {left} {right} uniformly recursive provided that

if a is an index of Y, then there exists an admissible subset A of T such that r E T and t E A implies { (rrp, trop) E a}{ (r(p, -rt-'gyp) E a}. The function cp is said to be {left} {right} weakly recursive provided that if

E is a compact subset of T and if a is an index of Y, then there exists an admissible subset A of T such that r (E E and t E A implies { (rio, trrp) E a { (rv, rt-'(P) E a } . The function -p is said to be bilaterally { uniformly } { weakly } recursive provided

that V is both left and right { uniformly } { weakly) recursive.

3.43. REMARK. We adopt the notation of 1.62. Let (p E 4,. Then c is a {left} {right} uniformly recursive function if and only if (P is a recursive point under (F, T, p). We adopt the notation of 1.63. Let a E 4. Then (p is a {left} {right} weakly recursive function if and only if (p is a recursive point under ((D, T, p).

TOPOLOGICAL DYNAMICS

28

3.44. DEFINITION.

[3.44]

Let 4 be a class of functions on a topological group T

to a uniform space Y. The function class 4) is said to be {left} {right} uniformly recursive provided

that if a is an index of Y, then there exists an admissible subset A of T such that r E T, t E A and cp E 4) implies { (r(p, trca) E a } { (r o, Tt-1c) E al. The function class 4) is said to be {left} {right} weakly recursive provided that if E is a compact subset of T and if a is an index of Y, then there exists an admissible subset A of T such that r E E, t E A and p E 4) implies { (rca, trop) E a } { (rcp, rt-,(P) E al. The function class 4) is said to be bilaterally {uniformly} {weakly} recursive

provided that (P is both left and right {uniformly} {weakly} recursive.

3.45. REMARK. We adopt the notation of 1.62. Let C 4) and let 4, be invariant under (4), T, p). Then the function class is { left } { right } uniformly recursive if and only if (4), T, p) is recursive on the set "Y.

We adopt the notation of 1.63. Let 4r C 4) and let k be invariant under (4), T, p). Then the function class Y is {left} {right} weakly recursive if and only if (4), T, p) is recursive on the set %F. 3.46. REMARK. In the definition of right uniformly recursive in 3.42 and 3.44, the expression (rco, rt-1(P) E a may be replaced by the expression (rcp, E a without changing the sense of the definition.

3.47. DEFINITION.

Let p be a function on X to a uniform space Y.

The function cp is said to be uniformly recursive relative to (X, T, a) provided

that if a is an index of Y, then there exists an admissible subset A of T such that x E X and t E A implies (xca, x7r` 'cp) E a. The function (p is said to be weakly recursive relative to (X, T, 7r) provided that

if E is a compact subset of X and if a is an index of Y, then there exists an admissible subset A of T such that x E E and t E A implies (xp, x7r`-'.p) E a. 3.48. REMARK.

Let rp be a function on a topological group T to a uniform

space Y. Then: (1) ca is uniformly recursive relative to the { left } { right } transformation group

of T if and only if (p is { left) [right) uniformly recursive. (2) cp is weakly recursive relative to the { left } { right) transformation group of T if and only if (p is (left } { right } weakly recursive.

3.49. REMARK. We adopt the notation of { 1.661 { 1.68 J. Let (p E (P. Then (p is a { uniformly } { weakly } recursive function relative to (X, T, ir) if and only if cp is a recursive point under (4), T, p).

Let 4) be a class of functions on X to a uniform space Y. The function class 4) is said to be uniformly recursive relative to (X, T, a) provided that if a is an index of Y, then there exists an admissible subset A of T such that x E X, t E A and (p (E 4) implies (xcp, x1r`-'(p) E a. 3.50. DEFINITION.

The function class 4) is said to be weakly recursive relative to (X, T, ,r) provided

RECURSION

13.551

29

that if E is a compact subset of X and if a is an index of Y, then there exists an

admissible subset A of T such that x E E, t E A and (p E 4' implies (xso, xa` ,.P) E a.

3.51. REMARK. Let 4) be a class of functions on a topological group T to a uniform space Y. Then:

(1) 4) is uniformly recursive relative to the {left} {right} transformation group of T if and only if 4' is {left} {right} uniformly recursive. (2) 4' is weakly recursive relative to the { left } { right } transformation group of T if and only if 4> is { left } { right } weakly recursive.

3.52. REMARK. We adopt the notation of 11.66111.681. Let ' C 4' and let I be invariant under (4', T, p). Then the function class ' is {uniformly} {weakly} recursive relative to (X, T, r) if and only if (4', T, p) is recursive on the set S.

Let X be a uniform space. Then: (1) If x E X, then the transformation group (X, T, 7r) is recursive at x if

3.53. REMARK.

and only if the motion 7rs : T -* X is left weakly recursive.

(2) If x E X, then the transformation group (X, T, 7r) is recursive on xT if and only if the motion 1r= : T -* X is right uniformly recursive. (3) The transformation group (X, T, 7r) is recursive if and only if the motion space [7ry : T ---> X I x E X] is right uniformly recursive.

3.54. REMARK. Let X be a uniform space and let (p be a function on X to a uniform space Y. Then: (1) If x E X, if the transformation group (X, T, 7r) is recursive at x and if ,p is continuous on xT, then the function ir.-p : T -* Y is left weakly recursive. (2) If x E X, if the transformation (X, T, 7r) is recursive on xT and if go is uniformly continuous on xT, then the function 7r,, p: T --* Y is right uniformly recursive.

(3) If the transformation group (X, T, a) is recursive and if gp is uniformly continuous, then the function class [7r.(p : T ---> Y I x E X] is right uniformly recursive. 3.55. NOTES AND REFERENCES.

(3.13) Use of the term recursive in studying simultaneously a number of the diverse recurrence phenomena which are of interest in the analysis of transformation groups occurs in Gottschalk and Hedlund [5]. The expression weakly almost periodic was introduced by Gottschalk [6]. (3.36) Cf. Gottschalk [2, 6, 8], Erdos and Stone [1], Gottschalk and Hedlund [5].

(3.38) The terms replete and extensive, as defined here, were introduced by Gottschalk and Hedlund [10]. If T is either 9 or (R, a subset A of T is extensive

if and only if A contains a sequence marching to + - and a sequence marching

to --.

30

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[3.55]

The expression almost periodic, as applied to a point, is a generalization of the term recurrent as used by G. D. Birkhoff ([1], vol. 1, pp. 654-672) for the case of a continuous flow in a compact space. It is not unrelated to the classic terminology of Bohr. A function which is almost periodic in the sense of Bohr is an almost periodic point of the continuous flow defined by translation if the function space under consideration is provided with the uniform (spaceindex) topology, and conversely. The expression almost periodic, as applied to a transformation group, is closely related to the same expression as used by Cameron [1] and Montgomery [1] (see, in this connection, 4.38). The phrase regularly almost periodic was introduced by G. T. Whyburn [1] in connection with a continuous transformation and its iterates. As applied to a transformation group, it is closely related to the nearly periodic of P. A. Smith [1]. In the case of a discrete or continuous flow, the property of recurrence has had a varied nomenclature (cf. Hedlund [4]). The term isochronous is to be found in Garcia and Hedlund [1].

4. ALMOST PERIODICITY 4.01. STANDING NOTATION.

Throughout this section (X, T, 7r) denotes a

transformation group. 4.02. REMARK.

Let x E X. Then the following statements are pairwise

equivalent:

(1) T is almost periodic at x; that is to say, if U is a neighborhood of x, then

there exists a left syndetic subset A of T such that xA C U. (2) If U is a neighborhood of x, then there exists a compact subset K of T

such that t E T implies xtK (1 U 0 0. (3) If U is a neighborhood of x, then there exists a compact subset K of T such that xT C UK. 4.03. THEOREM. Let x E X and let T be almost periodic at x. Then T is almost periodic at every point of xT. PROOF.

Use 3.21.

4.04. INHERITANCE THEOREM. Let T be locally compact and let S be a closed syndetic invariant subgroup of T. Then: (1) If x E X, then S is almost periodic at x if and only if T is almost periodic

at x.

(2) S is pointwise almost periodic if and only if T is pointwise almost periodic. PROOF.

Use 3.36.

4.05. THEOREM. Let M be a compact minimal orbit-closure under T. Then

T is discretely almost periodic at every point of M. PROOF. Let x E M and let U be an open neighborhood of x. By 2.12, M C UT. There exists a finite subset K of T such that M C UK whence xT C UK. By 4.02, x is discretely almost periodic.

4.06. THEOREM. Let X be compact. Then there exists a point of X which is

discretely almost periodic under T. PROOF.

Use 2.22 and 4.05.

4.07. THEOREM.

Let X be regular, let x E X and let T be almost periodic

at x. Then xT is minimal under T.

PROOF. Assume xT is not minimal. Then there exists y E xT such that x (t yT. Let U be a closed neighborhood of x for which U (1 yT = 0 and let K be a compact subset of T for which xT C UK. Since yK-' n U = 0, there exists by 1.18 (4) a neighborhood V of y such that VK-' (1 U = 0 whence V () UK = 0. However, y E xT, so that xT (1 V 34 0. Since also xT C UK, we have V (1 UK 0 0. This is a contradiction. The proof is completed. 31

TOPOLOGICAL DYNAMICS

32

[4.08]

Let X be regular and let T be pointwise almost periodic.

4.08. THEOREM.

Then the class of all orbit-closures under T is a partition of X. PROOF.

Use 4.07 and 2.23.

Let X be a T2-space, let x E X, let there exist a compact

4.09. THEOREM.

neighborhood of x and let T be almost periodic at x. Then xT is compact and T is discretely almost periodic at every point of xT. PROOF.

Let U be acompact neighborhood of x and let K be a compact

subset of T such that xT C UK. Since UK is compact, the conclusion follows from 4.05 and 4.07. Let X be a locally compact T2-space. Then the following

4.10. THEOREM.

statements are pairwise equivalent: (1) T is pointwise almost periodic. (2) T is discretely pointwise almost periodic. (3) The class of all orbit-closures under T is a decomposition of X.

PROOF. Use 4.05, 4.08 and 4.09. 4.11. THEOREM. Let X be regular, let x E X and let T be locally almost periodic at x. Then T is locally almost periodic at every point of xT. PROOF.

Use 3.25 and 4.07.

4.12. REMARK.

Let x E X. Then the following statements are pairwise

equivalent: (1) T is locally weakly almost periodic at x; that is to say, if U is a neighborhood of x, then there exist a neighborhood V of x, a left syndetic subset A of T and a compact subset C of T such that y E V implies the existence of a subset

B of T for which A C BC and yB C U. (2) If U is a neighborhood of x, then there exist a neighborhood V of x and

a compact subset K of T such that y E V implies the existence of a subset A of T such that T = AK and yA C U. (3) If U is a neighborhood of x, then there exist a neighborhood V of x and

a compact subset K of T such that y E V and t E T implies ytK (l U 54 0. (4) If U is a neighborhood of x, then there exist a neighborhood V of x and a compact subset K of T such that VT C UK. 4.13. THEOREM. Let x E X and let T be locally weakly almost periodic at x. Then T is locally weakly almost periodic at every point of xT. PROOF.

Use 3.21.

4.14. THEOREM.

Let the class of all orbit-closures under T be a star-closed

decomposition of X. Then T is discretely locally weakly almost periodic. PROOF. Let x E X and let U be an open neighborhood of x. Since xT C UT by 2.12 and xT is compact, there exists a finite subset K of T such that xT C UK.

ALMOST PERIODICITY

[4.20]

33

By 1.36 there exists a neighborhood V of x for which VT C UK. The conclusion now follows from 4.12. 4.15. THEOREM. Let X be a T2-space, let E be a compact subset of X, let there exist a compact neighborhood of each point of E and let T be locally weakly almost periodic at each point of E. Then ET is compact. PROOF.

For each x E E choose a compact neighborhood U. of x, a neighbor-

hood V. of x and a compact subset K. of T such that VVT C U=K2 . There exists a finite subset F of E such that E C UmEF V.. Since ET C UsEF VVT C UzEF U=Kz , the proof is completed. 4.16. THEOREM.

Let X be a locally compact T2-space and let T be locally

weakly almost periodic. Then the class of all orbit-closures under T is a star-closed decomposition of X.

PROOF. By 4.15 we may assume that X is compact. Let A be an orbitclosure under T and let U be an open neighborhood of A. It is enough to show that there exists a neighborhood V of A such that VT C U. Choose a closed

neighborhood W of A such that W C U. For each x E X - U there exists a

neighborhood N. of x and a compact subset K. of T such that N.T C (X - W)Ks . Select a finite subset E of X - U so that X - U C U.EE AT.. Define K = Kz . It follows that K is a compact subset of T for which (X - U)T C (X - W)K. Choose a neighborhood V of A such that C W. VK-1

Then VK-1 (1 (X - W) = 0, V (1 (X - W)K = 0, V n (X - U)T = 0, VT (1 (X - U) = 0 and VT C U. The proof is completed. 4.17. THEOREM.

Let X be a locally compact T2-space. Then the following

statements are pairwise equivalent: (1) T is locally weakly almost periodic. (2) T is discretely locally weakly almost periodic.

(3) The class of all orbit-closures under T is a star-closed decomposition of X. PROOF.

Use 4.14 and 4.16.

4.18. THEOREM.

Let X be a T2-space, let x E X, let there exist a compact

neighborhood of x and let T be almost periodic at x. Then T is discretely locally weakly almost periodic on xT. PROOF.

Use 4.07, 4.09 and 4.14.

4.19. THEOREM.

Let X be a compact T2-space, let X be minimal under T

and let S be a syndetic invariant subgroup of T. Then S is locally weakly almost periodic. PROOF.

Use 2.32 and 4.17.

4.20. STANDING NOTATION.

uniform space.

For the remainder of this section X denotes a

TOPOLOGICAL DYNAMICS

34

[4.211

4.21. REMARK. The following statements are pairwise equivalent:

(1) T is weakly almost periodic; that is to say, if a is an index of X, then there exist a left syndetic subset A of T and a compact subset C of T such that x E X implies the existence of a subset B of T for which A C BC and xB C xa.

(2) If a is an index of X, then there exists a compact subset K of T such that x E X implies the existence of a subset A of T for which T = AK and xA C xa. (3-4-5-6) If a is an index of X, then there exists a compact subset K of T such that x E X and t (E T implies { xtK n xa 54 0) { xK (1 xta

0 } { xt E xaK }

{xt E xKa}. (7-8) If a is an index of X, then there exists a compact subset K of T such that x E X implies { xT C xaK } { xT C xKa } . (9-10) If a is an index of X, then there exist an index 0 of X and a compact subset K of T such that x E X implies { xf T C xaK } { xT,6 C xKa } . 4.22. REMARK. Let T be weakly almost periodic. Then every orbit under T, and therefore every orbit-closure under T, is totally bounded. PROOF.

Use 4.21 (8).

4.23. REMARK. The following statements are valid: (1) If T is weakly almost periodic, then T is locally weakly almost periodic.

(2) If X is compact and if T is locally weakly almost periodic, then T is weakly almost periodic. 4.24. THEOREM. equivalent:

Let X be compact. Then the following statements are pairwise

(1) T is weakly almost periodic. (2) T is discretely weakly almost periodic. (3) The class of all orbit-closures under T is a star-closed decomposition of X. PROOF.

Use 4.17 and 4.23.

Consider the following statements: (I-II) If a is an index of X, then there exist an index f3 of X and a compact subset K of T such that x E X implies { xl3T C xKa } { xT,3 C xaK } . (III-IV) If a is an index of X, then there exists an index /3 of X such that x E X implies {x/3T C xTa} {xT# C xaT}. (V) T is weakly almost periodic. 4.25. REMARK.

Then:

(1) I is equivalent to II. (2) III is equivalent to IV. (3) I is equivalent to the conjunction of III and V. (4) If X is compact, then I through V are pairwise equivalent. 4.26. THEOREM. Let x E X, let there exist a compact neighborhood of x and let T be almost periodic at x. Then T is discretely weakly almost periodic on xT.

ALMOST PERIODICITY

[4.30]

PROOF.

35

Use 4.09, 4.18 and 4.23.

4.27. THEOREM. Let Y be a T-invariant subset of X and let T be weakly almost periodic on Y. Then T is weakly almost periodic on Y. PROOF.

Use 3.34.

4.28. THEOREM. Let X be complete, let x E X and let T be weakly almost periodic on xT. Then xT is compact and T is discretely weakly almost periodic

on xT. PROOF.

Use 4.22, 4.24 and 4.27.

4.29. THEOREM. Let X be compact, let X be minimal under T and let S be a syndetic invariant subgroup of T. Then S is weakly almost periodic. PROOF.

Use 2.32 and 4.24.

4.30. THEOREM. Let X be a compact minimal orbit-closure under T and let a be a class of nonvacuous subsets of T, called admissible, such that tcts C a

(t, s E T). Consider the following statements: (I) T is locally recursive. (II) T is locally recursive at some point of X. (III) If a is an index of X, then there exist x E X, a neighborhood V of x and an admissible subset A of T such that VA C xa. (IV) T is weakly, recursive. Then: (1) I is equivalent to II; II implies III; III is equivalent to IV. (2) If X is metrizable, then I, II, III, IV are pairwise equivalent. PROOF.

(1) By 3.25, I is equivalent to II. Clearly, II implies III.

Assume III. We prove IV. Let a be an index of X. Choose a symmetric index l3 of X for which 63 C a. By 4.21 (7) and 4.24 there exists a finite subset F of T such that x E X implies xT C x9F-1. Select an index y of X such that x E X and s E F implies xys C xs,6. There exist xo E X, a neighborhood V of x0 and an admissible subset A of T such that VA C x0-y- Choose a finite subset E of T for which X = VE-1. Define K = E U F. We show x C X implies the existence

of k, h E K such that xkAh C xa. Let x C X. Choose k E E such that xk E V

and then choose h E F such that xkAh (1 x/3 - 0. Since xkAh C VAh C xoyh C x0h,6, it follows that xkAh C x(33 C xa. This proves IV. Assume IV. We prove III. Let a be an index of X. Choose a symmetric index e of X for which (34 C a. There exist an admissible subset B of T and a finite

subset K of T such that x E X implies the existence of k, h E K such that xkBh C x fl. For k, h E K let { k, h } denote the set of all x E X such that xkBh C xx. Clearly X = Uk,AEK {k, h}. We show that if k, h E K and if x (=- {k, h}, then xkBh C x,63. Let k, h E K,

let x E { k, h) and let b E B. Choose an index y of X such that y C i3 and xykbh C xkbhl3. Now choose y E xy (1 { k, h 1. Then (x, xkbh) = (x, y) (y, ykbh) (ykbh, xkbh) E y,62 C 63 and xkbh E x$3.

36

TOPOLOGICAL DYNAMICS

[4.301

Since X = Uk,hEK {k, h} and K is finite, there exist k, h E K such that 0. Hence there exist k, h E K, x E { k, h } and a neighborhood V of x such that V X V C (3 and VC j k, h}. Now VkBh C xa since y E V and b E B implies (x, ykbh) = (x, y) (y, ykbh) E i *' C a whence ykbh E xa. Define A = kBh. Then A is an admissible subset of T such that VA C xa. This proves int { k, h }

III. (2) Assume III. We prove II. We first show: (L) If U is a nonvacuous open subset of X, then there exist x E U, a neighborhood V of x and an admissible subset A of T such that VA C U.

Let U be a nonvacuous open subset of X. Since X = UT, there exists an index a of X such that x E X implies the existence of t E T such that xa C Ut whence xat 1 C U. There exist y E X, a neighborhood W of y and an admissible subset B of T such that WB C ya. Choose t E T such that yat 1 C U. Define x = yt 1. Since xt = y, there exists a neighborhood V of x such that Vt C W. Hence VtBt 1 C WBt 1 C yat 1 C U. Define A = tBt 1. Since A is an admissible subset of T such that VA C U, the proof of (L) is completed. Let [ a I n = 1, 2, ] be a countable base of the uniformity of X. Define

Uo = X. For n = 1, 2,

, we proceed inductively as follows:

By (L) there exists a nonvacuous open subset U. of and an admissible subset A of T such that U X U C a , On C and U.A. C

It is clear that T is locally recursive at every point of fn=1 U . Since

nn,-,l U 54- 0, the proof is completed.

4.31. THEOREM. Let X be a compact minimal orbit-closure under T. Consider the following statements:

(I) T is locally almost periodic. (II) T is locally almost periodic at some point of X. (III) If a is an index of X, then there exist x E X, a neighborhood V of x and a left syndetic subset A of T such that VA C xa. (IV) T is weakly, almost periodic. Then: (1) I is equivalent to II; II implies III; III is equivalent to IV.

(2) If X is metrizable, then I, II, III, IV are pairwise equivalent. PROOF.

Use 4.30.

4.32. REMARK. The following statements are pairwise equivalent: (1) T is almost periodic; that is to say, if a is an index of X, then there exists

a left syndetic subset A of T such that x E X implies xA C xa. (2-3-4-5) If a is an index of X, then there exists a compact subset K of T such that to each t E T there corresponds k (=- K such that x E X implies {xtk E xa}{xk E xta}{xt E xak}{xt E xka}. 4.33. REMARK. The transformation group T is discretely almost periodic if and only if the transition group [ir` t E T] is totally bounded in its spaceI

index uniformity.

ALMOST PERIODICITY

[4.39]

37

4.34. THEOREM. Let Y be a T-invariant subset of X and let T be almost periodic on Y. Then T is almost periodic on Y. PROOF.

Use 3.33.

4.35. THEOREM.

Let X be compact and let T be almost periodic. Then T is

discretely almost periodic. PROOF. Let a be an index of X. Choose an index $ of X so that S2 C a. There exists a compact subset K of T such that t E T implies (xk, xt) E (x E X) for some k E K. Since it : X X K -* X is uniformly continuous, [ak I k E K] is equicontinuous. By 11.12, [Irk I k E K] is totally bounded in

its space-index uniformity. Hence, there exists a finite subset F of K such that k E K implies (xf, xk) E l3 (x (E X) for some f E F. If t C T, then there exist

k E K, f E F such that (xk, xt) E $ (x E X), (xf, xk) E 6 (x E X) whence (xf, xt) E a (x E X). The proof is completed. 4.36. DEFINITION. The transformation group T is said to be uniformly continuous provided that every transition a` (t E T) is uniformly continuous. 4.37. THEOREM. Let T be uniformly continuous and discretely almost periodic. Then T is uniformly equicontinuous. PROOF.

Use 4.33 and 11.12.

4.38. THEOREM. Let X be totally bounded and let T be uniformly continuous.

Then the following statements are pairwise equivalent: (1) T is discretely almost periodic.

(2) T is uniformly equicontinuous. (3) The transition group [7r` t E T] is totally bounded in its space-index uniformity. (4) The motion space [-irs I x E X] is totally bounded in its space-index uniformity. I

(5) If a is an index of X, then there exists a finite partition (t of X such that

AEaandtETimplies AtXAtCa.

(6) If a is an index of X, then there exists a finite partition (B of T such that

x E X and B E (B implies xB X xB C a. PROOF. By 4.37, (1) implies (2). By 11.12, (2) is equivalent to (3).

We show (3) implies (1). Assume (3). Let a be an index of X. There exists a finite subset F of T such that t E T implies (xf, xt) E a (x E X) for some f E F. This proves (1). By 11.06, (3) is equivalent to (5), and (4) is equivalent to (6). By 11.23 (1), (2) implies (4). By 11.23 (2), (4) implies (5). The proof is completed. 4.39. REMARK. The following statements are valid:

(1) If X is compact and if a is an index of X, then there exist an index of X and a neighborhood V of e such that x E X implies x$V X xfiV C a.

TOPOLOGICAL DYNAMICS

38

[4.391

(2) If X is compact and if T is provided with its left uniformity, then the motion space [1r, I x E X] is uniformly equicontinuous. 4.40. INHERITANCE THEOREM. Let X be [compact l {totally bounded } , let T be almost periodic} {discretely almost periodic} and let S be a subgroup of T. Then S is discretely almost periodic. PROOF.

Use 4.35 and 4.38.

4.41. LEMMA.

Let X be a uniform space and let A C X. Then A is totally

bounded if and only if for each index a of X there exists a totally bounded subset

E of X such that A C Ea. PROOF. The necessity is obvious. We prove the sufficiency. Let a be an index of X. Choose an index /3 of X such that ,132 C a. There exists a totally bounded subset E of X such that A C E,6. Select a finite subset F of E such that E C F13. Then A C EO C F$2 C Fa. The proof is completed. 4.42. THEOREM.

Let x E X, let x be almost periodic under T and let T be

equicontinuous at x. Then: (1) xT is totally bounded. (2) If T is abelian, then T is almost periodic on xT. PROOF. (1) We use 4.41. Let a be an index of X. There exists an index of X such that xfit C xta (t E T). For some compact subset K of T, we have xT C x$K. It follows that xT C x0K C xKa.

(2) Use 3.35. 4.43. THEOREM.

Let x E X and let T be uniformly equicontinuous. Then

the following statements are pairwise equivalent: (1) x is almost periodic under T. (2) xT is totally bounded. (3) T is discretely almost periodic on xT.

PROOF. Use 4.38 and 4.42. 4.44. THEOREM.

Let X be complete, let x E X and let T be uniformly equi-

continuous on xT. Then x is almost periodic under T if and only if xT is compact. PROOF.

Use 4.43.

4.45. THEOREM. Let X be compact, let 4 be the total homeomorphism group of X and let (k be provided with its space-index topology. Then the following statements are equivalent:

(1) (X, T, ir) is almost periodic. (2) The closure in (b of the transition group [a` : X -> X I t E T] is a compact topological group. PROOF.

Use 4.35, 4.38, 11.18 and 11.19.

4.46. REMARK. Let c be a continuous homomorphism of a topological

ALMOST PERIODICITY

[4.51]

39

group T into a compact topological group S and let (S, T, p) be the {left} {right} transformation group of S induced by T under gyp. Then: (1) (S, T, p) is almost periodic. (2) S is an almost periodic minimal orbit-closure under (S, T, p) if and only if Tcp = S. 4.47. DEFINITION. Let G be a topological group. A compactification of G is a couple (H, cp) consisting of a_compact group H and a continuous homo-

morphism co : G - H such that Up = H. 4.48. THEOREM. Let x E X and let T be abelian. Then the following statements are pairwise equivalent:

(1) X is an almost periodic compact minimal orbit-closure under (X, T, ir). (2) There exists a unique group structure of X which makes X a topological group such that (X, as) is a compactification of T. (3-4) There exist a compact topological group S and a continuous homomorphism 9 : T -* S such that Tcp = S and the {left} {right} transformation group of S induced by T under rp is isomorphic to (X, T, jr).

PROOF. Assume (1). We prove (2). Let 4) be the total homeomorphism group of X, let 45- be provided with its space-index uniformity and let 4 = t E T]. By 4.45, I = 4 is a compact abelian topological group. Define [7r` I

the continuous mapping f :'F onto X

by rpf = x

by yh = yf-' (y E X). The conclusion follows. To complete the proof, use 4.46.

4.49. DEFINITION. A topological group G is said to be {monothetic} {solenoidal}

provided there exists acontinuous homomorphism {(p : g -> G}

{(p : (R -+ G} such that {J(p = G) {&P = G}.

Let G be a topological group. Then: (1) If G is monothetic or solenoidal, then G is abelian. (2) G is monothetic if and only if there exists x E G such that [x° I n E 5]

4.50. REMARK.

is dense in G. 4.51. REMARK.

Let X be a compact uniform space. Then the following

statements are equivalent: (1) There exists a { discrete } { continuous } flow on X under which X is an almost periodic minimal orbit-closure.

TOPOLOGICAL DYNAMICS

40

[4.51]

(2) There exists a group structure of X which makes X a {monothetic} {solenoidal } topological group. 4.52. THEOREM. Let G be a compact abelian group. Then the following statements are pairwise equivalent: (1) G is monothetic. (2) G is separable and G/K is monothetic where K is the identity component of G.

(3) G is separable and if H is an open-closed finite-indexed subgroup of G, then G/H is cyclic. (4) The character group of G is algebraically isomorphic to a subgroup of the circle group. PROOF. Cf. Anzai and Kakutani [2]. 4.53. REMARK.

Let G be a compact connected separable abelian group.

Then G is monothetic. Let G be a compact abelian group. Then the following statements are pairwise equivalent: (1) G is solenoidal. (2) G is separable connected. (3) The character group of G is algebraically isomorphic to a subgroup of the line group R. PROOF. Cf. Anzai and Kakutani [2]. 4.54. THEOREM.

4.55. THEOREM.

Let X be a compact uniform space containing more than

one point and let X be an almost periodic minimal orbit-closure under a continuous flow (X, (1, 7r). Then X is not totally minimal under (X, (R, 7r). PROOF. Let x E X. By 4.47 we may suppose that X is a topological group such that 7rs : 61 -* X is continuous homomorphic. By the theory of characters

there exists a continuous homomorphism p of X onto the (additive) circle group C. There exists t E (R such that the subgroup of C generated by xtcp is finite. Since [x(tn) I n E g](p = [(tn)ir.,p I n E 9] = [(tar=(o)n I n E J] = [(xtcp)n

n E J], the orbit of x under the discrete flow generated by ir` is not dense in X. The proof is completed. 4.56. REMARK. The only universally valid two-termed implications among the almost periodicity properties are the obvious ones. These implications are summarized in Table 2. (See Part Two.)

4.57. REMARK. Let (p be a function on a topological group T to a uniform space Y. Then the following statements are pairwise equivalent: (1) cp is {left} {right} uniformly almost periodic. (2) If a is an index of Y, then there exists a left syndetic subset A of T such

rtv) E a) . (3) If a is an index of Y, then there exists a right syndetic subset A of T'

that r (E T and t E A implies { ( r r p , t r o p ) E a } { (rcp,

such that r E T and t E A implies { ( r c p , t r i p ) E a } { (rrp, rtlp) E a}.

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14.59]

41

(4) If a is an index of Y, then there exists a compact subset K of T such that t E T implies the existence of k E K such that r (E T implies { (krrp, trop) E a) { (rkcp, rtp) E a}. TABLE 2

T almost periodic

T locally almost periodic

) T locally almost periodic

at x

T weakly

almost periodic

T locally weakly almost periodic

T locally weakly almost periodic

at x

I

uniform almost periodicity properties

I

T poi se almost periodic

T almost periodic

pointwise almost periodicity properties

almost periodicity properties

at x

at a point

4.58. THEOREM. Let cc be a continuous {left} {right} uniformly almost periodic function on a topological group T to a uniform space Y. Then cc is {left} {right} uniformly continuous and bounded. PROOF. Suppose (p is continuous and left uniformly almost periodic. We show cp is left uniformly continuous. Let a be an index of Y. Choose a symmetric index ,6 of Y such that $8 C a. There exists a compact subset K of T such that t E T implies the existence of k E K such that (krcp, trop) (E,6 (r E T).

Select a neighborhood U of e such that kUcp C kcpe (k E K). It is enough to show that t E T implies tUcp C trpa. Let t E T. There exists k E K such that (krcp, trip) E ,B (r E T). Hence tUso C kUcp$ C

C tcc$ C tcpa and tUcp C trpa.

We use 4.41 to show that ip is bounded, that is, the range Tip of cp is totally bounded. Let a be an index of Y. There exists a compact subset K of T such that t E T implies the.existence of k E K such that (krcp, trip) C a (r (E T). Then Tip C Kioa. The proof is completed. 4.59. THEOREM. Let (4), T, p) be the {left} {right} uniform functional transformation group over a topological group T to a uniform space Y and let cp E -1). Then: (1) The orbit pT of (p is totally bounded if and only if (p is {left} {right} uniformly almost periodic.

_

(2) If Y is complete, then the orbit-closure cpT of ip is compact if and only if p

is { left } { right } uniformly almost periodic. PROOF. Use 4.43 and 11.05.

42

TOPOLOGICAL DYNAMICS

[4.601

4.60. REMARK. Let 'p be a complex-valued function on a discrete group. Then (p is { left almost periodic } { right almost periodic } { almost periodic } in

the sense of von Neumann if and only if rp is { left } { right } { bilaterally } uniformly

almost periodic. (Cf. von Neumann [1].) 4.61. THEOREM. Let 'p be a continuous function on a topological group T to a uniform space Y. Then the following statements are pairwise equivalent: (1) rp is bilaterally uniformly almost periodic. (2) cp is bilaterally uniformly almost periodic with respect to the discrete topology of T.

(3-4) p is { left } { right } uniformly almost periodic and { right } { left } uniformly continuous.

(5) If a is an index of Y, then there exists a finite partition 8 of T such that E E 8 and t, s E T implies tEs'p X tEs'p C a.

PROOF. We show (1) implies (2). Let 'p be bilaterally uniformly almost periodic and let (4>, T, p) be the {left} {right} uniform functional transformation group over T to Y. By 4.58, (p E 4), and by 3.43, 'p is an almost periodic point under (4), T, p). It follows from 4.43 that T is discretely almost periodic on rpT, whence, by 3.43, (p is { left If right) uniformly almost periodic with respect to the discrete topology of T. Clearly (2) implies (1). By 4.58, (1) implies (3) and (4). We show 1(3)) { (4) } implies (5). Assume 1(3)1 {(4)J. Let ('i, T, p) be the (left) { right I uniform functional transformation group over T to Y. By 4.58,

(p E T); by 3.43, cp is an almost periodic point under (4,, T, p); and by 4.43, (PT is totally bounded. Let a be an index of Y and let a. (cf. 11.01) be the correspond-

ing index of 4). By 4.38, there exists a finite partition 8 of T such that E E 8 and ¢ E rpT implies VE X ¢E C a*d . Let E E 8, let t, s E T and let Then PE X ,/'E C at implies tEscp X tEscp C a. This proves (5). We show (5) implies (2). Assume (5). Let a be an index of Y and let 8 be the corresponding finite partition of T. Let K be a finite subset of T such that E E 8

implies K n E # 0. Let t E T. Then there exists E E 8 and k E K such that t, k E E. Let r E T. It follows that ETlp X Ercp C a, whence (IcTcp, trop) E a, and rEEp X rE(p C a, whence (rkcp, rt(p) E a. By 4.57, 'p is bilaterally uniformly almost periodic with respect to the discrete topology of T. The proof is completed. 4.62. REMARK. Let cp be a continuous function on a topological group T to a uniform space Y and let the left and right uniformities of T coincide. Then 'p is left uniformly almost periodic if and only if 'p is right uniformly almost periodic. In such an event the words "left" and "right" may be omitted.

Let T be a topological group, let Y be a uniform space, and be a finite class of continuous bilaterally uniformly almost periodic functions on T to Y. Then the function class is bilaterally uniformly almost periodic. 4.63. THEOREM.

let

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[4.66]

43

PROOF. Let ('F, T, p) be the {left} {right} uniform functional transformation group over T to Y. By 4.61, 4) C'F. Hence, (PT = U, e 4, (pT is totally bounded. By 4.38, ('F, T, p) is almost periodic on 4T and therefore, by 3.45, the function class (PT, which contains -1), is {left} {right} uniformly almost periodic. The proof is completed.

4.64. THEOREM. Let T be a topological group, let Y be a complete uniform space, let n be a positive integer, let 0 be a continuous n-ary operation in Y, and let (p, , , (p,, be continuous bilaterally uniformly almost periodic functions on T to Y. Then 0((p, , , pn) is a continuous bilaterally uniformly almost periodic function on T to Y. PROOF.

Let a be an index of Y. Now E = TV, X ... X Tcn is compact

by 4.60 and therefore 0 is uniformly continuous on E. Hence, there exists an , n) with (x; , y;) E a (i = 1, index /3 of Y such that x; , y; E T(p; (i = 1, , xn), 0(y, , , y.)) E a. By 4.63, there exists a 2, , n) implies (O(x,, left syndetic subset A of T such that r E T and t E A implies { (T(p{ , tr(p;) E ,9 , n) 1. Hence, T/ E T and t E A , n) } { (r(p; , tr(p;) E 0 (i = 1, (i = 1, implies 0(rt(p, ,

{ (0 (7-(P., ,

...

,

O(tT4,l

,

...

tT con))

E a) 1 (0(T(pl

,

...

, T(Pn),

, rtp )) E a}. The proof is completed.

4.65. THEOREM.

Let (X, T, 7r) be a transformation group such that X is a

compact uniform space and T is almost periodic, let Y be a uniform space, let p be a continuous function on X to Y, and let a E X. Then the function 7ra(p : T -* Y is bilaterally uniformly almost periodic. PROOF. We show 7ra(p is right uniformly almost periodic. Let a be an index of Y. Choose an index /3 of X so that (x, y) E ,B implies (x(p, y(p) E a. There exists a left syndetic subset A of T such that xA C x,8 for all x E X. If r E T and t E A, then (r7ra , Tt7ra) = (aT, art) E / and (T7ra(p, Tt7ra(p) E a. By 4.39, Ira is left uniformly continuous. Hence, 7ra(p is left uniformly continuous. By 4.61, 7ra(p is also left uniformly almost periodic. The proof is completed. 4.66. LEMMA.

Let X be a compact uniform space, let a be an index of X,

and let Y be a uniform space which contains an arc. Then there exists an index ,l3 of Y and a finite set 4' of continuous functions on X to Y such that x, , x2 E X with (x,(p, x2(p) E ,e ((p E 4') implies (x, , x2) E a. PROOF. Let E be an are in Y, let yo , y, be the endpoints of E, and let Q be an index of Y such that (yo , y,) /3. Choose a symmetric open index y of X such that y3 C a. Select a finite subset A of X such that X = U.GA ay. Since X is normal, for each a E A there exists a continuous function 4a on X to E such that x(pa = yo (x (E ay) and x(pa = y, (x (E X - ay2). Define 4) = [(pa a E A]. Let x, , x2 E X with (x,(p , x2(p) E /3 ((p E 4'). Assume (x, , x2) a. aye since otherwise x2 E aye, Now x, E ay for some a E A. It follows that x2

TOPOLOGICAL DYNAMICS

44

[4.661

a E x,' , x2 E x,y3 and (x, , x2) E y3 C a. Hence x,cpa = yo , x2tpa = y, and (yo ,

y,) = (x,cpa , x2c .,) E ,6. This is a contradiction. The proof is completed.

4.67. THEOREM.

Let (X, T, 7r) be a transformation group such that X is a

compact uniform space which is minimal under T and let Y be a uniform space which contains an arc. Then the following statements are pairwise equivalent: (1) T is almost periodic. (2) If rp is a continuous function on X to Y and if a E X, then the function 7ra(p : T - Y is bilaterally uniformly almost periodic. (3) If p is a continuous function on X to Y, then there exists a E X such that the function 7rarp : T -* Y is right uniformly almost periodic.

PROOF. By 4.65, (1) implies (2). Clearly, (2) implies (3). Assume (3). We prove (1). We first show (I): If p is a continuous function on X to Y and if x E X, then 7rycp is a right uniformly almost periodic function. Let (p be a continuous function on X to Y. By hypothesis, there exists a E X such that 7rarp is a right uniformly almost periodic function. Let a be a closed

index of Y. There exists a left syndetic subset A of T such that (anp, artcp) = (riralp, Tt7r,jp) E a (r E T, t E A). By 3.32, (x(p, xtcp) E a (x E X, t E A). Hence (7-7r. (P, Tt7rep) = (x1- , xTtpo) E a (x (E X, T E T, t E A). This proves (I). Let a be a closed index of X. By 4.66, there exists an index 3 of Y and a finite set 4> of continuous functions on X to Y such that x, , x2 E X with (x,op, x,) E # (,p E I) implies (x, , x2) E a. Choose a E X. By 4.39 and (I), each Trap ('P (E 4))

is a left uniformly continuous right uniformly almost periodic function. By 4.63, the function class { 7ra5e I 'p E -' } is right uniformly almost periodic. Hence,

there exists a left syndetic subset A of T such that (a7', aTtrp) = (T7ra'p, rt7r.-P) E

$ (r E T, t E A, (p E (b). It follows that (ar, art) E a (T (E T, t E A) and, by

3.32, (x,xt)Ea(xEX,tET).

This proves (1). The proof of the theorem is completed. 4.68. DEFINITION.

Let T be a group, let Y be a uniform space, and let

'p : T -* Y. The map p is said to be stable provided that if a is an index of Y, then there exists an index ,6 of Y such that t, s E T with (tip, s(p) E i3 implies (tr', ST(p) E a (r (E T) and (Ttp, rs'p) E a (r E T). 4.69. THEOREM. Let T be a group, let Y be a topological group provided with its bilateral uniformity, and let v be a homomorphism of T into Y. Then 'p is stable. PROOF. if trp (s'p)

Let U be a neighborhood of the identity of Y. If t, s, r E T and

-' E U and (t'p) ' sp E U, then trip (sr'p) -' E U and (Tt(p) -' Tscp E U. The proof is completed. 4.70. REMARK. Let X be a set, let o be a group structure of X, let `U. be a uniformity of X, let 3 be the topology of X induced by 'U., and for each a E `U. suppose there exists,6 E `U. such that (x, y) E $ implies (x o z, y o z) E a (z E X).

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[4.74]

45

Then (X, o, 5) is a topological group such that the bilateral uniformity of (X, o, 3) coincides with `U.. 4.71. THEOREM.

Let T be a group, let Y be a separated uniform space, and

let (p be a map of T onto Y. Then (p is stable if and only if there exists a (necessarily unique) group structure o of Y such that (Y, o) is a topological group, (p : T -* (Y, o) is a homomorphism, and the bilateral uniformity of (Y, o) coincides with the given uniformity of Y. PROOF. The necessity follows from 4.69. We prove the sufficiency. Suppose (p is stable. Define G = ev(p '. We observe that if t, s E T, then the following statements are pairwise equivalent: t(p = s(p; tT(p = sT(p (r E T); Tt(p = Ts(p (T E T); is-' E G. Now G is a group since e E G and from t, s E G it follows that t(p = e(p = s(p and is-' E G. Also G is invariant in T since from t E G and T E T it follows that t(p = e(p, rt(p = Te(p = r(p, Ttr 1 = TT-'(p = e(p, and rtT-' E G. Since (p is

constant on each translate of G, there exists a map 0 : T/G - Y such that ,rip = (p where 7r is the projection of T onto T/G. Now 0 is a one-to-one map of T/G onto Y. Let o be the unique group structure of Y such that 0 is an isomorphism of T/G onto (Y, o). Clearly, (p = 7r( is a homomorphism of P onto (Y, o). To finish the proof we use 4.70. Let a be an index of Y. There exists an index l3 of Y such that t, s E T with (t(p, s(p) E /3 implies (t(p o T(p, s(p o r(p) = (trop, srop) E a (T E T) and (r(p o top, T(p o sop) = (Tt(p, Ts(p) E a (r E T). The proof is completed. 4.72. THEOREM. Let T be a discrete group, let Y be a topological group provided with its bilateral uniformity, and let (p be a homomorphism of T onto Y. Then .p is a uniformly almost periodic function if and only if Y is totally bounded. PROOF.

Use 4.57 and 4.69.

4.73. THEOREM. Let T be a discrete group, let Y be a totally bounded separated uniform space, and let (p be a stable map of T into Y. Then (p is a uniformly almost periodic function. PROOF.

Use 4.71 and 4.72.

4.74. STANDING NOTATION.

For the remainder of this section we adopt the

following notation.

Let N be the set of all positive integers. Let X be a set. If n E N, then X° or X" denotes the nth cartesian power of X. The total power of X, denoted X*, is UnEN Xn. The phrase "finite family" shall mean "nonvacuous finite family". We consider any finite ordered family in X to be an element of X*. We define

the binary operation composition in X* if a = (x1 , b = (y, , , y,n) E X*, then ab = (x1 , , xn , y, ,

,

xn) E X* and if

yn). Composition is associative but not commutative. If n E N and if x E X, then xn or x`n' denotes the n-tuple (x, , x). If (a, c E I) is a finite ordered family in X*, then I

,

46

TOPOLOGICAL DYNAMICS

[4.74]

J,EI a, denotes the continued composition a a,, where I = [c, , , L"] and c, < . < t" . If a C X X X, then a* denotes the set of all couples (a, b) such that for some n E N we have a = (x, , , x") E X" , b = (y, ,

.,y")EX"and

4.75. DEFINITION.

1,...,n).

Let X be a uniform space. An averaging process in X

is defined to be a function µ : X* -* X such that:

(1) If x E X and if n E N, then x"µ = X. (2) If a E X* and if b is a permutation of a, then aµ = bµ. (3) If n E N and if (a, c E I) is a finite ordered family in X", then (a,µ I

c E I)µ = (1 1,., ajµ. (4) If a is an index of X, then there exists an index ,e of X such that (a, b) E (3*

implies (aµ, bµ) E a. 4.76. STANDING NOTATION. Let X be a uniform space, let µ be an averaging process in X, let T be a discrete group and let c : T --> X.

4.77. REMARK. By 4.75(2), if n E N and if (x, E I) is a finite family in X, then (x, c E I)µ is uniquely defined. Similarly, if E is a finite subset of X, then Eµ is uniquely defined. I

Let a and ,3 be indices of X such that (a, b) E (3* implies (aµ, bµ) E a. It follows from 4.75(1) that if x E X and if (x, c E I) is a finite family in x3, then (x, c E I),u E xa. I

I

4.78. DEFINITION. Let a be an index of X. An a-mean of c is a point x of X such that taspµ E xa (t, s E T) for some a E T*. The set of all a-means of

,p is denoted by acp. 4.79. REMARK. If a and (3 are indices of X such that (3 C a, then & C a
Let a be an index of X. Then:

(1) There exists an index (3 of X such that (3

(1) Let 3 be an index of X such that (32 C a. Let x E No. Choose

y E & n x,8. There exists a E T* such that tas,pµ E yo (t, s E T). Since Y,8 C x,32 C xa, we have tascpµ E xa (t, s E T) and x E acp. The proof is com-

pleted.

(2) Let y be a symmetric index of X such that y2 C a. Let 0 be an index of X such that if x E X and if (x), I A (E A) is a finite family in x,8, then (x), I A E A)µ E xy. Let x, y E (3(0. There exists a, b E T* such that tas,pµ E x,3 (t, s E T) and tbscpµ E y(3 (t, s (E T). Let a = (r, c (E I) and let b = (v, I K C- K). Now ar,)rpA = aq,cpµ E x,B (K C- K) and r,b(pµ E y(3 (c E I). It follows that (ll K au,co)µ = (aQKTA I K C K)µ E xy and (l,EI r,b)(pµ = (11E7 r,bcp)µ = (rAPµ I t E I)µ E yy. Since 11KEK aQK and 11,E1 r,b are permutations of each other, (lKEK ao,)cpµ = (JJ,EI r,b)cpµ and we conclude that xy n yy , 0 whence (x, y) E y2 C a. The proof is completed. I

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[4.86]

4.81. DEFINITION.

47

Let x E X. The point x is said to be a mean of (P provided

that if a is an index of X, then there exists a E T* such that t, s E T implies tascpµ E xa. 4.82. REMARK. Let x E X. Then x is a mean of (p if and only if x is an a-mean of (p for each index a of X. 4.83. LEMMA. Let X be complete. Then there exists a mean of

Let it be the uniformity of X. Assume aS' Ft 0 (a E ca). By 4.79

and 4.80(2), [acp I aE `U.] is a cauchy filter base on X. Since

by 4.80(1), it follows that I

IaE1l a
I

IaE`t acp = I

IaE91 ago

0. The converse is obvious. The proof

is completed. 4.84. LEMMA.

Let (X,

i

c E I) be a finite family of sets. Then there exists a

one-to-one choice function of (X,

I

c E I) if and only if K C I implies crd K

crd U.Ex X. . PROOF. The necessity is obvious. We prove the sufficiency. The theorem is true when crd I = 0 or 1. Let n be an integer such that n > 1. Assume the theorem is true when crd I < n. We show the theorem is true when crd I = n.

Suppose erd I = n. If K C I with 0 < crd K < n implies crd K <

crd U.EK XK , then choosing µ E I and x E X it follows that A C I - µ implies crd A < UREA (X), - x), whence (X, - x I c (E I - µ) has a one-to-one choice function by the induction assumption and the proof is completed. Suppose,

on the contrary, there exists K C I such that 0 < crd K < n and crd K = crd X where X = U.EK X.. By the induction assumption (X, I K C- K) has a one-to-one choice function. It is therefore enough to show (X, - X I t (E I - K) has a one-to-one choice function. To do this it suffices by the induction assump-

tion to show that A C I - K implies crd A 5 crd UXEA (X), - X). However, if A C I - K with crd A > crd UXEA (X), - X), then crd (K U A) = crd K -[crd A > crd X + crd UaEn (X), - X) > crd UµExvn X which contradicts the hypothesis. The proof is completed. 4.85. LEMMA.

Let X be a set and let 8,

be partitions of X with the same

finite cardinal. Then the following statements are pairwise equivalent:

(1) a C 8, B C F and U a C U B implies crd a 5 crd B. (2) a C 8, B C iF and U B C U a implies crd B < crd a. (3) There exists a common choice set of a and B. PROOF.

Use 4.84.

4.86. THEOREM. Let X be complete and let p be uniformly almost periodic. Then there exists a mean of gyp. 0. PROOF. Let a be an index of X. It is enough by 4.83 to show that acp Let (3 be an index of X such that (32 C a. Choose an index y of X such that

48

TOPOLOGICAL DYNAMICS

[4.861

(c, d) E y* implies (cµ, dµ) E (3. By 4.61(5) there exists a finite partition 8 of T with least cardinal such that tEsso X tEscp C y (t, s E T; E E 8). We observe that if p = (p, c E I) and q = (q, c (=- I) are choice functions of 8, then (tp&rµ, tgscpA) E /3 (t, s E T); for we may suppose that for each c E I the points p, and q, belong to a common member of 8, whence t, s E T implies (tp,&p, tq,scp) C -,y (, E I), (tpsp, tgsp) C y* and (tpsrpµ, tgssoµ) E /3. Let a = (T, I , (E I) be a choice function of 8 and define x = We show x E acp. Let t, s E T. By 4.85 there exists a common choice function b = (Q, E I) of 8 and t-18s 1. I

I

I

Since a and tbs are choice functions of 8, we have (acpi, tbscpg) E /3. Since b and a are choice functions of 8, we have (tbssoµ, tas-pA) E /3. Thus (a(pµ, tas(pjA) E 02

E a and tasrpµ E xa. The proof is completed. 4.87. NOTES AND REFERENCES.

A number of the results of the early part of this section can be found in Gottschalk [3, 6, 7]. (4.48) For related results, see Stepanoff and Tychonoff [1]. (4.52 and 4.53) Cf. Halmos and Samelson [1]. (4.55) This theorem is due to E. E. Floyd (Personal communication).

(4.59) The connection between compactness and almost periodicity of functions was first observed by Bochner (cf. Bochner [1]).

(4.61(5)) This characterization of an almost periodic function is due to W. Maak (cf. Maak [1, 2]) (4.65 and 4.67) Forms of these theorems were originally proved by J. D. Baum (cf. Baum [1]). (4.73) The connections between stability and almost periodicity of functions have been observed and studied many times (cf. Franklin [1], Markoff [2], Bohr [1, C32], Hartman and Wintner [1]). (4.85) Cf. Halmos and Vaughan [3].

5. REGULAR ALMOST PERIODICITY 5.01. STANDING NOTATION.

Throughout this section (X, T, 7r) denotes a

transformation group.

5.02. REMARK. Let x C X. Then: (1) If T is regularly almost periodic at x, then T is isochronous at x. (2) If T is isochronous at x, then T is almost periodic at x. 5.03. THEOREM.

Let x (E X and let T be {regularly almost periodic) { iso-

chronous } at x. Then T is [regularly almost periodic l { isochronous } at every point of xT. PROOF.

Use 3.21.

Let T be a discrete group. Then: (1) If G is a syndetic subgroup of T, then there exists a syndetic invariant sub-

5.04. LEMMA.

group H of T such that H C G. (2) If G and H are syndetic subgroups of T, then G n H is a syndetic subgroup of T, of G, of H. PROOF.

(1) For t E T let gyp, : T/G -p T/G be the permutation of T/G defined

by Ep, = t-'E (E E T/G). Let P be the permutation group of T/G. Define the homomorphism cp : T --> P by tcp = (p, (t E T). Define H to be the kernel of cp. The conclusion follows. (2) If G and H are subgroups of T, then t(G n H) = tG (1 tH (t (E T) whence

T/G (1 H C T/G (1 T/H. The conclusion follows. 5.05. REMARK. Let T be discrete and let x E X. Then the following statements are equivalent: (1) T is regularly almost periodic at x. (2) If U is a neighborhood of x, then there exists a syndetic subgroup A of

T such that xA C U. 5.06. REMARK. Let T be discrete and let x E X. Then the following statements are equivalent: (1) T is isochronous at x.

(2-3) If U is a neighborhood of x, then there exist a syndetic subgroup A of T and t (E T such that {xtA C U} {xAt C U}. 5.07. INHERITANCE THEOREM.

Let T be discrete and let S be a syndetic sub-

group of T. Then:

(1) If x E X, then S is

{ regularly almost periodic isochronous } at x if and

only if T is { regularly almost periodic } { isochronous } at x. 49

TOPOLOGICAL DYNAMICS

50

[5.071

(2) S is pointwise { regularly almost periodic } { isochronous I if and only if T is pointwise { regularly almost periodic } { isochronous 1.

PROOF. From 5.04 the first reading is obvious.

Let x E X. To prove the second reading it is enough to show that if T is isochronous at x, then S is isochronous at x. By 5.04(1) we assume without loss that S is also invariant in T. We first show that if T is isochronous at x, then S. is isochronous at x. Suppose T is isochronous at x. Let U be a neighborhood of x. By 2.10(2), x (E x(T - Si).

Hence we may assume that U n x(T - Sam) = 0. There exist t E T and a syndetic subgroup A of T such that xtA C U. Therefore xtA () x(T - Sam) = 0,

to C SS , t E S. and A C t-'Sx = Sz by 2.10(1). Thus t E S. and A is a syndetic subgroup of S. such that xtA C U. This shows that S. is isochronous at x. We next show that if S is isochronous at x, then S is isochronous at x. Suppose S. is isochronous at x. Let U be an open neighborhood of x. By 2.10(3) there

exists a finite subset M of T such that xM C U and S. C SM'. Let V be a neighborhood of x for which VM C U. There exist t E S. and a syndetic subgroup A of S. such that A is invariant in T and xtA C V. Choose s E S and

m E M such that t = sm'. Define B = S (1 A. Then xsB = xsm-'Bm C xtAM C VM C U. Thus s E S and B is a syndetic subgroup of S such that xsB C U. This shows that S is isochronous at x. The proof is completed. 5.08. THEOREM.

Let X be regular, let T be discrete and let T be pointwise

regularly almost periodic. Then every orbit-closure under T is zero-dimensional. PROOF. Let x E X. We assume without loss that X = xT. Let U be a neighborhood of x. There exists a subgroup A of T and a finite subset E of T such that T = AE and xA C U. By 5.07(2), A is pointwise regularly almost periodic. By 4.08, the class e of all orbit-closures under A is a partition of X. Since X = xT = xAE, e is finite. Hence xA is an open-closed neighborhood of x. The proof is completed.

5.09. THEOREM. Let X be a compact metrizable minimal orbit-closure under T, let T be discrete, let R be the set of all points of X at which T is regularly almost periodic and let R -- 0. Then R is a T-invariant residual Ga subset of X.

Clearly, R is T-invariant. Let `U be a countable base of the uniformity of X such that every element of ti is a closed index and let B be the class of all PROOF.

syndetic invariant subgroups of T. We use the notation of 3.30. If B E B, then by 2.25 the class of all orbit-closures under B is a finite partition of X. It follows that if (3 E V and if B E B, then E(B, 3) is a union of orbit-closures under B and hence E(B, a) is open in X. Since R= no... UBEB E(B, 0) and `U is countable, we conclude that R is a residual Ga subset of X inasmuch as R is represented as a countable intersection of everywhere dense open sets. 5.10. STANDING NOTATION.

a uniform space.

For the remainder of this section X denotes

REGULAR ALMOST PERIODICITY

15.17]

51

5.11. REMARK. The following statements are valid: (1) T is regularly almost periodic if and only if T is isochronous. (2) If T is isochronous, then T is almost periodic.

Let Y be a T-invariant subset of X and let T be regularly

5.12. THEOREM.

almost periodic on Y. Then T is regularly almost periodic on Y. PROOF.

Use 3.33.

Let T be discrete. Then the following statements are equi-

5.13. REMARK.

valent: (1) T is regularly almost periodic.

(2) If a is an index of X, then there exists a syndetic subgroup A of T such

that x E X implies xA C xa. 5.14. INHERITANCE THEOREM. Let T be discrete and let S be a syndetic subgroup of T. Then S is regularly almost periodic if and only if T is regularly almost

periodic. PROOF.

Use 5.04.

5.15. THEOREM. Let X be locally compact, let x E X, let T be isochronous at x and let T be equicontinuous. Then T is regularly almost periodic on xT.

PROOF. By 4.09 we may suppose that X is compact. By 5.12 it is enough to show that T is regularly almost periodic on xT. Let a be an index of X. Choose a symmetric index a of X such that (32 C a. There exists an index y of X such

that y E X and t E T implies yyt C yti3. There exist a syndetic invariant subgroup A of T and s E T such that xsA C xy. If t E T, then xtA = xsAs 't C xys' t-' C xs' t '3, xtA X xtA C 162 and xtA C xt32 C xta. Thus t E T implies xtA C xta. The proof is completed. 5.16. LEMMA.

Let X be compact and suppose that if x, y E X with x 9 y,

then there exist a neighborhood U of x, a neighborhood V of y and an index a of

X such that t E T implies (Ut X Vt) (l a = 0. Then T is uniformly equicontinuous. PROOF.

tEB].

For A C X X X and B C T, define AB = [(xt, yt) (x, y) E A, I

Let # be an open index of X. For each z E X X X - ,Q there exists a neighborhood W, of z and an index a, of X such that W,T (1 a, = 0, whence a,T C

X X X - W. . Choose a finite subset E of X X X such that X X X - 8 C UZEE W, , whencen=EE (X X X - W,) C fl. Define a= naEE a, . Now a is an index of X. Since aT C fl xEE a,T C n,eE (X X X - W,) C 0, it follows that aT C ,6. The proof is completed. Let X be compact. Consider the following statements: (I) T is pointwise isochronous and equicontinuous.

5.17. THEOREM.

52

TOPOLOGICAL DYNAMICS

[5.17]

(II) T is pointwise regularly almost periodic and S is weakly almost periodic for every syndetic invariant subgroup S of T.

(III) T is regularly almost periodic. Then:

(1) I is equivalent to II; III implies I and II. (2) If T is discrete, then I, II, III are pairwise equivalent. PROOF. By 5.15 and 4.38, I implies II. Assume II. We prove I. Clearly, T is pointwise isochronous. It remains to prove that T is equicontinuous. Let x, y E X with x 0 y. By 5.16, it is enough to show that there exist a neighborhood U of x, a neighborhood V of y and an index a of X such that (U X V) T (1 a = 0. Since T is regularly almost periodic at x, there exists a syndetic invariant subgroup S of T such that y (t xS. By 4.24 the class a of all orbit-closures under S is a star-closed decomposition of X.

By 2.30 and 2.37, a is star-indexed. Since y ($ xS, we have xS n yS = 0, that is, xa (1 ya = 0. Choose an index a of X such that xal3 n ya(3 = 0. Let y be an index of X such that y2 C ,8. Since a is star-indexed, there exists an index

5 of X such that xoa C xay and y6a C yay Provide a with its partition uniformity, which induces its partition topology by 2.36. Clearly, a is compact. Let (a, T, p) be the partition transformation group of a induced by (X, T, 7r). Since AS = A (A E a), it follows that (a, T, p) is periodic and thus almost periodic. By 4.35, (a, T, p) is discretely almost periodic and hence, by 4.38, ((t, T, p) is uniformly equicontinuous. If µ is an index of X, then µ* = [(A, B) I A,

B E a, A C Bµ, B C Aµ] is an index of a. Since (a, T, p) is uniformly equicontinuous, there exists an index 0 of a such that (A, B) E 0 and t E T implies (At, Bt) E y*. Since the projection of X onto A is uniformly continuous, there exists an index a of X such that (p, q) E a implies (pA, qA) E 0. Define U = x5 and V = yS. Let x, E U, y1 E V and t E T. We must show (x,t, y1t) a.

Assume (x,t, y1t) E a. Then (x1at, ylat) = (x1ta, y1ta) E 0, whence (x1a, y1(i) E y*, x1a C y,ay and xlay (1 y,ay 0 0. Now xlay C x8ay C xay2 C xai8

and y,ay C yoay C yay2 C ya,6. Since xa(3 (1 ya,3 = 0, it follows that xlay n y,ay = 0. This is a contradiction. This proves I. Now assume T discrete. By 4.37, III implies I. Assume II. We prove III. Let a be an index of X. Choose a symmetric index # of X such that #4 C a. For each x E X there exists a syndetic invariant subgroup A. of T such that xA. C x/3. By 4.25, for each x E X there exists a neighborhood U. of x such that U.A. C xA,z,3. Select a finite subset E of X for which

X = UxEE U.. Define A= n.EE A.. Then A is an invariant subgroup of T and by 5.04, A is syndetic in T. We show that x0 E X implies x0A C xoa. Let x0 E X. Choose x E E for which xo E UU . Since xo E x0A. C xA.$ C x/32 it follows that x E x002 and xoA C xoA. C x,82 C x0,64 C xoa . Hence x0A C xoa. The proof is completed. 5.18. THEOREM. Let X be a compact minimal orbit-closure under T. Then T is regularly almost periodic if and only if T is pointwise regularly almost periodic.

REGULAR ALMOST PERIODICITY

[5.23]

PROOF.

53

Use 4.29, 5.15 and 5.17

5.19. THEOREM.

Let X be locally compact separated and let T be pointwise

regularly almost periodic. Then T is regularly almost periodic on each orbit-closure under T. PROOF.

Use 5.18 and 4.09

5.20. REMARK. The following statements are equivalent: (1) T is weakly isochronous.

(2) If a is an index of X, then there exist a syndetic invariant subgroup A of T and a finite subset K of T such that x E X implies the existence of k E K such that xkA C xa. 5.21. REMARK. Let T be discrete. Then the following statements are equivalent: (1) T is weakly isochronous. (2-3) If a is an index of X, then there, exist a syndetic subgroup A of T and a finite subset K of T such that x E X implies the existence of k E K such that

{ xkA C xa l { xAk C xa } . 5.22. THEOREM. Let Y be a T-invariant subset of X and let T be weakly isochronous on Y. Then T is weakly isochronous on Y. PROOF.

Use 3.33.

5.23. THEOREM. Let X be a compact minimal orbit-closure under T. Consider

the following statements:

(I) T is regularly almost periodic at some point of X. (II) T is isochronous at some point of X.

(III) If a is an index of X, then there exist x° E X and a syndetic invariant subgroup A of T such that xA C xa. (IV) T is weakly isochronous. Then:

(1) I implies II; II, III, IV are pairwise equivalent. (2) If X is metrizable, then I, II, III, IV are pairwise equivalent. PROOF.

(1) Clearly, I implies II; II implies III; IV implies II. Assume III.

We prove IV. Let a be an index of X. Choose a symmetric index l3 of X for which

R3 C a. By 4.21 and 4.24 there exists a finite subset F of T such that x E X implies xT C x,BF-1. Select an index y of X such that x E X and s E F implies xys C xs,B. There exists an index S of X such that S2 C 'Y. Choose xo E X and a syndetic invariant subgroup A of T such that x0A C x0&. By 4.25 and 4.29 there exists a neighborhood U of xo such that UA C x0A8. Select a finite subset

E of T for which X = UE-1. Define K = EF. We show x E X implies the existence of k E K such that xkA C xa. Let x E X. Choose t E E such that xt E U and then choose s E F such that xts E x13. Since xtAs C UAs C x0A6s C

TOPOLOGICAL DYNAMICS

54

[5.23]

x0626 C xoys C x0s(3 and xtAs n x,6 54 0, it follows that xtAs C x,33 C xa. Define

k = ts. Then k E K and xkA C xa. The proof of (1) is completed. (2) Assume III. We prove I. We first show: (L) If U is a nonvacuous open subset of X, then there exist x E U and a syndetic invariant subgroup A of T such that xA C U. Let U be a nonvacuous open subset of X. Choose xo E U and an index a of X such that xoa2 C U. There exists a finite subset E of T such that X = xoaE, -'.

Select an index i3 of X such that (x, , x2) E $ and t E E implies (x,t, x2t) E a. There exists y E X and a syndetic invariant subgroup A of T such that yA X yA C ,9. Choose t E E for which yt E xoa. Define x = yt. Since xA X xA =

yAt X yAt C a and xA n xoa

0,

it

follows

that xA C xoa2 and

xA C xoa2 C U. This proves (L). Let [a,. I n = 1, 2, ) be a countable base of the uniformity of X. Define

V, = X. For n = 1, 2,

, we proceed inductively as follows:

There exists a nonvacuous open subset U. of V. such that U. X U. C a,. By (L) there exist x,. E U. and a syndetic invariant subgroup A. of T such that of x such that x A,. C U . By 2.32 there exists an open neighborhood C U.. V,.+, C U. and It is clear that T is regularly almost periodic at every point of fn=, U

Since f ,-, U.

0, the proof is completed.

5.24. THEOREM. Let X be a compact metrizable minimal orbit-closure under T and let T be isochronous at some point of X. Then T is locally almost periodic. PROOF.

Use 4.30 and 5.23.

5.25. THEOREM.

Let X be separated, let x E X, let there exist a compact

neighborhood of x and let T be isochronous at x. Then T is weakly isochronous on xT. PROOF.

Use 5.02, 4.09, 4.07 and 5.23.

5.26. LEMMA. Let G be a zero-dimensional locally compact topological group such that the left and right uniformities of G coincide. Then every neighborhood of the identity of G contains an open-closed invariant subgroup of G. PROOF. Let U be an open-compact symmetric neighborhood of e. Choose an open-closed symmetric neighborhood V of e such that UV C U. Let H be

the subgroup of G generated by V. Define K= n.,,,, xHx '. Then K is an open-closed invariant subgroup of G such that K C U. 5.27. LEMMA. Let X be a compact uniform space, and let 4 be a zero-dimensional compact topological homeomorphism group of X. Then -D is regularly almost periodic. PROOF.

Use 5.26.

5.28. LEMMA. Let G be a discrete group, let H be a topological group with identity e and let cp be a homomorphism of G into H such that Gyp = H and for

each neighborhood U of e there exists a syndetic subgroup A of G such that Arp C U. Then H is zero-dimensional.

REGULAR. ALMOST PERIODICITY

15.331

55

PROOF. Let U be a closed neighborhood of e. There exists a syndetic subgroup A of G such that A(P C U. Define K = App. Now K is a subgroup of finite index in H. Thus K is an open-closed neighborhood of e such that K C U. The proof is completed.

5.29. THEOREM. Let X be compact, let T be discrete, let 4) be the total homeomorphism group of X and let 4) be provided with its space-index topology. Then

the following statements are equivalent:

(1) (X, T, a) is regularly almost periodic. (2) The closure in 4' of the transition group [7r` : X -' X I t E T] is a zerodimensional compact topological group. PROOF.

Use 4.44, 5.27, 5.28.

5.30. REMARK.

Let p be a continuous homomorphism of a topological

group T into a zero-dimensional compact topological group S and let (S, T, p) be the {left} {right} transformation group of S induced by T under (p. Then: (1) (S, T, p) is regularly almost periodic. (2) S is a regularly almost periodic minimal orbit-closure under (S, T, p) if and only if Tcp = S. 5.31. THEOREM.

Let x E X and let T be discrete abelian. Then the following

statements are pairwise equivalent.

(1) X is a regularly almost periodic compact minimal orbit-closure under (X, T, jr). (2) X is an almost periodic zero-dimensional compact minimal orbit-closure under (X, T, 7r). (3) There exists a unique group structure of X which makes X a topological group such that (X, Try) is a zero-dimensional compactification of T. (4-5) There exist a zero-dimensional compact topological group S and a homomorphism (P : T -* S such that Tip = S and the {left} {right} transformation group of S induced by T under (p is isomorphic to (X, T, a). PROOF.

Use 4.47 and 5.08.

5.32. LEMMA. Let G be a compact metrizable abelian group and let H be the set of all x E G such that the closure of [x" I n E 9] is zero-dimensional. Then H is a dense subgroup of G.

This result is from the theory of Lie groups. 5.33. THEOREM. Let X be a compact metric space with metric p, let cp be an almost periodic homeomorphism of X onto X and let e be a positive number. Then there exists a regularly almost periodic homeomorphism P of X onto X such that x E X implies p(xrp, x¢) < e; indeed, i' 1may be chosen to be uniform limit of a

sequence of (positive) {negative} powers of gyp. PROOF.

Use 5.32.

56

TOPOLOGICAL DYNAMICS

[5.34]

5.34. THEOREM. Let cp be a regularly almost periodic homeomorphism of a two-dimensional manifold X onto X. Then p is periodic.

PROOF. Let a be the partition space of all orbit-closures. By 5.08 the projection P : X -+ A is light interior. It is known (Whyburn [1], p. 191) that a light interior mapping on a manifold is locally finite-to-one. Hence (P is pointwise periodic. It is known (Montgomery [2]) that a pointwise periodic homeomorphism

on a manifold is periodic. The proof is completed.

5.35. REMARK. The only universally valid implications among the regular

almost periodicity properties are the obvious ones. These implications are summarized in Table 3. TABLE 3

T regularly almost periodic

I

T pointwise regularly almost periodic

T regularly almost periodic

at x

T isochronous

I

I.

T weakly isochronous

T pointwise isochronous

T isochronous

uniform regular almost periodicity properties

pointwise regular almost periodicity properties

regular almost periodicity properties

at x

at a point

5.36. NOTES AND REFERENCES.

(5.04) Since T is discrete, this is a purely group-theoretic result and (2) may be found in Kurosch [1]. (5.08) Cf. P. A. Smith [1]. (5.09) This theorem and its proof are due to E. E. Floyd. (Personal communication.) (5.18) Subject to the additional hypotheses that T is discrete and abelian, this theorem was proved by Garcia and Hedlund [1]. A proof of 5.18 was obtained by S. Schwartzman [1]. (5.33) Cf. P. A. Smith [1].

6. REPLETE SEMIGROUPS 6.01. DEFINITION. A topological group T is said to be generative provided that T is abelian and is generated by some compact neighborhood of the identity element of T. 6.02. STANDING NOTATION.

Throughout this section T denotes a generative

topological group.

6.03. REMARK. The assumption that T is generative ensures the existence of "sufficiently many" replete semigroups in T. The generality of generative topological groups T is indicated by the structure theorem [Weil [1], p. 110] that T is isomorphic to C X9' X (R' where C is a compact abelian group and n, m are nonnegative integers. The theorems of this section have lemma value. Some of them do not use all of the hypothesis that T is generative. 6.04. THEOREM.

Let P be a semigroup in T. Then:

(1) If P is replete in T, then T = P-1P. (2) If T = P-1P and if T is discrete, then P is replete in T. PROOF. (1) If t E T, then there exists s E T such that st, st2 E P whence t = (st)_1(st2) E P_1P (2) Let n be a positive integer and let t1 , - , t E T. For eaeh i = 1, ,n p . Then tt; E P choose p; , q; E P such that t; = p; 1q; . Define t = pi

(i = 1,

-

,

n). The proof is completed.

6.05. REMARK.

Let P be a semigroup in (R such that P is maximal with

respect to the property of containing only positive nonintegral numbers. Then P is replete relative to the discrete topology of CR but P is not replete relative to the natural topology of (R. 6.06. THEOREM. Let P and Q be replete semigroups in T. Then PQ and P-1

are replete semigroups in T. PROOF. Obvious. 6.07. THEOREM. Let P be a replete semigroup in T and let K be a compact subset of T such that e E K. Then nkEK kP is a replete semigroup in T. PROOF. Since Q= f kEK kP = I IkER (P n kP) and each P (l kP (k E K) is a semigroup, it follows that Q is a semigroup. Let C be a compact subset of T and define D = C V K-1C. There exists t E T such that tD C P. Now k E K

implies C C D (1 kD whence tC C tD 0 kt D C P 0 kP. Thus tC C Q and the proof is completed. 6.08. THEOREM.

Let E be a subset of T such that E contains an open symmetric

neighborhood of e which generates T and let t E T. Then U.=1 t'E' is a replete semigroup in T. 57

TOPOLOGICAL DYNAMICS

58

PROOF.

[6.08)

Obvious.

The class of all replete semigroups in T has a countable

6.09. THEOREM.

of replete semigroups in T such base, that is, there exists a sequence P, , P2 , that each replete semigroup in T contains P. for some positive integer n.

PROOF. By 6.03 there exist a compact subgroup K of T and a separable ] be a countable closed subgroup S of T such that T = KS. Let [s, , s2 , dense subset of S and let U be a compact symmetric neighborhood of e whose ). Each P" interior generates T. Define P. = Ui=1 s;,U` (n = 1, 2, (n = 1, 2, ) is a replete semigroup in T. Let P be a replete semigroup in T.

There exist a neighborhood V of e and t E T such that VtKU C P. Choose k E K and s E S for which t = ks. There exists a positive integer n such that sn E Vs. Hence s.U C VsU = Vtk 'U C P and P C P. The proof is completed. Let S be a closed syndetic subgroup of T and let K be a compact

6.10. THEOREM.

subset of T. Then there exists a compact subset H of S such that K' n S C H" for all integers n.

PROOF. We may suppose that T = SK, e E K and K = K-'. Define H = K3 n S. Now H is a compact subset of S and to prove the theorem it is enough to show that K" 0 S C H", for all positive integers n. Let n be a positive integer and let k, , , k" E K such that k, . . . kn E S. If t E T, then there

exists s E S such that t E sK-' whence s E W. Thus for each integer i (1 < kn . i < n) there exists s; E S such that s; E k, k;K. Define sn = k, Clearly sn E S and sn E k, k,K. Now se's;., E Kk;+,K C K3 and s;'s;+1 E

S (1 < i < n) whence sz's;+, E H and s;+1 E s;H (1 < i < n). Also s, E k,K C K3 and s, E S whence s, E H. We conclude that k, kn = sn E sn_,H C sn_2H2 C . . C s,Hn-' C H". The proof is completed. Let S be a closed syndetic subgroup of T. Then S is a genera-

6.11. THEOREM.

tive topological group. PROOF.

Use 6.10.

6.12. REMARK.

Actually 6.11 remains true when "syndetic" is omitted.

6.13. THEOREM. Let S be a closed syndetic subgroup of T and let Q be a replete semigroup in S. Then there exists a replete semigroup P in T such that P (1 S C Q. PROOF.

Let K be a compact symmetric neighborhood of e whose interior

generates T. By 6.10 there exists a compact subset H of S such that K" (1 S C H"

for all integers n. For some s E S we have sH C Q. Define P = Un°1 s"K". Now P is a replete semigroup in T and

P n S C U (s"K" () S) C U s"H" C Q. n=1 n=1 The proof is completed.

REPLETE SEMIGROUPS

[6.191

6.14. THEOREM.

59

Let S be a closed syndetic subgroup of T and let P be a replete

semigroup in T. Then P n S is a replete semigroup in S. PROOF.

Clearly P 0 S is a semigroup. Let H be a compact subset of S

and let K be a compact subset of T for which T = SK. Since K-'H is compact,

there exists t E T such that tK-'H C P. Choose s E S and k E K such that t = sk. Now sH = tk-'H C P and sH C S. Hence sH C P (1 S and the proof is completed. 6.15. THEOREM. equivalent.

Let A C T. Then the following statements are pairwise

(1) A is extensive in T, that is, A intersects every replete semigroup in T. (2) A intersects every translate of every replete semigroup in T. (3) T = AP for every replete semigroup P in T. (4) T = AtP for every replete semigroup P in T and every t E T. PROOF. Assume (1). We prove (2). Let P be a replete semigroup in T and let t E T. By 6.07, P (1 tP is a replete semigroup in T. Hence A intersects P (1 tP and consequently tP. Assume (2). We prove (3). Let P be a replete semigroup in T and let t E T. Since A (1 tP-' 9 0, there exist a E A and p E P such that a = tp '. Thus

t=apEAP.

Clearly, (3) implies (4). Assume (4). We prove (1). Let P be a replete semigroup in T. Since T = AP-',

there exist a E A and p E P such that e = ap'. Hence a = p and A (1 P

0.

The proof is completed. 6.16. THEOREM. Let A be an extensive subset of T and let t E T. Then to and A-' are extensive in T. PROOF.

Use 6.15 and 6.06.

6.17. THEOREM.

Let A be a syndetic subset of T. Then A is extensive in T.

PROOF. Let P be a replete semigroup in T. Choose a compact subset K of T such that T = AK and then choose t E T such that tK C P. Since T =

AtK C AP, the conclusion follows from 6.15. 6.18. THEOREM.

Let A, B, K C T such that A is extensive in T, K is compact,

and A C BK. Then B is extensive in T. PROOF. Let P be a replete semigroup in T. Choose t E T such that tK C P. From 6.15 we conclude that T = AtP C BtKP C BP and that B is extensive

in T. 6.19. THEOREM.

Let S be a closed syndetic subgroup of T and let A C S.

Then A is extensive in S if and only if A is extensive in T. PROOF.

Use 6.13 and 6.14.

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60

[6.20]

6.20. THEOREM. Let S be a closed syndetic subgroup of T, let Q be a replete semigroup in S and let K be a compact subset of T such that e E K. Then there exists a replete semigroup P in T such that PK (1 S C Q.

PROOF. By 6.13 there exists a replete semigroup R in T such that R n S C Q.

Define P= n ,,K Rk-'. By 6.07, P is a replete semigroup in T. Now PK C R whence PK n S C R n S C Q. The proof is completed. 6.21. THEOREM. Let S be a closed syndetic subgroup of T, let U be a neighborhood of e and let t E T. Then there exists a positive integer n such that t" E SU. PROOF. Let K be a compact subset of T for which T = SK. Choose a neighborhood V of e such that VV-' C U. Let a be a finite class of translates of V which covers K. To each positive integer n there correspond s E S and k E K such that t" = s k . Select positive integers p, q such that p > q and k, , k, E Vo

for some Vo E ff. Then tD The proof is completed.

= ssQ'k,kq' E Skka' C SV,,Vo' C SVV-' C SU.

6.22. THEOREM. Let S be a closed syndetic subgroup of T, let U be a neighborhood of e, let K be a compact subset of T and let k, , ka , be a sequence of elements of K. Then there exist finitely many positive integers it , , i (n > 1)

such that i, < PROOF.

subsets U1,

< in and k;,

k;, E SU.

It follows readily from 6.21 that there exist finitely many open ,

U. of T and positive integers p,

,

, p such that K C U 1

_1

U;

, m). There exists an integer j (1 < j < m) such that and U;' C S U (j = 1, k; E U; for infinitely many positive integers i. Define n = p.. Choose positive < in and k;, , , k;. E U1 . Hence integers i, , , in such that it < k;, k;w E U; C SU. The proof is completed.

6.23. STANDING NOTATION.

For the remainder of this section (X, T, r)

denotes a transformation group. 6.24. THEOREM. Let Y be a subset of X such that every replete semigroup in T contains a compact set E such that Y C YE. Then there exists a compact subset C

of T such that YT = YC. PROOF.

Let U be an open symmetric neighborhood of a such that U generates

T and U is compact. Define H = Ua and K = U3 = UH. We first show that there exists a positive integer n such that if k E K, then Y C U:-1 Yk(kH)'. To show this it is enough to prove that if ko E K, then there exists a positive integer m and a neighborhood V of e such that k E k0V implies Y C UT 1 Yk(kH)'. Now suppose ko E K. Define P = U i ko(koU)'. The set P is an open replete semigroup in T. Hence P contains a compact set E such that Y C YE. Choose a compact symmetric neighborhood V of e for which V C U and EV C P. Since EV is compact, there exists a positive integer m such that EV C U 'j-1 ko(koU)' and hence YV C YEV C U-1 Yko(ioU)'.

REPLETE SEMIGROUPS

[6.27]

61

Let k E k0V and y E Y. Choose v E V such that ko = kv. Then m

m

m

yv E U Yko(koU)` = U Ykv(kvU)' C U Ykv(kH)' i=1

i=1

i=1

and y E U-1 Yk(kH)'. This completes the proof that there exists a positive integer n such that if k E K, then Y C U;=1 Yk(kH)'. Let n denote such an integer. Choose a positive integer p (p => n) so large that if k1 , for some n + 1 of the elements k1 , , k9+1 , let us say k1 ,

kn+1 E K, then , we have k1-'k1 E U (i, j = 1, , n + 1). We now show that YT C YKD, which will complete the proof. Assume YT d YKD. Then c YK" for otherwise YT C Ui-=1 YK' C U 9 YK' C YKD. Select y E Y and kl , , k,., C K for which yk1 YKD. There exist n + 1 of the elements k1 , k9+1 , k,+l , let us say k,, , ks+1 , such that k:1k; C U (i, j = 1, , n + 1). Let r be a positive integer such that r <_ n. It follows that yk1 . k,+1 (E YK' and yk1(k1u2) . . . (k1u,+1) YK' where u2 , , u,+1 are elements of U for which Yk11(k-11H)'. We YH' and y k2 = k1u2 I* . , k,+, = klu,+1 . Thus yk;+1 ,

.

YKD+1

conclude that y E U:=1 Ykl-'(k-l1H)'. Since k1-1 E K, this contradicts the definition of n. The proof is completed. 6.25. THEOREM.

Let U be an open subset of X such that U is compact and

suppose that for every compact subset K .of T there exist x E U and t E T such that

xtK (1 U = 0. Then there exist y E U and a replete semigroup P in T such that

yP (1 U = 0. PROOF. Assume the conclusion is false. Then for each x E U and each replete semigroup P in T we have xP C\ U 0 0 whence x E UP-'. Since the inverse of a replete semigroup in T is also a replete semigroup in T, it follows that for each replete semigroup P in T we have U C UP; since U is open and U is compact, we can choose a finite subset F of P such that U C UF. Thus

each replete semigroup in T contains a finite set F for which U C UF. By 6.24

there exists a compact subset C of T such that UT = UC. Hence x E U and t E T implies xt E UC and proof is completed.

xtC-1 (1

U 0 0. This contradicts the hypothesis. The

6.26. THEOREM. Let A be a nonvacuous subset of T such that every replete semigroup in T contains a compact set E such that A C AE. Then A is syndetic in T.

PROOF. Apply 6.24 to the transformation group (T, T, p) where p : T X T -* T is defined by (t, s)p = is (t, s E T). 6.27. THEOREM. PROOF. t-1

Let P be an extensive semi group in T. Then P is syndetic in T.

Let Q be a replete semigroup in T. Choose t E P n V. Then

E Q and P C Pt-'. The conclusion now follows from 6.26.

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62

[6.28]

6.28. THEOREM. Let M be a closed subset of X, let P be a replete semigroup in T, let MP C M and let x E M_- MP. Then there exists a replete semigroup xQ. Q in T such that Q C P and x PROOF.

Let K be a compact subset of T such that K contains an open

symmetric neighborhood of e which generates T. Select t E T such that H =

tK C P. Define Q = Un=, H". Now MH C MP C M whence MH" C MH (n = 1, 2, - _) and MQ C MH. Since M is closed and H is compact, MH is closed. Then xQ C MQ C MH C MP and x (JE xQ. Since Q is a replete semigroup in T, the proof is completed. 6.29. THEOREM.

Let S be a subgroup of T which is not syndetic in T, and

let K be a compact subset of T. Then there exists a compact subset C of T such that every translate of C contains a translate of K disjoint from S.

PROOF. We may suppose that K contains an open symmetric neighborhood of e which generates T. Assume the conclusion is false. Then for each positive integer n there exists t" E T such that S intersects every translate of K which ). is contained in t"K". It follows that there exists s" E t"K n S (n = 1, 2, Since S is a group, every translate of K contained in s,-'t"K" for some positive

integer n intersects S. We show S is replete in T. Let Ko be an arbitrary translate of K. Choose a positive integer m such that K0K C Km. Since stmt-.' E K, we have Kosmtm' C Km and Ka C sm'tmKt. It follows that K,, n S 3-4 0. Thus S

is syndetic in T, contrary to hypothesis. 6.30. THEOREM. Let n be a positive integer, let S, , . , S" be subgroups of T such that each is not syndetic in T, let P be a replete semigroup in T and let Ko be a compact subset of T. Then there exists a translate of Ko contained in P and disjoint U":-I S:from PROOF. By 6.29 there exist compact subsets K, , , K. of T such that for

each i = 1,

, n every translate of K; contains a translate of K;_, disjoint from Si . Since P contains some translate of K. , the conclusion follows. 6.31. THEOREM.

Suppose T is not compact. Then there exists a replete semi-

group P in T such that e ($ P. PROOF. For otherwise [e] would be syndetic in T by 6.26 and thus T would be compact. 6.32. THEOREM. Suppose T is not compact, let P be a replete semigroup in T and let K be a compact subset of T. Then there exists a replete semigroup Q in T

such that Q C P and Q ( \ K = 0. PROOF. By 6.31 we may suppose that P 5,6 T. It is enough to show that pP (' K = 0 for some p E P and then to take Q = pP. Assume that p E P implies pP (1 K 0 whence p E KP-1. Thus P C KP-', T = PP-' C KP-'P-1 C KP-' and T = PK-'. Choose t E T such that K-'t C P; we have T = Tt = PK-'t C PP C P. This is a contradiction. The proof is completed.

[6.37]

REPLETE SEMIGROUPS

63

6.33. DEFINITION. Let x E X and let P C T. The P-limit set of x, denoted Pz , is defined to be n,ET xtP. Each point of P. is called a P-limit point of x.

Let x E X, and let P be a replete semigroup in T. Then: (1) If t E T, then P. = Px, = Pxt.

6.34. REMARK.

(2) P. is closed invariant. (3) P. = f,EP xpP C XP(4) If X is compact, then P. 5-6 0. (5) If T is connected, then xT U Ps is connected. (6) If P is connected, then xP U Pz is connected. (7) If X is compact and if P is connected, then P. is connected. 6.35. DEFINITION.

Let S be a replete semigroup in T and let A C T. The

set A is said to be S-extensive provided that if P is a replete semigroup in T such that P E S, then A (1 P 54 0. The set A is said to be S-syndetic provided there

exists a compact subset K of S such that S C AK. 6.36. DEFINITION.

Let S be a replete semigroup in T. If in 3.13 the term

admissible is replaced by { S-extensive } { S-syndetic 1, then the term recursive is replaced by { S-recurrent } { S-almost periodic } . 6.37. NOTES AND REFERENCES.

A number of the theorems of this chapter can be found in Gottschalk and Hedlund [10].

7. RECURRENCE 7.01. STANDING NOTATION.

Throughout this section (X, T, 7r) denotes a

transformation group whose phase group T is generative. 7.02. REMARx.

Let x E X. Then- the following statements are pairwise

equivalent: (1) T is recurrent at x; that is to say, if U is a neighborhood of x, then there

exists an extensive subset A of T such that xA C U. (2) If U is a neighborhood of x and if P is a replete semigroup in T, then

xPn U - 0.

(3) If U is a neighborhood of x, if P is a replete semigroup in T and if t E T, then xtP (1 U Fl- 0. (4) If U is a neighborhood of x and if P is a replete semigroup in T, then

_

xT E UP.

(5) If P is a replete semigroup in T, then x E xP. (6) If P is a replete semigroup in T and if t E T, then x E xtP. (7) If P is a replete semigroup in T, then xT = xP. 7.03. THEOREM.

Let x E X and let T be recurrent at x. Then T is recurrent

at every point of xT. PROOF.

Use 3.21.

7.04. INHERITANCE THEOREM.

Let S be a closed syndetic subgroup of T. Then:

(1) If x E X, then S is recurrent at x if and only if T is recurrent at x. (2) S is pointwise recurrent if and only if T is pointwise recurrent. PROOF.

Use 3.36.

7.05. THEOREM. Let X be locally compact, let T be pointwise recurrent and let the class of all orbit-closures under T be a partition of X. Then T is pointwise

almost periodic.

PROOF. We may suppose that X is minimal under T. Assume some point x of X is not almost periodic. Then there exists an open neighborhood U of x such that U is compact and such that for each compact subset K of T there exists t E T for which xtK (l U = 0. By 6.25 there exists y E U and a replete semigroup P in T such that yP (l U = 0. Hence x yP. Since y is recurrent, yP = yT. Therefore x ($ yT. This contradicts the minimality of X. The proof is completed. 7.06. THEOREM. Let T = .4 or 6t, let S be a locally compact topological group and let cp be a continuous homomorphism of T into S. Then exactly one of the following

statements is valid: (1) cp is a homeomorphic isomorphism of T into S. 64

RECURRENCE

[7.121

65

(2) Tcp is compact and for each neighborhood V of the identity of S the set Vi o' is syndetic in T. PROOF.

Apply 7.05 and 4.09 to the

{ left } { right I

transformation group of

S induced by T under gyp. 7.07. THEOREM.

Let X be locally compact zero-dimensional and let T be

pointwise recurrent. Then T is locally weakly almost periodic. PROOF. Let x E X and let U be a neighborhood of x. Choose an open compact neighborhood V of x such that V C U. It is enough to show that there exists a

compact subset K of T such that y E V and t E T implies ytK (1 U 0 0. Assume this to be false. Then by 6.25 there exists y E V and a replete semigroup

P in T such that yS (1 V = 0. Hence y is not recurrent. This is a contradiction and the proof is completed. 7.08. THEOREM.

Let X be a compact zero-dimensional uniform space and

let T be pointwise recurrent. Then T is weakly almost periodic. PROOF.

Use 7.07 and 4.23(2).

7.09. REMARK.

Let x E X and let T be almost periodic at x. Then T is

recurrent at x. PROOF.

Use 6.17.

7.10. THEOREM. Let X be a compact zero-dimensional uniform space, let T be discrete and let T be pointwise regularly almost periodic. Then T is regularly

almost periodic. PROOF.

Use 5.17, 7.04 and 7.08.

7.11. THEOREM. Let X be a compact metric space and let T be discrete. Then T is regularly almost periodic if and only if T is pointwise regularly almost periodic and weakly almost periodic.

PROOF. The necessity is trivial. We prove the sufficiency. Let S be a syndetic

subgroup of T. By 4.24 and 5.17 it is enough to show that the class a of all orbit-closures under S is a star-closed partition of X. By 4.08 and 5.07, a is a partition of X. We show d is star-closed. Let xo , x, , x2 , be a sequence of points of X such that lim, x = xo . Since x0T by 1.38 and 4.24, Y = Um=0 xmT is closed in X. By 5.08, each set xmT (m = 0, 1, ) is zerodimensional and therefore Y is a T-invariant compact zero-dimensional subset of X. It follows from 5.07 that S is pointwise regularly almost periodic on Y and it then follows from 7.08 that S is weakly almost periodic on Y. Hence x,S = x0-S. The proof is completed. 7.12. REMARK.

equivalent:

Let x E X. Then the following statements are pairwise

TOPOLOGICAL DYNAMICS

66

[7.12]

(1) T is regionally recurrent at x; that is to say, if U is a neighborhood of x, then there exists an extensive subset A of T such that a E A implies U (`l Ua

0.

(2-3) If U is a neighborhood of x and if P is a replete semigroup in T, then {Ur\ UP O}{x E UP}. (4-5) If U is a neighborhood of x, if P is a replete semigroup in T and if

t E T, then { U n UtP 54 011 x E UtP}. (6) If U is a neighborhood of x and if P is a replete semigroup in T, then

xTCUP.

Let R be the set of all regionally T-recurrent points of X.

7.13. THEOREM.

Then R is a T-invariant closed subset of X. PROOF.

Use 3.22 and 3.28.

Let x E X and let T be regionally recurrent at x. Then T

7.14. THEOREM.

is regionally recurrent at every point of xT. PROOF.

Use 7.13.

Let T be regionally recurrent and let R be the set of all T-recur-

7.15. THEOREM.

rent points of X. Then: (1) If X is metrizable, then R is a T-invariant residual G5 subset of X. (2) If X is a complete metric space, then X = R. PROOF.

Use 3.31 and 6.09.

7.16. THEOREM. Let X be a complete metric space. Then T is regionally recurrent if and only if the set of all T-recurrent points of X is a dense subset of X. PROOF.

Use 3.28 and 7.15.

7.17. DEFINITION. The center of the transformation group (X, T) is defined to be the greatest T-invariant subset C of X such that T is regionally recurrent on C, that is, such that the transformation group (C, T) is regionally recurrent; the existence of C is readily seen, for C is the union of the class of all T-invariant subsets of X on which T is regionally recurrent.

Let C be the center of (X, T). Then: (1) C is a closed subset of X. (2) C is contained in the set of all T-regionally recurrent points of X.

7.18. THEOREM.

PROOF.

Obvious.

7.19. REMARK.

For a T-invariant subset Y of X let Y* denote the set of

all points of Y at which the transformation group (Y, T) is regionally recurrent. For each ordinal a define R. by transfinite induction as follows:

(I) Ro = X. (II) If a is a successor ordinal, then R. = R*._1 . (III) If a is a limit ordinal, then Ra = na<« Ra

[7.24]

RECURRENCE

67

Then: (1) There exists a least ordinal y such that R,. = R. for all a > y. (2) If X is second-countable, then -y is countable.

(3) R., is the center of (X, T). 7.20. THEOREM. Let X be a complete metric space. Then the center of (X, T) coincides with the closure of the set of all T-recurrent points of X. PROOF.

Use 3.28 and 7.16.

7.21. INHERITANCE THEOREM.

Let S be a closed syndetic subgroup of T.

Then S is regionally recurrent if and only if T is regionally recurrent. PROOF.

It follows from 6.19 that the regional recurrence of S implies the

regional recurrence of T.

Now assume that T is regionally recurrent. Let x E X, let U be an open neighborhood of x and let Q be a replete semigroup in S. There exist an open neighborhood Uo of x and a compact symmetric neighborhood V of e in T such

that U0V C U. By 6.20 we can find a replete semigroup P in T such that PV () S C Q. For i = 1, 2, we proceed inductively as follows: 0. There exists x, E U;_1 There exists p; E P such that U;_1 (1 U;-1p; such that x;p; E U;_1 . There exists an open neighborhood U. of x; such that U. C U;_1 and U;p; C U;_1 . There exists a compact subset K of T such that T = SK. For each positive integer i there exist s; E S and k; E K such that p; = s;k; . By 6.22 we can < in and , i. (n > 1) for which i1 < find positive integers n, i1 , , n) and U;,p1 C Uo, we kin E SV. Since U;;p;, C U;,_, (j = 2, k;, p;, C Uo . Also U;. C Uo whence U0 (1 Uop,,, conclude that U;,p;. k;, where s = sin . . s;, E S. Choose p;, = skin p;, 54 0. Now pi p;, = ssov. Define k;, = sov. Then pin so E Sand v E V such that k;,, q = sso, q E PV, sso E S and PV (1 S C Q. 1. We observe that q = pin p;.v 1 (1 p;,v 1 0 0, Uo C U and Uov C U. Hence q E Q. Now Uov Uop;,, Thus U (1 Uq 0 0 and the proof is completed. 7.22. DEFINITION. Let x E X. The transformation group T is said to be 1

wandering at x and the point x is said to be wandering under T or T-wandering provided that T is not regionally recurrent at x; that is, there exist a neighborhood U of x and a replete semigroup P in T such that U () UP = 0.

7.23. REMARK. Let W be the set of all T-wandering points of X. Then W is a T-invariant open subset of X. Let X be a complete metric space and let { R W } be the set of all {T-recurrent} {T-wandering} points of X. Then R U W is a residual subset of X. 7.24. THEOREM.

Since _T is regionally recurrent on X - W, it follows from 7.16 that R D X - W, whence X = R U W and X (R U W) C (R - R) U PROOF.

68

TOPOLOGICAL DYNAMICS

[7.241

(W - W). By 7.15, k - R is a first category subset of R and therefore of X. Since W is open in X, W - W is nowhere dense in X. The conclusion follows. 7.25. NOTES AND REFERENCES.

(7.02) If T = 6t, and thus (X, T, r) is a one-parameter continuous flow, the point x E X is recurrent if and only if, given any neighborhood U of x, there exists a sequence ... < t_, < to < t, < in 6t with limn._m t = - co and lim,+m t _ + c such that xt. E U (n (E J). The classic term for recurrent is stable in the sense of Poisson.

(7.12) If T = 6t, the point x E X is regionally recurrent under T if and only if x is a non-wandering point in the sense of G. D. Birkhoff [2]. (7.15) This is the topological analogue of the Poincar6 Recurrence Theorem (cf., eg., Caratheodory [1]). See also Hilmy [4].

(7.17-7.20) If T = 6t, the center coincides with Birkhoff's set of central motions. The process of 7.19, whereby the center is obtained by transfinite induction, is due to Birkhoff. (7.21) Cf. Erdos and Stone [1].

8. INCOMPRESSIBILITY 8.01. STANDING NOTATION.

Throughout this section (X, T, r) denotes a

transformation group whose phase group T is generative. The following statements are pairwise equivalent: (1) T is pointwise periodic. (2) If M is a nonvacuous subset of X, then there exists an extensive subset A

8.02. THEOREM.

of T such that a E A implies M (1 Ma 0 0. (3) If M is a subset of X, if P is a replete semigroup in T and if MP C M, then MP = M. PROOF. That (1) implies (2) follows from 6.17. Clearly (2) implies (3). Assume (3). We prove (1). Let x E X and let E be the period of x under T. By 6.27 it is enough to show that E is extensive in T. Let P be a replete semigroup in T. Define M = x U xP. Then MP C M whence MP = M and xp = x for some p E P. Therefore E () P 0 0. The proof is completed. 8.03. REMARK.

Let T = 9 and let 0 0 t E T. Then the following statement

may be adjoined to 8.02.

(4) If M C X such that Mt C M, then Mt = M. 8.04. LEMMA. Let S be a semigroup in T and let D = [x I x E X, x xS]. Then there exists C C X such that C (1 CS = 0 and D = CT. PROOF. Let C be a maximal subset of X for which C (1 CS = 0. It is easily verified that D = CT.

The following statements are pairwise equivalent: (1) The set of all T-periodic points of X is a residual subset of X. (2) If M is a second category subset of X, then there exists an extensive subset

8.05. THEOREM.

A of T such that a E A implies M (1 Ma 0 0. (3) If M is a subset of X, if P is a replete semigroup in T and if MP C M, then M - MP is a first category subset of X.

Let E be the set of all T-periodic points of X. Assume (1). We prove (3). Let M C X, let P be a replete semigroup in T, PROOF.

and let MP C M. Since M - MP C X - E, it follows that M - MP is first category.

Assume (3). We prove (2). Let M C X, let P be a replete semigroup in T and let M n MP = 0. It is enough to show that M is first category. If N = M U MP, then NP C N whence M = N - NP is first category. Assume (2). We prove (1). By 6.09 there exists a countable base P, , Pa for the replete semigroups in T. Let D. = [x I x E X, x xP;] (i = 1, 2,

Since E = X - U;-'l D; , it is enough to show that each D;(i = 1, 2, first category. 69

,

).

) is

TOPOLOGICAL DYNAMICS

70

[8.05]

Let i be a positive integer. Write P. = P and D; = D. By 8.04 there exists C C X such that C r) CP = 0 and D = CT. Let V be a compact symmetric neighborhood of e. Define Q= n:ev. tP. By 6.07, Q is a replete semigroup in T. Now CV n CVQ = 0. If M = CV U CVQ, then MQ C M whence CV = M - MQ is first category. By 6.03 there exists a countable subset B of T such that T = VB. Hence D = CT = CVB is first category. The proof is completed. 8.06. REMARK. Let T = g and let 0 54 t E T. Then the following statement may be adjoined to 8.05: (4) If M C X such that Mt C M, then M - Mt is a first category subset of X.

The following statements are pairwise equivalent. (1) T is pointwise recurrent. (2-3) If M is { an open } { a closed) subset of X, if P is a replete semigroup in

8.07. THEOREM.

T and if MP C M, then MP = M. PROOF. If M C X and if P C T, then IMP C M) IMP = MI if and only if {M'P-' C M'} {M'P-' = M'}. This shows the equivalence of (2) and (3). By 6.28, (1) implies (3). Assume (3). We prove (1). Let x E X and let P be a replete semigroup in T. Define M = x U xP. Now MP C xP C M whence MP = M, M = xP and x E xP. The proof is completed.

8.08. REMARK.

Let T = 9 or (R and let 0 0 t E T. Then the following

statement may be adjoined to 8.07: (4) If M is an open or closed subset of X such that Mt C M, then Mt = M. 8.09. THEOREM. Let X be a complete metric space. Then the following statements are pairwise equivalent: (1) The set of all T-recurrent points of X is a residual subset of X. (2-3) If M is a second category { open I { closed } subset of X, then there exists

an extensive subset A of T such that a (=- A implies M (1 Ma 0 0. (4-5) If M is Ian open) { a closed) subset of X, if P is a replete semigroup in T and if MP C M, then M - MP is a first category subset of X.

PROOF. By 7.15 and 7.16, (1) is equivalent to regional recurrence of T. Clearly (2) and (3) are each equivalent to regional recurrence of T. Assume (1). We prove (4). Let M be an open subset of X, let P be a replete semigroup in T and let MP C M. Suppose M - MP is a second category subset of X. Then M - MP is a second category subset of M. Since M - MP is closed in M, there exists a nonvacuous open subset U of M, and therefore of X, such

that U rl MP = 0. This contradicts (1). By 6.28, (1) implies (5). Assume (4). We prove T is regionally recurrent. Suppose T is not regionally

recurrent. Then there exist a nonvacuous open subset U of X and a replete semigroup P in T such that U r) UP = 0. Define M = U U UP. Since M is

INCOMPRESSIBILITY

[8.131

71

open and MP C M, it follows that U = M - MP is first category. This is a contradiction. Assume (5). We prove T is regionally recurrent. Suppose T is not regionally

recurrent. Then there exist a nonvacuous open subset U of X and a replete

semigroup P in T such that U (1 UP = 0 whence U n UP = 0. Define M = U U UP. Since M is closed and MP C UP C M, it follows that M - MP is first category, U C M - MP and U is first category. This is a contradiction. The proof is completed. 8.10. REMARK.

Let T = 5 or cR and let 0 54 t E T. Then the following

statements may be adjoined to 8.09: (6-7) If M is { an open } { a closed } subset of X such that Mt C M, then

M - Mt is a first category subset of X. 8.11. DEFINITION.

Let x E X and let P C T. The point x is said to be

compactive under P or P-compactive provided that xP is compact. The antonym of "compactive" is "noncompactive." 8.12. DEFINITION.

Let x E X. The point x is said to be totally noncompactive

(under T) provided that x is P-noncompactive for every replete semigroup P in T. 8.13. THEOREM. Let X be a T2-space, let R denote the set of all T-recurrent points of X, let N, denote the set of all totally noncompactive points of X and let N2 denote the set of all T-noncompactive points of X. Consider the following state-

ments:

(I) RUN, = X. (I') R U N, is a residual subset of X. (II) If M is a compact subset of X, if P is a replete semigroup in T and if

MP C M, then MP = M. (II') If M is a compact subset of X, if P is a replete semigroup in T and if MP C M, then M - MP is a first category subset of X.

(III) R U N2 = X. (III') R U N2 is a residual subset of X. Then:

(1) I implies II; II implies III; I' implies II'. (2) If X is a locally compact separable metrizable space, then II' implies III'. (1) That I implies II, and II implies III follows from 6.28. Assume I'. We prove II'. Let M be a compact subset of X and let P be a replete semigroup in T such that MP C M. If x E M - MP, it follows from 6.28 that x ($ R and it follows from xP C MP C M = M that x ($ N, . Thus PROOF.

M - MP C X - R U N, and M - MP is a first category subset of X. The proof that I' implies II' is completed. (2) Let X be a locally compact separable metrizable space. Assume II'. We prove III'.

TOPOLOGICAL DYNAMICS

72

[8.13]

Let p be a compatible metric in X. Let S, , S2 , - be a sequence of replete semigroups in T such that if S is any replete semigroup in T, then S. C S for some positive integer n. The existence of such a sequence is assured by 6.09. Let Xl C X2 C . be an expanding sequence of compact subsets of X such that if M is any compact subset of X, then M C X. for some positive integer n. Let e, > e2 > .. be a decreasing sequence of positive numbers such that lim;-m e; = 0. For positive integers i, j, k let E(i, j, k) = [x I x E X, xT C X; ,

xS; n xek = 0]. It is easily verified that E(i, j, k) is a closed set and that X - R U N2 = U;:;,k-1 E(i, j, k). To prove III' it is sufficient to prove that if i, j, k are positive integers, then int E(i, j, k) = 0. Suppose that there exist positive integers i, j, k such that int E(i, j, k) 0 0. Then there exists an open subset Y of X such that 0 0 Y C E(i, j, k) and diam Y < ek . Now Y (1 YS, = 0. For otherwise there would exist s E S; and y E Y such that ys E Y; but then p(y, ys) < ek and yS; (1 yek 0 0, which is not the case. Since Y is open, Y 0 YS= 0. Let M = Y U YS,.

Then M C X. , M is a compact subset of X, MS, C YS, C M and Y C M MS, . It follows from II' that Y is a set of the first category, which is not the case, since Y is nonvacuous open in X. The proof is completed.

8.14. REMARK. Let X be a locally compact separable metrizable space, let R be the set of all T-recurrent points of X, let N be the set of all totally noncompactive points of X and let T = g or CR. Then it may be proved that the following statements are pairwise equivalent:

(1) R U N is a residual subset of X. (2-3) If M is { an open } { a closed } subset of X such that M is compact, if t E T and if Mt C M, then M - Mt is a first category subset of X. 8.15. NOTES AND REFERENCES.

Many of the theorems of this section can be found in C. W. Williams [1]. See also Whyburn [1], Chapter XII. (8-14) Cf. Hilmy [3] and E. Hopf [1].

9. TRANSITIVITY 9.01. STANDING NOTATION.

Throughout this section (X, T, r) denotes a

transformation group. 9.02. DEFINITION.

Let x E X. The transformation group (X, T) is said

to be transitive at x and the point x is said to be transitive under (X, T) provided

that if U is a nonvacuous open subset of X, then there exists t E T such that

xtE U. The transformation group (X, T) is said to be { pointwise} { point} transitive provided that (X, T) is transitive at { every } { some } point of X. The transformation group (X, T) is said to be (regionally) transitive provided

that if U and V are nonvacuous open subsets of X, then there exists t E T 0. such that Ut (1 V Let x E X. The transformation group (X, T) is said be be extensively transitive

at x and the point x is said to be extensively transitive under (X, T) provided that if U is a nonvacuous open subset of X, then there exists an extensive subset

A of T such that xA C U. The transformation group (X, T) is said to be universally transitive provided

that if x, y E X, then there exists t E T such that xt = y. The transformation group (X, T) is said to be extensively (regionally) transitive

provided that if U and V are nonvacuous open subsets of X, then there exists an extensive subset A of T such that t E A implies Ut (l V 5-4 0. The transformation group (X, T) is said to be { pointwise } { point } extensively

transitive provided that (X, T) is extensively transitive at

{ every } { some }

point of X. The transformation group (X, T) is said to be (regionally) mixing provided that if U and V are nonvacuous open subsets of X, then there exists a compact

subset K of T such that t E T - K implies Ut (1 V 0 0. 9.03. REMARK. The following statements are pairwise equivalent.

(1) (X, T) is universally transitive; that is to say, if x, y E X, then there exists t E T such that xt = y. (2) xT = X for every x E X. (3) xT = X for some x E X. 9.04. REMARK.

Let S be a subgroup of a topological group T. Then the

{left} {right} transformation group of {T/S} {T\S} induced by T is universally transitive. 9.05. THEOREM.

Let (X, T, a) be universally transitive, let x E X, let P be

the period of x and let the motion 1r= : T -* X be open. Then (X, T, 7r) is isomorphic

to the right transformation group of T\P induced by T. PROOF.

Use 3.08. 73

TOPOLOGICAL DYNAMICS

74 9.06. REMARK.

[9.061

Let x E X. Then the following statements are pairwise

equivalent.

(1) (X, T) is transitive at x; that is to say, if U is a nonvacuous open subset

of X, then there exists t E T such that xt E U. (2-3) If U is a nonvacuous open subset of X, then { xT () U z 0) { x E UT 1.

(4) xT = X. 9.07. REMARK. The transformation group (X, T) is pointwise transitive if and only if X is minimal under (X, T). 9.08. LEMMA.

Let (X, cu) be a uniform space, let U be a base of `U, and let

A. be a dense subset of X. Then [UaEA af3 X a# 10 E `U] is a base of `U,. PROOF.

Let /3 (E V. There exists T E `U such that y-'y C a. Then UaEA ay X

ay C y 1y C 0. We show y C UaEA as X a(3. Let (x, , x2) E y. There exists

a E A such that a E x1y. Hence (a, x,) E y' C 0, x, E ali, (a, x2) = (a, x,) (x, , x2) E -['y C 3, x2 E ali and (x, , x2) E afl3 X a(3. The proof is completed. 9.09. THEOREM. Let X be a topological space and let 4' be a pointwise transitive homeomorphism group of X. Then the following statements are equivalent. (1) There exists a compatible uniformity `U, of X such that 4' is uniformly equicontinuous relative to cu. (2) For each x E X and each neighborhood U of x there exists a neighborhood

V of x such that 'p E 4' with x E V' implies VSo C U. If such a uniformity `U. exists, that is, if (1) holds, then [UwE4. Nso X Nip I N E 9i] is a base of %, where 91 is any neighborhood base of any point of X, and `U, is therefore unique. PROOF.

For A C X let A* = U. A(p X A. We remark that if x E X

and if A C X, then xA* = U [Acp I (p E 4', x E Arp].

Assume that for each x E X and each neighborhood U of x there exists a neighborhood V of x such that p E 4' with x E Vp implies Vip C U. Let x6 E X, let 91 be an open-neighborhood-base of xo , and let `u be the filter on X X X generated by the filter-base [N* I N E 9Z]. Let U E 91. Then there exists V E 91 such that p E 4) with x0 E Vip implies VVo C U. To prove `U is a uniformity of X we need only show V*V* C U*. Let cp, ¢ E 4' with Vip (1 V¢ 0 0. It now suffices to prove that Vp U V C Upo for some (po E 4'. Choose Vo E 4' so that x,po E Vcp (1 Vti. Since xo E Vpro 1 (l V¢vo', Vpro-I U V¢vo' C U and VVp U V,,G C Ucco . Hence cu is a uniformity of X. It is clear that 4' is uniformly equicontinuous relative to %. We show 'U is compatible with the topology of X. If x E X and if N E 9r, then xN* = U [N'p I (p E 4', x E N'p] is open. If x E X and if U is a neighborhood of x, then there exists N E 91 such that (p E 4' with

x E N'p implies Nrp C U, whence xN* C U. This completes the proof of the sufficiency.

Assume there exists a uniformity cu of X such that 1L is compatible with the topology of X and 4) is uniformly equicontinuous relative to U. If x E X and if a E cu, then there exist f3, y E `U, so that 0 is symmetric, R2 C a and xw C

TRANSITIVITY

[9.141

75

xv$ ((p E 4)), whence cp E (b with x E xyp implies x E xrpj9, xcp E xa, xy(p C xcp,6 C x132 C xa and xycp C xa. Let x E X and let X be a neighborhood-base of x. There exists a base U of `U such that xp13 = x,Qrp (3 E eu). By 9.08, [UEs'by13

X yfi 113 E `U] = [U,E4. x13cp X x1cp 113 E V] is a base of 'a. Clearly, each of [U9E, xl3cp X x,Bcp

a E U1, [UcE,, Np X Nc I N E 91] refines the other. The

proof is completed.

9.10. REMARK. The following statements are pairwise equivalent: (1) (X, T) is transitive; that is to say, if U and V are nonvacuous open sub-

sets of X, then there exists t E T such that Ut n V 0. (2) If U and V are nonvacuous open subsets of X, then UT n V (3) If U is a nonvacuous open subset of X, then UT = X.

0.

(4-5) Every invariant {nonvacuous open) {proper closed} subset of X is

{ everywhere } { nowhere) dense in X.

(6-7) If E and F are invariant {open }{closed } subsets of X such that

{EnF= O){EUF=X},then {E= OorF= 01{E=XorF=X1.

(8-9) If E and F are invariant subsets of X such that E n F = 0 and E U F = X, then lint E = 0 or int F = 01 {E = X or F = X}. (10) If E is an invariant subset of X, then int E = 0 or k = X. 9.11. DEFINITION. A Baire subset of a topological space X is defined to be

a subset E of X such that E = UDI for some open subset U of X and some first category subset I of X, where A denotes symmetric difference. 9.12. THEOREM.

Let X be a complete metric space and let T be countable.

Then the following statements are equivalent:

(1) (X, T) is transitive.

(2) If E is an invariant Baire subset of X, then either E or X - E is a first category subset of X.

PROOF. Assume (1). We prove (2). Let E be an invariant second category Baire subset of X. It is enough to show that E is a residual subset of X. There exist an open subset U of X and a first category subset I of X such that E = UDI

whence U = EiI = (E - I) U (I - E) and UT = (E- n,,, it) U (I - E)T. Thus UT = (E - J) U K where J and K are first category subsets of X such

that J C E and (E - J) n K = 0. It follows that E = (UT - K) U J.

Since U 0 0, we have UT = X. Hence UT is residual in X and E is residual in X. This proves that (1) implies (2). Assume (2). We prove (1). Let U be an invariant nonvacuous open subset of X. It is enough to show that U = X. Since U is a second category Baire subset of X, we have that U is residual in X and U = X. The proof is completed. 9.13. STANDING HYPOTHESIS.

In 9.14-9.24 we assume that the phase group

T is generative.

9.14. REMARK. Let x E X. Then the following statements are pairwise equivalent:

TOPOLOGICAL DYNAMICS

76

[9.141

(1) (X, T) is extensively transitive at x; that is to say, if U is a nonvacuous open subset of X, then there exists an extensive subset A of T such that xA C U. (2-3) If U is a nonvacuous open subset of X and if P is a replete semigroup

_

in T, then {xP (1 U 54 o} {x E UP}. (4) If P is a replete semigroup in T, then xP = X. 9.15. REMARK. Let x E X. Then (X, T) is extensively transitive at x if and only if (X, T) is both transitive and recurrent at x.

9.16. REMARK. The transformation group (X, T) is pointwise extensively transitive if and only if (X, T) is both pointwise transitive and pointwise recurrent. 9.17. REMARK. The following statements are pairwise equivalent: (1) (X, T) is extensively transitive; that is to say, if U and V are nonvacuous open subsets of X, then there exists an extensive subset A of T such that a E A implies Ua ( \ V 5,61 0.

(2) If U and V are nonvacuous open subsets of X and if P is a replete semigroup in T, then UP n V 5-6 0. (3) If U is a nonvacuous open subset of X and if P is a replete semigroup

in T, then UP = X. (4-5) If E is a { nonvacuous open } { proper closed } subset of X, if P is a replete

semigroup in T and if EP C E, then E is (everywhere) { nowhere } dense in X. (6-7) If E and F are { open } { closed } subsets of X such that { E (1 F = 0 }

J E U F = X}, if P is a replete semigroup in T, if EP C E and if FP-' C F, then {E _ O or F = O} {E = X or F = X}. (8-9) If E and F are subsets of X such that E (1 F = 0 and E U F = X, if P is a replete semigroup in T and if EP C E whence FP-' C F, then { int E

or int F = O } I k = X or F = X 1. (10) If E is a subset of X, if P is a replete semigroup in T and if EP C E,

then int E = 0 or E = X. 9.18. REMARK. Certain universally valid implications among the transitivity properties are summarized in Table 4. TABLE 4

T pointwise extensively transitive

T extensively

T pointwise transitive

T transitive at x

T point

transitive

extensively

at x

transitive

T point transitive

9.19. REMARK. Let Y be the set of all transitive points. Then:

(1) If Y ; 0, then (X, T) is transitive.

T transitive

TRANSITIVITY

19.241

77

(2) Y is invariant. (3) If `U is a base of the topology of X such that 0 Er `U, then y = (1 VEV VT. (4) If X is second-countable, then Y is a G, subset of X. (5) If X is second-countable and if (X, T) is transitive, then Y is a residual subset of X. 9.20. THEOREM. Let X be a complete separable metric space. Then the following statements are pairwise equivalent:

(1) (X, T) is transitive. (2) (X, T) is point transitive. (3) The set of all transitive points is an invariant residual G, subset of X. PROOF.

Use 9.19.

Let Z be the set of all extensively transitive points. Then: (1) If Z 0 0, then T is extensively transitive. (2) Z is invariant. (3) If `U is a base of the topology of X such that 0 V and if a is a base of the class of all replete semigroups in T, then Z = (1 vev floe e VQ. (4) If X is second-countable, then Z is a G, subset of X. (5) If X is second-countable and if T is extensively transitive, then Z is a residual subset of X. 9.21. REMARK.

9.22. THEOREM. Let X be a complete separable metric space. Then the following statements are pairwise equivalent: (1) (X, T) is extensively transitive. (2) (X, T) is point extensively transitive. (3) The set of all extensively transitive points is an invariant residual G, subset of X.

(4) (X, T) is transitive and regionally recurrent. PROOF.

Use 9.21.

9.23. THEOREM. Let X be a complete separable metric space, let T = J or 6t and let every orbit under (X, T) be a first category subset of X. Then (X, T) is point extensively transitive if and only if (X, T) is transitive. PROOF. Let (X, T) be transitive. Then (X, T) is regionally recurrent. Now

apply 9.22. 9.24. THEOREM.

Let X be second-countable, let n be a positive integer, let

(X, (R', x) be an extensively transitive n-parameter continuous flow and let E be the set of all t E (R" such that the one-parameter discrete flow on X generated by 7r° is extensively transitive. Then E is a residual GS subset of W. PROOF. Let V be a countable base of the topology of X such that 0 V. Let (P be the class of all replete semigroups in J. For V, W E *0 and P E (P define D (V, W, P) to be the set of all t E (R" such that V (p t) (l W F6 0 for some p E P. Then E= n,. wev nPEA D(V, W, P). It remains to show that E

78

TOPOLOGICAL DYNAMICS

is a residual Ga subset of

(R".

[9.24]

Let V, W E `U and let P E (P. Clearly

D = D(V, W, P) is open in R. Since (P is countable, E is a Ga subset of W. We show D is dense in 6t". Let U be a nonvacuous open subset of a'. Define

H = [p t I p E P, t E U]. Since H contains a replete semigroup in (R", we have VH n W - 0 whence D n U 0. The proof is completed. 9.25. STANDING HYPOTHESIS. For the remainder of this section the condition that T is generative is omitted. 9.26. DEFINITION. A homeomorphism cp of X onto X is said to be motion preserving relative to (X, T, -jr) or (X, T, ir)-motion preserving provided that x E X implies 7r ,,p = ay , . 9.27. REMARK. Let p be a homeomorphism of X onto X. Then the following statements are pairwise equivalent: (1) So is motion preserving; that is to say, arcp = 7r., (x E X).

(2) tascp = tirs, (x E X, t E T). (3) (x, t)irp = (xc, t)ir (x E X, t E T). (4) xtcp = xcpt (x E X, t E T).

(5) x,r`cp=xg1r`(xEX,tET). (6) 7r`,p = cp7r` (t E T). 9.28. DEFINITION. The motion preserving group of (X, T, Tr) or the (X, T, 7r)motion preserving group is defined to be the group of all motion preserving homeo-

morphisms relative to (X, T, 7r).

9.29. DEFINITION. A homeomorphism p of X onto X is said to be orbit preserving relative to (X, T, w) or (X, T, 7r)-orbit preserving provided that x E X implies T,rxcp = Tam, , or in different notation, xTso = xcpT. 9.30. DEFINITION. The orbit preserving group of (X, T, 7r) or the (X, T, 7r)orbit preserving group is defined to be the group of all orbit preserving homeomorphisms relative to (X, T, 7r). 9.31. DEFINITION.

Let A be a subset of a group G. The { centralizer } { nor-

malizer} of A in G is defined to be the group of all x E G such that {a E A implies xa = ax } { xA = Ax).

9.32. REMARK. Let 4) be the total homeomorphism group of X, let G be the transition group [7r` t E T] of (X, T, 7r) and let { H) { K } be the { motion } {orbit} preserving group of (X, T, ir). Then: (1) H coincides with the centralizer of G in (P. (2) K contains the normalizer of G in 4). I

(3) G U H C K. (4) If T is abelian, then G C H. (5) If G C H and if (X, T, 7r) is effective, then T is abelian. 9.33. REMARK.

Let X be a compact uniform space, let 4) be the total homeo-

TRANSITIVITY

[9.361

79

morphism group of X, let be provided with its space-index topology, let G be a subgroup of 4) and let H be the centralizer of G in 4). Then: (1) H is closed in -P.

(2) If G is universally transitive abelian, then G = H. (3) If G is equicontinuous transitive abelian, then G = H and H is equicontinuous universally transitive abelian. 9.34. DEFINITION. Let ((p. I n (=- I) be a sequence of functions on a topological space X to a uniform space Y, let cp be a function on X to Y and let x E X. The sequence (.p I n E I) is said to converge to c uniformly at x provided that if a is an index of Y, then there exist a neighborhood U of. x and a positive integer N such that y E U and n E .4 with n > N implies (yip, ycp,,) E a. 9.35. LEMMA. Let X be a topological space, let Y be a metric space, let (,p I n = ) be a sequence of continuous functions on X to Y which converges pointwise 1, 2,

to a function rp on X to Y and let E be the set of all x E X such that (ip I n = 1, 2,

) converges to (p uniformly at x. Then E is a residual Ga subset of X.

PROOF. Let p be the metric of Y. For each positive number a define A (e) to be the set of all x E X such that p(ycp, yyp ,) < e for all elements y of some neighborhood of x and all integers m greater than some positive integer. Clearly A(e) is open in X for all positive numbers e and E= nn-1 A(1/n). Let e be a positive number. It remains to show that B = X - A(e) is a first category subset of X. Define a = e/5. For each positive integer n define C. to be the set of all x E X such that p(xrp, xg,,.) < S for all integers m such that

m > n. By hypothesis X = Una1 C whence B = Un=1 B (1 C . Let n be a positive integer and define D = B 1 1 C . The proof will be completed when we show that D is nowhere dense in X. Assume that D is somewhere dense in X, that is, there exists a nonvacuous open subset U of X such that U C D. If x E U and if p is an integer such that p > n, then there exists an integer q such that q >_ n and p(xcp, x1p,) < 6 and there exists y E U (1 D such that p(x,p, , yrp,) < 6 and p(xcp, , y(p,) <

xg ) + p(xSo, , whence p(xcp, ycq) + p(yso,, , yw) + p(ycp, ycvn) + p(yc,, , xpn) < 56 = e. To summarize, if x E U

and if p is an integer such that p > n, then p(xp, xrp,) < e. By definition there-

fore, U C A (e) = X - B whence U (1 B = 0. However, since U C D C B and U is nonvacuous open, U (1 B

0. This is a contradiction. The proof is

completed.

be the total homeo9.36. THEOREM. Let X be a compact metric space, let morphism group of X, let G be a universally transitive subgroup of (b and let H be the centralizer of G in 4. Then H is equicontinuous. h2 , be a sequence in H. By 11.13 it is enough to show has a uniformly convergent subsequence. Choose xo c X. converges. We may suppose lim,,,, xah Some subsequence of xoh, , xoh2 , exists; call this limit yo . If x E X and if g E G such that xog = x, then yog = PROOF.

Let h1

that h, , h2 ,

,

80

TOPOLOGICAL DYNAMICS

[9.391

lim xh . Hence h, , h2 , converges pointwise to some function h on X to X. By 9.35 there exists x, E X such that h, , h2 , converges to h uniformly at x, . We show h, , h2 , converges to h uniformly at all points of X. Let x E X and let a be a positive number. There exists g E G such that x,g = x. Choose a positive number S such that z, , z2 E X with p(z, , z2) < S implies p(z,g, z2g) < e.

There exists a neighborhood U, of x, and a positive integer N such that z all integers n with n > N, where p is the metric of X. Define U = U,g. Now U is a neighborhood of x. If y E U and if n is an integer with n > N, then yg 1 E U, and p(yhg-', p(yg 'h, yg 'hn) < S

p(zh,

whence p(yh,

converges to h uniformly at x. converges uniformly to h. The proof is com-

E. This shows that h, , h2 ,

It now follows that h, , h2 , pleted.

9.37. THEOREM. Let X be a compact metric space, let T be abelian and let (X, T, r) be transitive. Then the following statements are equivalent: (1) X is an almost periodic minimal orbit-closure under (X, T, r). (2) The motion preserving group of (X, T, r) is universally transitive. PROOF.

Use 4.35, 4.38, 9.33 and 9.36.

9.38. THEOREM. Let X be a T2-space, let X be minimal under T, let rp be a (X, T)-motion preserving homeomorphism such that xxp E xT for some x E X and let T be abelian. Then (p = r` for some t E T. PROOF. Let x E X and t E T such that xv = xr'. Then xsrp = xsr' (s E T), y,p = yr` (y E xT) and yp = yrt (y E X). The proof is completed.

9.39. NOTES AND REFERENCES.

(9.02) The term transitive was used by G. D. Birkhoff in 1920 (cf. Birkhoff [1], vol. 2, p. 108 and p. 221) to denote regionally transitive as defined here. The term is commonly used with the significance of the expression universally transitive as defined here. (9.11) Cf., e.g., Kuratowski [1]. (9.12(2)) This is, in a sense, the topological analogue of metric transitivity. Cf. E. Hopf [2]. (9.17) Cf. Hilmy [2]. (9.24) Cf. Oxtoby and Ulam [2]. (9.35) Cf. Hausdorff [1], pp. 385-388. (9.36) Cf. Gottschalk.[9] and Fort [1]. Interesting examples of transitivity are to be found in Birkhoff [1], vol. 3, p. 307, Seidel and Walsh [1] and Oxtoby [1].

10. ASYMPTOTICITY 10.01. STANDING HYPOTHESIS.

In 10.02-10.10 we assume that X is a

separated uniform space, and that (X, T, 7r) is a transformation group. 10.02. DEFINITION.

Let x, y E X. The points x and y are said to be sep-

arated (each from the other) under T provided there exists an index a of X such that t E T implies (xt, yt) a. The points x and y are said to be nonseparated

(each from the other) under T provided that x 76 y and the pair x, y is not separated under T, that is, if a is an index of X, then there exists t E T such that (xt, yt) E a. Let A and B be orbits under T. The orbits A and B are said to be { separated } {nonseparated }

(each from the other) provided that A 0 B and there exist

x E A and y E B such that x and y are {separated }{nonseparated} under T. 10.03. INHERITANCE THEOREM. Let X be compact, let x, y E X and let S be a syndetic subgroup of T. Then x and y are nonseparated under S if and only if x and y are nonseparated under T. PROOF. Suppose x and y are nonseparated under T. Let a be an index of X

and let K be a compact subset of T such that T = SK. There exists an index $ of X such that (x1 , x2) E ,B and k E K implies (x,k-', x2k-') E a. Select t E T such that (xt, yt) E fl. Now select s E S and k E K such that t = sk. It follows that (xs, ys) = (xtk-', ytk-') E a. The proof is completed. 10.04. THEOREM. Let X be a compact minimal orbit-closure under T and let x, y E X such that x and y are nonseparated under T. Then the T-traces of x and y

coincide. PROOF.

Use 10.03 and 2.43.

10.05. THEOREM.

Let X be a compact minimal orbit-closure under T and

let T be regularly almost periodic at some point of X. Then the following statements are equivalent: (1) x and y are nonseparated under T. (2) The T-traces of x and y coincide.

PROOF. 'By 10.04, (1) implies (2). Assume (2). We prove (1). Let a be an index of X. Choose a symmetric index /3 of X such that 32 C a. Let T be regularly

almost periodic at z E X. There exists a closed syndetic invariant subgroup A of T such that zA C z/3. Let K be a compact subset of T such that T = AK. Since X = zAK = UkEK zkA, by 2.42 there exists k E K such that x, y E zkA whence xk-', yk-' E zA C z,6 and (xk-', yk-') E a. The proof is completed. 10.06. THEOREM. Let X be compact, let X be minimal under T and let x, y E X such that x - y and xcp = y for some (X, T)-motion preserving homeomorphism cp. Then x and y are separated under T. 81

82

TOPOLOGICAL DYNAMICS

[10.061

PROOF. Assume x and y are nonseparated. Choose open neighborhoods U, V of x, y such that U Th V = 0 and Lice C V. There exists a finite subset E of T such that X = UE. Select t E T and s E E such that xt, yt E Us whence

xts-1, yts 1 E U. It follows that yts ' =

xcpts-'

= xts 'cp E Ucp C V and

yts-' E U (1 V. This is a contradiction. The proof is completed. 10.07. THEOREM. Let X be compact, let X be minimal under T, let x, y E X such that x 3-!5 y and y E xT, and let T be abelian. Then x and y are separated under T. PROOF.

Use 9.32(4) and 10.06.

Let X be compact, let x, y E X, let both x and y be almost

10.08. THEOREM.

periodic under T, let x and y be nonseparated under T and let T be locally compact. Then neither x nor y is regularly almost periodic under T. PROOF. Assume x is regularly almost periodic under T. Then there exists a closed syndetic invariant subgroup S of T such that y Er xS. Since x and y are almost periodic under S, we have that xS (1 yS = 0. It follows from 10.03 that the pair x, y is separated under T. This contradicts the hypothesis and the

proof is completed. 10.09. REMARK.

Let X be compact, let x, y E X and let .p be a (X, T)-

motion preserving homeomorphism. Then xcp and ycp are separated under T if and only if x and y are separated under T. 10.10. THEOREM. Let X be compact, let X be minimal under T, let there exist exactly one nonseparated pair of orbits under T and let T be abelian. Then every (X, T)-motion preserving homeomorphism is a transition of (X, T). PROOF.

Let x, y E T such that x, y are nonseparated and xT 5 yT. Let

be a motion preserving homeomorphism. Since xceT, yPT are nonseparated

orbits, we have xcpT = xT or xppT = yT. It is enough by 9.38 to show that x,pT = xT. Assume xcoT = yT. Then xcpa` = y for some t E T. By 10.06, x and y are separated. This is a contradiction. The proof is completed. 10.11. STANDING NOTATION.

Throughout the remainder of this section X

denotes a compact metric space with metric p and cp denotes a homeomorphism of X onto X. It is somewhat more convenient to speak of the homeomorphism ce than the discrete flow generated by p. 10.12. DEFINITION.

Let x E X. The orbit of x, denoted 0(x), is the set

U.'-'-. x(P". The {negative} {positive} semiorbit of x, denoted {0-(x)} {0+(x)}, is the set {Un=o xcp"}{Un=0 xcp"}. To indicate dependence on P we may suffix the phrase "under cp" and adjoin a subscript cp to 0. 10.13. REMARK.

Let x E X. Then the negative semiorbit of x under cc

coincides with the positive semiorbit of x under rp '.

ASYMPTOTICITY

[10.201

83

Let x E X. Then: 0(x) = 0-(x) U 0+(x).

10.14. REMARK. (1)

0(x)c0-' = 0(xco-') = 0(x) = O(xgo) = O(x)cp. 0-(x)go

= 0-(xcp-') C 0-(x) C 0-(XP) = 0-(x)co.

0+(x),P-' = 0+(xgo ') D 0+(x) D 0+(xgP) = 0+(x),P. (2)

0(x) = 0-(x) U 0+(x).

5(x),' = 0(xg ') = 0(x) = 0(xgo) = 0(x)g. 0 (x)go '= 0-(xgo ') C 0-(x) C 0-(XP) = 0-(x)go. 0+(x),P-'= 0+(xgo ') D 0+(x) D O+(xgo) = 0+(x)v. 10.15. DEFINITION. Let x E X. The { a-limit } { w-limit } set of x, denoted {a(x) } [.(x)), is the set of ally E X such that lim1. 0. x(pn' = y for some sequence 1. Each n, , n2 , of integers such that { n, > n2 > ... } In, < n2 < point of { a (x) } { w (x) } is called an { a-limit } { w-limit } point of x. To indicate dependence on v we may suffix the phrase "under go" and adjoin a subscript gp to a and co. This definition agrees with 6.33.

Let x E X. Then the a-limit set of x under rp coincides

10.16. REMARK.

with the w-limit set of x under go '.

Let x E X. Then: 0(x) = 0(x) U a(x) U w(x). 0-(x) = 0-(x) U a(x) . 0+(x) = 0+(x) V w(x) . (2) a(x) and w(x) are nonvacuous closed invariant. 10.17. REMARK.

(1)

a(xgo ') = a(x) = a(xcp) . w( xcp') = w(x) = w(xg) . ' xr ) (4) a(x) = nn--,O w(x) = I n,-. (U (y ma-n .,,= xgm) (5) Every point of a(x) U w(x) is regionally recurrent.

(3)

(/

f

10.18. DEFINITION. Let x E X. The homeomorphism gp is said to be {negatively) { positively } recurrent at x and the point x is said to be (negatively) { posiof integers tively} recurrent under rp provided there exists a sequence n, , n2 , such that In, > n2 > . . . } In, < n2 < . . } and lim;._+ xgn' = x. The homeo-

morphism (p is said to be recurrent at x and the point x is said to be recurrent under (p provided that go is both negatively and positively recurrent at x. This definition agrees with 3.36. 10.19. REMARK.

Let x E X. Then x is negatively recurrent under rp if and

only if x is positively recurrent under go '. 10.20. REMARK.

Let x E X. Then:

(1) The following statements are pairwise equivalent: (I) x is negatively recurrent.

(II) X E a(x).

TOPOLOGICAL DYNAMICS

84

[10.201

(III) X E 0-(xP-'). (IV) 0(x) = 0(x) = a(x) D w(x). (2) The following statements are pairwise equivalent: (I) x is positively recurrent. (II) X E w(x). (III) X E 0+(x(p).

(IV) 0(x) = 0+(x) = w(x) D a(x). (3) The following statements are pairwise equivalent: (I) x is recurrent. (II) X E a(x) (1 w(x). _ (III) X E 5-(XP-I) (1 0+(x(P).

(IV) 0(x) = 0-(x) = 0+(x) = a(x) = WW. 10.21. DEFINITION.

Let x E X and let B be a closed invariant subset of X.

The point x is said to be { negatively } { positively } asymptotic to B under c provided that x ($ B and p(xo", B) = 01 {lim, .+0. p(xcp", B) = 01.

Let A be an orbit under op and let B be a closed invariant subset of X. The orbit A is said to be { negatively } { positively ) asymptotic to B provided there

exists x E A such that x is {negatively} {positively} asymptotic to B under (p. Let x, y E X. The points x and y are said to be {negatively} {positively} asymptotic (each to the other) undercp provided that x F-4 y and {lim, p(x(p", yip") = 0} { Jim,+m (xga", yip") = 0 } .

Let A and B be orbits under (p. The orbits A and B are said to be { negatively } { positively) asymptotic (each to the other) provided that A 0 B and there exist x E A and y E B such that x and y are {negatively} {positively} asymptotic under (p.

The term { asymptotic } f doubly asymptotic) means negatively for) { and } positively asymptotic. 10.22. REMARK.

Let x E X and let A be an invariant closed subset of X.

Then x is negatively asymptotic to A under (p if and only if x is positively asymptotic to A under '.

Let x, y E X. Then x and y are negatively asymptotic under o if and only if x and y are positively asymptotic under cp '. 10.23. REMARK.

Let x E X, let A be an invariant closed subset of X and

let n be an integer. Then x'p" is { negatively } { positively) asymptotic to A under cp if and only if x is {negatively} {positively} asymptotic to A under cp.

Let x, y E X and let n be an integer. Then x0" and ycp" are { negatively } { positively } asymptotic under cp if and only if x and y are { negatively) { positively} asymptotic under cp.

Let x E X. Then: (1) x is negatively asymptotic to a(x) if and only if x ($ a(x).

10.24. REMARK.

w(x). (2) x is positively asymptotic to w(x) if and only if x a(x) U w(x). (3) x is doubly asymptotic to a(x) U w(x) if and only if x

ASYMPTOTICITY

[10.30]

85

10.25. REMARK. Let A be the set of all regionally recurrent points of X. Then A is a nonvacuous closed invariant subset of X such that every point of

X - A is doubly asymptotic to A. Let n be a positive integer. Then: (1) If x E X and if A is an invariant closed subset of X, then x is {negatively} 10.26. INHERITANCE THEOREM.

{ positively) asymptotic to A under cp" if and only if x is { negatively } { positively } asymptotic to A under rp.

(2) If x, y E X, then x and y are { negatively } { positively } asymptotic under co' if and only if x and y are {negatively} {positively} asymptotic under (p. PROOF.

Obvious.

10.27. THEOREM. Let x E X and let A be a closed invariant subset of X such A. Then the following statements are pairwise equivalent: that x (1) x is {negatively} {positively} asymptotic to A.

(2) {a(x) C A}{w(x) C Al. (3) If U is a neighborhood of A, then there exists an integer n such that {

Um-n

xIpm C U} { Um-n xcpm C U}.

PROOF.

Obvious.

10.28. THEOREM. Let A be a closed non-open invariant subset of X and let there exist a neighborhood U of A such that x E U - A implies 0(x) ( U. Then there exists y E X such that y is asymptotic to A. PROOF. Let V be an open neighborhood of A such that V C U. Choose a sequence x, , x2 , of points of X - A such that lim;-.Fm x; E A. For each

positive integer i -let n; be the integer with least absolute value such that xsp ni E X - V. We may assume that 0 < n1 < n2 ... and hm,-+. xtcp"` -y E X - V. It follows that Una , y(p" C V. By 10.24(1) it is enough to show that a(y) C A. Assume there exists x E a(y) - A. Then 0(x) C a(y) C V C U. This contradicts the hypothesis. The proof is completed. 10.29. THEOREM.

Let A be a subset of X such that Ace C A and let U be an

open neighborhood of A. Then at least one of the following statements is valid:

(1) There exists a closed subset E of X such that A C E C U, EIp C E and

EC (2) There exists an open subset V of X such that A C V C U and Vv-' C V. PROOF. Define E= UIp". Now E is closed, E(p C E and E C U/p 1. Since A C Ace ' C Ac-' C , we have A C A1p" C Uco " (n = 1, 2, ) and A C E. If E r U, then (1) holds. Assume E C U. There exists a positive

integer m such that nn, c U. Define V= fln, U(p-". Now V is open and A C V C U. Since V =u (l nn,-, UP ", we have Vcp 1 =nn-1 C V. The proof is completed. 10.30. THEOREM.

Let X be self-dense and let a be a positive number. Then

there exist x, y E X with x 0 y such that n >= 1 implies p(xcp", y1p") < e.

TOPOLOGICAL DYNAMICS

86 PROOF.

[10.301

Define the homeomorphism

XXX° XXX by (x, y),p = (xp, yco) (x, y E X). The diagonal A of X X X is a closed non-open

#i-invariant subset of X X X. For n a positive integer, define U. = f (x, y) x, y E X, p(x, y) < 1/n1. Clearly U is an open subset of X X X and A C U . Now apply 10.29. If conclusion (1) holds for infinitely many positive integers n, the proof of the theorem is completed. If this is not the case, there exists a positive integer N such that n > N implies the existence of an open subset V of X X X such that A C V C U and V"(P-' C V . It follows that the set -1) = [p ' 1 is an equicontinuous set of homeomorphisms of X onto X, i = 0, 1, 2, and thus, by 11.31, 4) is totally bounded in its compact index uniformity. Let 4,+ = [,p' I i = 0, 1, 2, 1. Then 4) = 4)+ U D- is a group of unimorphisms of X onto X and it follows from 11.18 that the space-index uniformity coincides with the inverse space-index uniformity of 1. Thus the set (+ is totally bounded in its compact index uniformity, hence equicontinuous, and corresponding to

e > 0 there exists d > 0 such that x, y E X with p(x, y) < S and n > 1 imply p(xv", ycp") < E. Since X is self-dense we can determine x, y E X with x y and p(x, y) < S. The proof of the theorem is completed. 10.31. DEFINITION. The homeomorphism (p is said to be expansive provided

there exists a positive number d such that if x, y E X with x y, then there exists an integer n such that p(xcp", y(p") > d. Let x E X. The homeomorphism cp is said to be expansive at x, and the point x is said to be expansive under cp provided there exists a positive number d such that if y E X with y 54 x, then there exists an integer n such that p(xo", ySo") > d. 10.32. INHERITANCE THEOREM.

Let n be a nonzero integer. Then:

(1) cp" is expansive if and only if

0. There exists a positive number d such that x, y E X with x 5;6 y implies

p(x(o', ycp') > d for some integer m. Choose a positive number c such that x, y E X with p(x, y) > d implies p(xp', y-o') > c (i = 0, , n - 1). Let x, y E X with x y. There exists an integer m such that p(xcp", yco') > d.

Select integers p, r such that np = m + r and 0

r < n. It follows that

p(x0"p, yip"D) = p(xrp"'+', yip"'+') > c. This proves that co" is expansive.

The converse is obvious. The proof is completed. (2) Similar to (1). 10.33. DEFINITION. Let x E X. The homeomorphism v is said to be periodic at x and the point x is said to be periodic under rp provided there exists a positive integer n such that xSo" = x. If cp is periodic at x, then the period of c at x and

the period of x under cp is defined to be the least positive integer n such that

ASYMPTOTICITY

[10.38]

87

x0" = x. The homeomorphism (p is said to be fixed at x and the point x is said to be fixed under rp provided that xrp = x, that is, the period of (P at x is 1. The homeomorphism cp is said to be pointwise periodic provided that (p is periodic at every point of X. The homeomorphism cp is said to be periodic provided there exists a positive integer n such that x E X implies xvn = x. If cp is periodic, then the period of c is defined to be the least positive integer n such that x E X implies xc? = x. It is clear that this definition agrees essentially with 3.06. Let cp be expansive. Then: (1) If n is a positive integer, then the set of all points of X with period n is finite. (2) The set of all periodic points of X is countable. 10.34. THEOREM.

PROOF.

It is clear that the set of all fixed points is finite. Now use 10.32.

Let X be self-dense and let cp be expansive. Then: (1) (p is not pointwise periodic. (2) (p is not almost periodic.

10.35. THEOREM.

PROOF. To prove (1) use 10.34(1). To prove (2) use 4.35 and 4.37. 10.36. THEOREM. Let X be self-dense and let c be expansive. Then there exists a pair of points of X which are positively asymptotic under (P and a pair of points of X which are negatively asymptotic under rp. PROOF. It is sufficient to prove the existence of a pair of points of X which are positively asymptotic under (p. Since (p is expansive, there exists a positive number d such that if x, y E X with x 5,6 y, then there exists an integer m such that p(xcp', ycp'") > d. According to 10.30, there exist xo , yo E X with xo 0 yo

such that n > 1 implies p(xocP?, yocv") < d/2. We show that xo and yo are positively

asymptotic under cp. If this is not the case, there exists a positive number a and a sequence of integers n, < n2 < . . . such that i > 1 implies p(xorp°, yocp") > e. We can assume that lim,-. xocp"` = x, and lim,_ yo(p"` = y, . Then p(x, , yl) > e

and thus xl 0 y . Let Ic be any integer. Then lim

xo
=

P , lim xik

n ''+k

= yi
k

and since for i sufficiently large, n; + k > 0, it follows that p(xl(pk, yicpk) < d, contrary to the hypothesis that 0 is expansive. The proof of the theorem is completed. 10.37. THEOREM. Let x E X, let x be non-isolated and let cp be periodic at x and expansive at x. Then there exists y E X such that y is asymptotic to x. PROOF.

Use 10.32, 10.28 and 10.26.

10.38. DEFINITION.

Let x, y E X. The points x and y are said to be {nega-

tively} {positively} separated (each from the other) under cp provided there exists

a positive number e such that if n is a {negative} {positive} integer, then

88

TOPOLOGICAL DYNAMICS

[10.381

p(x(p", ycp") > e. The points x and y are said to be {negatively} {positively} nonseparated (each from the other) under (p provided that x - y and if a is a positive

number, then there exists a {negative} {positive} integer n such that p(xcp",

y?) < e. The term separated means negatively and positively separated. The term nonseparated means negatively or positively nonseparated. The term doubly nonseparated means negatively and positively nonseparated. Let A and B be orbits under cp. The orbits A and B are said to be admissible (each from the other) provided that A 5,46 B and there exist x E A and y E B such that x and y are admissible under (p, where "admissible" is replaceable by one of the seven terms defined above. This definition agrees with 10.02. 10.39. REMARK.

Let x, y E X. Then:

(1) x and y are negatively nonseparated under (p if and only if x and y are positively nonseparated under c 1. (2) If n is an integer, then xcp" and yyo" are [negatively) { positively l nonseparated under cp if and only if x and y are [negatively) {positively} non.

separated under cp. (3) If x and y are { negatively } { positively } asymptotic under (P, then x and y are { negatively } { positively } nonseparated under (p. 10.40. INHERITANCE THEOREM.

Let x, y E X and let n be a positive integer.

Then x and y are { negatively } { positively } nonseparated under cp" if and only if x and y are { negatively } { positively } nonseparated under so. PROOF.

Obvious.

10.41. THEOREM. Let X be a locally recurrent minimal orbit-closure and let x, y E X. Then x and y are negatively nonseparated if and only if x and y are

positively nonseparated. PROOF. Let x and y be negatively nonseparated. It is enough to show that x and y are positively nonseparated. Let e be a positive number. There exist a positive number S and an extensive subset E of 9 such that U"ea xScp" C xE. Choose a finite subset F of 5 such that X = UmEF x&pm. There exists a negative integer k such that =pk, yck E xSSom for some m E F whence xlpk-m, YIP k-m E xS. Now select n E E such that p = k - m + n > 0. It follows that xcp9, ycp' E xe and p(xq?, yq?) < 2e. The proof is completed.

10.42. THEOREM.

Let X be a minimal orbit-closure and let x, y be a nonseparated

pair of points of X. Then neither x nor y is regularly almost periodic. PROOF. Use 10.08. 10.43. THEOREM.

Let X be a minimal orbit-closure. Then:

(1) rp is weakly almost periodic.

[10.45]

ASYMPTOTICITY

89

(2) The following statements are pairwise equivalent: (I) Some point of X is locally almost periodic. (II) gp is locally almost periodic. (III) So is weakly,, almost periodic. (3) The following statements are pairwise equivalent: (I) Some point of X is isochronous. (II) (p is weakly isochronous. (III) The set of all regularly almost periodic points of X is an invariant residual Ga subset of X. (4) If rp is weakly isochronous, then cp is locally almost periodic. (5) rp is regularly almost periodic if and only if (p is pointwise regularly almost periodic. PROOF.

Use 4.24, 4.31, 5.09, 5.23, 5.24, 5.18.

10.44. THEOREM. Let X be a weakly isochronous minimal orbit-closure and let A be the set of all x E X such that x is nonseparated from some point of X.

Then A is an invariant first category subset of X. PROOF.

Use 10.42 and 10.43.

10.45. NOTES AND REFERENCES.

(10.15) This terminology is due to G. D. Birkhoff [2]. (10.29) This theorem is due to Montgomery [3]. See also Kerekj irtd [1]. (10.30 and 10.36) These theorems are due to S. Schwartzman [1]. A weaker form of 10.36 was proved by Utz [1]. (10.32, 10.34, 10.37) Of. Utz [1].

(10.02, 10.38) The terms distal and proximal for separated and nonseparated, respectively, have been proposed and are being used by Gottschalk.

11. FUNCTION SPACES 11.01. DEFINITION.

Let X be a set, let Y be a uniform space with uniformity

I and let 4) be a set of mappings of X into Y. For a C Y X Y define ad _ [(go,+L) Ip,,p (E 4) &(xgo,x'G) Ea(xEX)].Define

ti = [a,, I aEI]. It is readily verified that `0 is a uniformity-base of -1). The uniformity 9t of cP generated by 10 is called the space-index uniformity of '. The topology of (D induced by iL is called the space-index topology of 4). 11.02. THEOREM. Let X be a topological space, let Y be a uniform space, let -D be a set of continuous mappings of X into Y, let be provided with its space-

index topology and let 7r : X X Then 7r is continuous.

-* Y be defined by (x, (p) -7r = x(p (x E X, So E I)).

PROOF. Let x E X, let gp E (b and let V be a neighborhood of (x, v)ir = xgo. Choose an index ,6 of Y and a neighborhood U of x such that xso82 C V and U(p C xgp,6. We show (U X cp/i,,)a C V. Let y E U and let t E (p/ij, . Then

(xso, yv) E ,3, (go, sG) E l34 , (ygo, Yk) E 3, (xgv, yi&) _ (xgv, y(p) (ygp, y,') E ,62 and (y, ')7r = y¢ E xgp$2 C V. The proof is completed.

yso E xgc/3,

11.03. THEOREM. Let X be a set, let Y be a complete uniform space and let 4) be the set of all mappings of X into Y. Then 4) is complete in its space-index uniformity. PROOF.

Let (P be provided with its space-index uniformity. Let F be a cauchy

filter on 4). For each x E X, [xF I F E a] is a cauchy filter-base on Y. Define go : X -* Y by xgo E n FE 5 xF (x E X). We show 5 -> ,p. Let a be a closed index

of Y. There exists F E such that F X F C a,,. Then xF X xF C a (x E X),

xFXxFCa(xEX),[xgo]XxFCa(xEX),[go]XFCa,and FCgoa4'. The proof is completed. 11.04. THEOREM. Let X be a {topological} {uniform} space, let Y be a uniform space, let (D be the set of all mappings of X into Y, let 4) be provided with its space-

index topology and let 4' be the set of all { continuous } { uniformly continuous } mappings of X into Y. Then ' is a closed subset of (1). PROOF.

First reading. Let (p E 'I'. We show rp is continuous. Let xo E X

and let a be an index of Y. Choose a symmetric index,6 of Y such that ,93 C a. There exists ¢ E `I' (1 gp#,b . Select a neighborhood U of xo such that U,, C xo p,6. If x E U, then (xorp, xoi') E a, (xoiP, *) E a, (xi', x(p) E 0, (xo(p, x(p) E #3 C a and xrp C xogpa. The proof is completed. Second reading. Let (p E *. We show (p is uniformly continuous. Let ,6 be an

index of Y. Choose a symmetric index y of Y such that y3 C 6. There exists 90

FUNCTION SPACES

[11.08]

E `I' (1 SPY,

.

91

Select an index a of X such that (x1 , x2) E a implies

(xok, x4) E y. If (x, , x2) E a, then xi') E 'Y, (x,', x2#G) E 7, (x2Y', x2'p) E 7 and (x,,p, x20 E 'Y3 C (3. The proof is completed. 11.05. THEOREM.

Let X be a { topological } { uniform } space, let Y be a complete

uniform space and let - be the set of all {continuous} {uniformly continuous} mappings of X into Y. Then 4) is complete in its space-index uniformity. PROOF.

Use 11.03 and 11.04.

11.06. THEOREM.

Let X be a set, let Y be a totally bounded uniform space

and let 4) be a set of mappings of X into Y. Then the following statements are equivalent:

(1) (D is totally bounded in its space-index uniformity.

(2) If a is an index of Y, then there exists a finite partition a of X such that A E a and 'p (E 4) implies A'p X A' C a. PROOF. We show that (1) implies (2). Assume (1). Let a be an index of Y. Choose a symmetric index (3 of Y such that a3 C a. There exists a finite subset

F of 4) such that 4 _ UfEF faj, . Since Y is totally bounded, for each f E F there exists a finite partition a, of X such that A f X A f C (3 (A E a1). Define

a = ()rep ar . Clearly a is a finite partition of X such that Af X Af C a (A E a, f (E F). Let A E a and let (p E 4). We show App X A(o C a. Choose

f E F such that p E f(3. . Then x'p E x f f3 (x E A), Arp C A f (, App X Acp C A f,6 X A f,3 = 16(A f X A f ),B C ,33 C a. Hence (1) implies (2).

We show that (2) implies (1). Assume (2). Let a be an index of Y. Choose a symmetric index l3 of Y such that ,3` C a. There exists a finite partition a of X such that Aso X Acp C,6 (A E a, (p E 4)). Select a finite subset E of Y such that Y = E,B. If A E a and if p E 4), then there exists y E E such that y,3 ('1 A(p 96 0 whence Acp = y/(Acp X A(p) C y$2. That is to say, if 'p E 4), then to each A E a there corresponds at least one y E E such that A'p C y,62. Each p E 4) thus determines a nonvacuous set ,p* of mappings of a into E as follows: t E 'p* if and only if Ace C At,32 (A (E a). Since the set of all mappings of a into E is finite, there exists a finite subset F of 4) for which U,E4 rp* = UIEF f*. We show = UfEF fao . Let 'p (E 4). Select t E rp*. Now t E f* for some f E F.

Hence A'p C At,82 (A (E a) and Af C At#2 (A E a). If x E X, then x E A for some A E a whence xcp E At,2, xf E At$2 and (xf, x(p) = (xf, At) (At, x(p) E ,34 C a. Thus (xf, x') E a (x E X) and 'p E fa .. The proof is completed.

Let X be a topological space, let Y.be a uniform space and let - be a set of mappings of X into Y. If x E X, then 4, is said to be equicontinuous at x provided that if ,B is an index of Y, then there exists a neighborhood U of x such that p C 4) implies Ucp C xtp(3. The set (D is said to be equicontinuous provided that 4) is equicontinuous at every point of X. 11.07. DEFINITION.

11.08. DEFINITION. Let X, Y be uniform spaces and let 4) be a set of mappings of X into Y. The set 4) is said to be uniformly equicontinuous provided

TOPOLOGICAL DYNAMICS

92

[11.08]

that if a is an index of Y, then there exists an index a of X such that x E X and cp E 4) implies xap C x(o,3.

Let X and Y be uniform spaces and let 4' be a set of mappings of X into Y. Then: (1) If 4' is uniformly equicontinuous, then 4) is equicontinuous. (2) If X is compact and if 4' is equicontinuous, then t is uniformly equi11.09. REMARK.

continuous. 11.10. REMARK. Let X be a {topological} {uniform} space, let Y be a uniform space, let 4' be the set of all mappings of X into Y, let 4' be provided with

its space-index topology and let 4, be a {equicontinuous} {uniformly equicontinuous } set of mappings of X into Y. Then I is { equicontinuous } set of mappings of X into Y. Then NP is { equicontinuous I { uniformly equicontinuous 1. 11.11. DEFINITION. Let X be a set, let Y be a uniform space and let 4) be a set of mappings of X into Y. The set 4' is said to be bounded provided that U,E. Xcp is a totally bounded subset of Y. 11.12. THEOREM.

Let X and Y be uniform spaces and let 4' be a set of uniformly

continuous mappings of X into Y. Then: (1) If 4' is totally bounded in its space-index uniformity, then 4' is uniformly equicontinuous.

(2) If X is totally bounded, then 4' is totally bounded in its space-index uniformity if and only if 4' is uniformly equicontinuous and bounded. PROOF. (1) Let # be an index of Y. Choose a symmetric index y of Y for which y3 C t3. Since - is totally bounded in its space-index uniformity, there exists a finite subset F of P such that p E 4' implies (xf, xcp) E y (x E X) for some f E F. Select an index a of X such that (x, , x2) E a and f E F implies (x, f , x2 f) E y We show that (x, , x2) E a and *p E 4' implies (x,cp, x2c) E fl. Let (x, , x2) E a and let (p E 4'. Choose f E F such that (xf, xrp) E -y (x E X). Then (x,(o, xw) = (xAP, x, f) (x, f , x2f) (x2f, x2c) E ?'3 C a. This proves (1). (2) Assume that X is totally bounded and that 4' is uniformly equicontinuous and bounded. We show that 4' is totally bounded in its space-index uniformity. It is enough by 11.06 to show that for each index a of Y there exists a finite partition a of X such that A E a and (p E 4' implies Acp X Acp C ,3. Let ,3 be an index of Y. Choose an index a of X such that (x, , x2) E a and (p E 4' implies (xAp, x2c') E 3. Since X is totally bounded, there exists a finite partition a of

X such that A E a implies A X A C a. Hence A E a and c' E 4) implies A(p X Acp C $.

Now assume that X is totally bounded and that 4' is totally bounded in its space-index uniformity. By (1), 4) is uniformly equicontinuous. We show that 4) is bounded, that is, U,Ej) Xcp is totally bounded. Let /3 be an index of Y. Choose an index y of Y for which y2 C a. There exists a finite subset F of 4) such that (p E 4' implies (xf, xp) E y (x E X) for some f E F. Since each f E F

is uniformly continuous, UIEF Xf is totally bounded. Hence there exists a

FUNCTION SPACES

[11.171

93

finite subset E of Y such that U fE F X f C Ey. Since p E 4) implies X'P C X f y, it follows that U,E* X

Let X and Y be compact uniform spaces, let 4' be the set of

all continuous mappings of X into Y, let 4' be provided with its space-index topology

and let I C 4). Then 4, is compact if and only if 4, is equicontinuous. PROOF.

Use 11.05, 11.09 and 11.12.

11.14. THEOREM. Let X be a uniform space and let 4) be a semigroup of uniformly continuous mappings of X into X. Then the semigroup multiplication of

4) is continuous in the space-index topology of -. PROOF. Let coo , ¢o E 4' and let y be an index of X. Choose an index l3 of X such that fl' C y. There exists an index of a of X such that (x, y) E a implies

(x#o , y+Go) E /3. We show (Sooa*) (,koip) C (,poPo)ys . Let So E -poa4, and let vp (=- &o(3 . If x E X, then (xco , xso) E a, (xSoo}Go , x4 o) 0) E /3, (x4o , xrp+G) E /3 and (xSoo+G0 , x4) E #2 C y. Hence 4 E ((poto)yq, . The proof is completed. 11.15. DEFINITION.

Let X be a set. A permutation of X is defined to be a

one-to-one mapping of X onto X. 11.16. DEFINITION.

Let X be a uniform space with uniformity I and let

4) be a set of permutations of X. For a C X X X define a*, = [(co, P) cp, 4, E 4 and (xco, x#) E a (x E X)], a** = [(so, P) p,' E 4' and (x

&=a*(lat .

Define

[a* I a E I],

`V

`0* = [a**4

V

aEI],

= [c* I a E I].

It is readily verified that V, V*, v are uniformity-bases of 4'. The uniformities RL, CtL*, i of (D generated by V, CO", V are called the space-index, the inverse space-index, the bilateral space-index uniformities of 4). The topologies 5, 5*, 3 of 4' induced by `U, IIL*, 'U are called the space-index, the inverse space-index, the bilateral space-index topologies of (D. It is clear that: (1) If 4' is a group, then { V } { 3* } is the image of { lu } { 3 } under the group inversion of 4).

(2) cU,=`l1V`LL*,

_3V5*.

11.17. REMARK. Let G be a group, let { 3 } { `U. } be a { topology } {uniformity } of G, let { 3* 1190 } be the image of { a } {'U. } under the group inversion of G and let {3 = 3 V 3* } { `t1 = Ri. V `11* 1. Then:

(1) The group inversion of G is a homeomorphism of { (G, 3) onto (G, 3*) } { (G, 3*) onto (G, 3) } 1 (G, 3) onto (G, 3) 1.

94

TOPOLOGICAL DYNAMICS

(2) The group inversion of G is a unimorphism of { (G, `U.) onto (G, `U,*) } { (G, `U.*) onto (G, %)) { (G, `U.) onto (G, `U.) } .

(3) The following statements are pairwise equivalent: (I) The group inversion of G is 5-continuous and hence 5-homeomorphic. (II) The group inversion of G is 3*-continuous and hence 3*-homeomorphic.

(III) 3 = 3* and hence 5 = 3* = 3. (4) The following statements are pairwise equivalent: (I) The group inversion of G is %-uniformly continuous and hence `U-unimorphic. (II) The group inversion of G is 9.1.*-uniformly continuous and hence `tt-unimorphic. (III) `U = `U.* and hence `U, = `U.* _ `U.. (5) If the group multiplication of G is 3-continuous, then (G, 3) is a topological group. (6) If 'a induces 3, then {'U.* induces 3* } { RL induces 5 } . (7) If G provided with some topology is a topological group and if '1LL , CUR , CUB

are the left, right, bilateral uniformities of G, then: (I) `U. C `U,L is equivalent to `U.* C `uR , and either condition implies `U. C CUB . (II) `U. J `U.L is equivalent to `U.* D `U.R , and either condition implies it J CUB . (III) `U, = CILL is equivalent to `U* = `U.R , and either condition implies

it

dl .B .

11.18. THEOREM. Let X be a uniform space and let be a group of unimorphisms of X onto X. Then: (1) The space-index, the inverse space-index, the bilateral space-index topologies

of 4) all coincide; call this topology 5. (2) (4), 5) is a topological group. (3) The { left } { right } { bilateral } uniformity of (1, 5) coincides with the {spaceindex { inverse space-index } { bilateral space-index) uniformity of I'. (4) ((b, 3) is a topological homeomorphism group of X. PROOF.

(1) By 11.17 (3) it is enough to show that the group inversion of

-b is continuous with respect to the space-index topology of 4). Let (p (E 4) and let a be a symmetric index of X. There exists an index a of X such that (x, y) E / implies (xp-', ycp ') E a. We show (c06*)-' C 'ap . Let ¢ E pa* . Then

(xv, x+G) E 0 (x E X), (x, x¢cp ') = (xw ', zoo) E a (x E X), (x¢-', XP-') = (x¢-', xV'i ') E a (x E X), (xcp ', x¢-') E a (x (=- X) and ¢-' E c 'a* . This proves (1). (2) Use 11.14 and 11.17 (5). (3) Let ,p, ,, (E 4), let 0 be the identity mapping of X and let a be an index of X. The following statements are readily verified:

(s',4')Ea t-*,P-lv,Goa*, (s', 0 ) E a*

9'

((P, 0 E a*

50-1# E Ba*&cp¢-' E Ba.

1 E Oat

, .

FUNCTION SPACES

[11.24]

95

The conclusion follows. Cf. also 11.17 (7). (4) Use 11.02. 11.19. THEOREM.

Let X be a complete separated uniform space and let (b

be the group of all unimorphisms of X onto X. Then 4) is complete in its bilateral space-index uniformity. PROOF. Let `U, be the bilateral space-index uniformity of 4) and let 5 be a `U.-cauchy filter on 4). Let T be the semigroup of all uniformly continuous mappings of X onto X and let `U be the space-index uniformity of I. By 11.05 there exist (p, ¢ E `' such that F -* (p in ti and f 1 ---> 4, in V. By 11.14. FF 1 (p¢ in V and 1`.f OSo in V. Let 0 be the identity mapping of X. Since o E n ff5 1 and 0 E n iF % we have So0 = 0 = 4,Sc. Hence (p, ¢ E 4), (p -3L = 41 and ff --+ 1p in 91. The proof is completed.

11.20. THEOREM.

Let X be a complete uniform space and let 4) be the group

of all unimorphisms of X onto X. Then (b is a bilaterally complete topological group in its space-index topology. PROOF.

Use 11.18 (2) and 11.19.

11.21. THEOREM.

Let X be a totally bounded uniform space and let 4) be a

symmetric set of unimorphisms of X onto X. Then 4) is totally bounded in its bilateral space-index uniformity if and only if 4) is uniformly equicontinuous. PROOF.

Use 11.18 and 11.12 (2).

11.22. THEOREM. Let X be a compact uniform space, let (P be the group of all homeomorphisms of X onto X, let 4) be provided with its space-index topology and let ! be a symmetric subset of 4). Then 1k is compact if and only if' is equicontinuous. PROOF.

Use 11.18 and 11.21.

11.23. REMARK. Let X and Y be sets, let Z be a uniform space and let

7r:X X Y->Z.Write (x,y)ir=xy(x(E X,yE Y).For xEXdefine irr: Y-+Z by yr. = xy (y E Y). For y E Y define a" : X ---) Z by x7r" = xy (x (E X). Then: (1) If X is a totally bounded uniform space and if [7r" I y E Y] is uniformly equicontinuous, then [1r I x E X] is totally bounded in its space-index uniformity. (2) If [ate I x E X] is totally bounded in its space-index uniformity, then for

each index a of Z there exists a finite partition (t of X such that A E a and y E Y implies Ay X Ay C a. 11.24. DEFINITION. Let X be a topological space, let a be the class of all compact subsets of X, let Y be a uniform space with uniformity I and let 4) be a set of mappings of X into Y. For A C X and a C Y X Y define

(A, a) o = [(gyp,') gyp, 4, E 4', (xv, xi') E a (x E A)]. I

Define

v= [(A,a),IAEa,aEI].

96

TOPOLOGICAL DYNAMICS

[11.241

It is readily verified that `U is a uniformity-base of 4). The uniformity 91 of (P generated by U is called the compact-index uniformity of 4). The topology of
Let X be a locally compact topological space, let Y be a

uniform space, let - be a set of continuous mappings of X into Y, let 4) be provided with its compact-index topology and let it : X X 4) -* Y be defined by (x, -p)ir = xcp (x E X, p (E
11.27. THEOREM. Let X be a topological space, let Y be a complete uniform space and let - be the set of all mappings of X into Y. Then (P is complete in its

compact-index uniformity. PROOF. Let 4) be provided with its compact-index uniformity. Let 5 be a cauchy filter on k. For each x E X, [xF I F E `.F] is a cauchy filter-base on Y.

Define (p : X -* Y by xso E n.., xF (x E X). We show 5 -* (p. Let A be a compact subset of X and let a be a closed index of Y. There exists F E `.F such

that FXFC(A,a)..Then xFXxFCa(xEE A),xFXxFCa(xEA), [x-p] X xF C a (x E A), [gyp] X F C (A, a) 4, and F C p(A, a),p

.

The proof is

completed. 11.28. THEOREM.

Let X be a locally compact topological space, let Y be a

uniform space, let 4) be the set of all mappings of X into Y, let 4) be provided with its compact-index topology and let * be the set of all continuous mappings of X into Y. Then I' is a closed subset of (P. PROOF. Let c E'P. We show cp is continuous. Let xo E X and let a be an index of Y. Choose a symmetric index a of Y such that 03 C a and choose a compact neighborhood V of xo . There exists' E'I' n (p(V, 0),, . Select a neighborhood U of xo such that U C V and U¢ C x4,6. If x E U, then (xoc', xo+G) E

0, (xo¢, x&) E 0, (xvi, x(p) E 3, (xocv, x4p) E 03 C a and x(o E xocoa. The proof is completed. 11.29. THEOREM.

Let X be a locally compact topological space, let Y be a

complete uniform space and let - be the set of all continuous mappings of X into Y. Then
Use 11.27 and 11.28.

FUNCTION SPACES

[11.34]

11.30. DEFINITION.

97

Let X be a topological space, let Y be a uniform space

and let 4) be a set of mappings of X into Y. The set 4) is said to be locally bounded

provided that if x E X, then there exists a neighborhood U of x such that U,E1, U(p is a totally bounded subset of Y. 11.31. THEOREM. Let X be a locally compact T2-space, let Y be a uniform space and let 4) be a set of continuous mappings of X into Y. Then P is totally bounded in its compact-index uniformity if and only if (D is equicontinuous and

locally bounded. PROOF.

Apply 11.12 to the restrictions of ( to the compact subsets of X.

11.32. THEOREM. Let X be a locally compact T2-space, let Y be a complete uniform space, let 4) be the set of all continuous mappings of X into Y, let 4) be

provided with its compact-index topology and let ' C -. Then ' is compact if and only if' is equicontinuous and locally bounded. PROOF.

Use 11:29 and 11.31.

Let X be a locally compact uniform space and let 4) be a

11.33. THEOREM.

semigroup of continuous mappings of X into X. Then the semigroup multiplication of 4) is continuous in the compact-index topology of -P. PROOF. Let cpo , ko C 4', let A be a compact subset of X and let a be an index of X. Choose a compact subset C of X and an index y of X such that

Acpoy C C. Then choose an index ,6 of X such that 0 C y and such that (x1, x2) C Q n (C X C) implies (x1'0 , x2¢0) E y. Define B = A U C. Let (cpo , rp) E (B, 0)*

and let (¢o ,') E (C, y) ,j. . We show (cpoGo , pV,) E (A, a).. Let x E A. Then x E B and (xp0 , x,p) E 3. Since (a,po , awp) E l3 C y (a E A), we have Acp C Acp0y C C and xcp0 , xcp C. Hence (xcpo , xcp) E R (1 (C X Q. We conclude that (x,po#o , xvGo) E y, (xvOo , x4) E y and (xcpoi o , x90) E y2 C a. This shows that (,po+Go

,

cvO) C (A, a)* . The proof is completed.

11.34. DEFINITION. Let X be a uniform space with uniformity I, let a be the class of all compact subsets of X and let 4) be a set of permutations of X.

For A C X and a C X X X define (A, a), = [(cv, 4,) cv, 4, E - and (x,v, x4,) C a (x E A)J, (A, a) [(cc,') So, 4, C - and (xcc ', x¢-') E a (x E A)J, I

I

(A, a); = (A, a),, n (A, a)4* Define

U_ [(A, a)* I A E a and a E IJ,

I A E a and aCIJ, _[(A,a),,IAEa and a(EIJ.

ti * _ [ (A , i3

a) ,*

It is readily verified that '0, `0*, `ll are uniformity-bases of 4). The uniformities

98

TOPOLOGICAL DYNAMICS

[11.34]

`U., `UK, iii of b generated by V, V*, 3 are called the compact-index, the inverse compact-index, the bilateral compact-index uniformities of
Let X be a compact uniform space and let 4) be a group of

permutations of X. Then the

{ space-index } { inverse space-index } { bilateral

space-index) uniformity of 4) coincides with the { compact-index } { inverse compact-index} {bilateral compact-index} uniformity of
Let X be a locally compact uniform space, let - be a group of homeomorphisms of X onto X and let - be provided with its bilateral compactindex topology. Then: (1) 4) is a topological group. (2)
PROOF.

Use 11.33.

11.37. THEOREM.

Let X be a locally compact complete separated uniform

space and let
in its bilateral compact-index uniformity. PROOF.

Let Cu be the bilateral compact-index uniformity of 4) and let if

be a `U.-cauchy filter on 4). Let * be the semigroup of all continuous mappings of X into X and let `U be the compact-index uniformity of 'Y. By 11.29 there exist cp,

t' E ' such that if -*

G

in V and -> i/icp in V. Let 0 be the identity mapping of X. Since o E n Fi ' and o E no;-15:, we have co G = B = ip. Hence, (p, ¢ E
Let X be a locally compact separated uniform space and let


Let 4) be provided with its bilateral compact-index topology. By

11.36,
We first show that if A is a compact subset of X, then there exists F E if such that AF U AF-' is a conditionally compact subset of X. Let A be a compact subset of X. Choose an index a of X such that Aa is a conditionally compact

subset of X. Then there exists F E if such that p, ¢ E F implies (x, x(p-'¢) E a (x E A) and (x, x'-') E a (x E A). Select (p,, E F. Then (p E F implies xc 'cpo E xa (x E A), x'pvo' E xa (x E A), Acp lppo C Aa, AV90-' C Aa, A( p-' C App', and Arp C Aasoo . Hence AF U conditionally compact.

AF-' C Aapo' U Aapo and AF U AF-' is

FUNCTION SPACES

[11.40]

99

Let A be a compact subset of X and let a be an index of X. Choose a compact subset B of X such that AF, U AFB' C B for some F, E F. There exists F2 E £ such that gyp, E F2 implies (y, yip `#) E a (y E B) and (y, yvo-') E a (y E B).

Define F = F, (1 F2 E F. Then rp, ¢ E F implies (xop, x¢) E a (x E A) and

(Xlp',xt )Ea(xEA).

Hence, T is a cauchy filter in the bilateral compact-index uniformity of -.

The proof is completed. 11.39. THEOREM. Let X be a locally connected locally compact uniform space and let - be a group of homeomorphisms of X. Then the compact-index, the inverse compact-index, and the bilateral compact-index topologies of 4) all coincide. PROOF. It is enough to show that the group inversion of (D is continuous in the compact-index topology of 4). Let A be a compact connected subset of X such that int A 0, let a be an index of X and let (P E 4). It is enough to show

there exist a compact subset B of X and an index j3 of X such that (.p, ¢) E (B,,3)

implies (co ', ¢-') E (A, a). . We first show there exist a compact subset C of X and an index y of X such that (gyp, ¢) E (C, y), implies A¢-' C C. Choose a compact neighborhood U of A and then choose a symmetric index y of X such that Ay C int U and ay C A for some a E A. Define C = Up-'. Let ((p, ,t) E (C, y)q, . We show A¢-' C C. Suppose A,,-' (Z C. Then A Q C4,. Since app' E C and arp 'ii E a,p 'rpy = ay C A, we have A (1 C¢ 0 cp. Since A is connected, there exists x E A (1 bdy (C¢). Define y = x¢-'. It follows that yip E (bdy C)cp = bdy U and y¢ = x E A. Since y E C, we have (yip, y#) E 'Y and yrp E y*Gy C Ay C int U. We now have yip E bdy U and yip E int U. This is a contradiction. Hence

A¢-' C C. Select a compact subset C of X and an index y of X such that (gyp, ¢) E (C, y),, implies A¢-' C C. Define B = C U Ccp. Choose a symmetric index # of X such

that # C y and such that (x, , x2) E $ n (B X B) implies

x,, p- E a.

Let (gyp, ¢) E (B, ,$)i, . We show (f ', 01 E (A, a)4. . Let x E A. Since (,p, rp) E (C, -1),p

, we have App-' C C, x E A C Csp C B and x E B. Since (gyp, +') E (C, y) 4.

,

we have A¢-' C C, x4,-' E C, pp-'(p E Cp C B and xP-'gyp E B. Since x¢-' E B, and (gyp, P) E (B, 0),, , we have (xV,-'(p, x) Eli and (x, x¢-'(p) E $. Thus x,i-') E a. Hence ((p', ¢-') E (x, xp-',p) E $ n (B X B). It follows that (A, a). . The proof is completed. 11.40. DEFINITION. Let X and Y be topological spaces, let a be the class of all compact subsets of X, let S be the class of all open subsets of Y and let 4) be a set of mappings of X into Y. For A C X and E C Y define

(A, E) 4 _ [ , p I p G

and Ae C E].

Define

s = [(A, E)4, I A E a and E E C ] . The topology 5 of fi generated by 8 is called the compact-open topology of (P.

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100

11.41. THEOREM.

[11.41]

Let X be a topological space, let Y be a uniform space and

let 4) be a set of continuous mappings of X into Y. Then the compact-index topology of CF coincides with the compact-open topology of (b. PROOF.

Let 3, be the compact-index topology of D and let 32 be the compact-

open topology of CF.

We show 3, C 32 . Let A be a compact subset of X, let a be an index of Y and let p E 4). It is enough to show that U C cp(A, a)4, for some 32-neighborhood U of cp. Since cp is continuous on the compact set A, there exist a finite family (A, I L E I) of closed subsets of A and a family (E, c E I) of open subsets of I

Y such that A = U,E1 A, , A,(p C E, (c (E I) and E, X E, C a (L E I). Define U = n, E I (A, , E,) 4, . Now U is a 32-neighborhood of cp. If J, E U and if x E A,

then x E A, for some a E I whence (xcp, x4,) E A,,p X A,V, C E, X E, C a. Therefore ¢ E U implies (xv, x¢) E a (x E A) and 4, E qp(A, a).. Thus U C . This shows 3, C 32 We show 32 C 3, . Let A be a compact subset of X, let E be an open subset

cp(A, a)4,

of Y and let cp E (A, E)4, . It is enough to show that U C (A, E)4, for some 3,-neighborhood U of gyp. Since Ace C E and Ace is compact, there exists an index

a of Y such that Acpa C E. Define U = cp(A, a)4, . Now U is a 3,-neighborhood

ofcp.If,'E U, then (x(p,*) Ea(xEA),x'Ex'paCE(x(=- A),AV,CE and ¢ E (A, E). . Thus U C (A, E) o . This proves that 32 C 3, Hence 3, = 3, and the proof is completed.

.

11.42. REMARK. We conclude from 11.41 that for a set of continuous mappings, the compact-index topology depends only on the topology of the range space. 11.43. THEOREM. Let X be a locally compact space, let Y be a topological space, let CF be a set of continuous mappings of X into Y and let a : X X CF -> Y be defined by (x, p) 7r = xrp (x E X, cp E 4'). Then the compact-open topology of CF is the least

topology of - which makes a continuous.

Let 3 be the compact-open topology of 4). We show that 3 makes 7r continuous. Let x E X, let (p E - and let W be an open neighborhood of (x, cc) = xcp. There exists a compact neighborhood U of x such that Ucp C W. Define V = (U, W), . Now V is a 3-neighborhood of cc and (U X V),r C W. This shows that 3 makes a continuous. PROOF.

Now let 3o be a topology of CF which makes it continuous. We show 3 C 30 Let A be a compact subset of X, let E be an open subset of Y and let p E (A, E), . It is enough to show that U C (A, E), for some 3o-neighborhood U of V. Since (A X [cp])7r C E and A is compact, there exists a 3o-neighborhood U of cp such that (A X U)x C E whence U C (A, E), . The proof is completed. 11.44. DEFINITION.

Let X be a topological space, let a be the class of all

compact subsets of X, let 8 be the class of all open subsets of X'and let CF be a set of permutations of X. For A, E C X define

FUNCTION SPACES

111.471

101

(A, E) 1, = [p I p E 4' and Aco C E], (A, E)4* = [ c o I p E 4'

and Aco 1 C E].

Define

S_ [(A, E),,I A E Ct and E E 8],

5* = [(A,E)*bIAE(t and EE 8],

i=5l)5*. The topologies 5, Y*, 3 of fi generated by 5, 5*, 9 are called the compact-open, the inverse compact-open, the bilateral compact-open topologies of 4). It is clear

that: (1) If 4) is a group, then 3* is the image of 3 under the group inversion of 'D.

(2) 5 = 5 V 3*. 11.45. THEOREM. Let X be a uniform space and let 4' be a group of homeomorphisms of X onto X. Then the { compact-index } { inverse compact-index } { bi-

lateral compact-index) topology of 4) coincides with the { compact-open } { inverse compact-open) { bilateral compact-open) topology of 4). PROOF.

Use 11.41.

11.46. THEOREM. Let X be a locally compact T2-space and let - be a group of homeomorphisms of X. Then the bilateral compact-open topology of 4' is the least topology of 4) which makes 4) a topological homeomorphism group of X. PROOF.

Use 11.43 and 11.36.

11.47. NOTES AND REFERENCES.

The purpose of this section is to set forth those developments of the theory of function spaces needed elsewhere in the book. It can be read independently of the other sections. Most of the results can be found in Bourbaki [4], where some references to the pertinent literature can be found. See also Areas [1, 2].

PART II. THE MODELS 12. SYMBOLIC DYNAMICS 12.01. STANDING NOTATION. Let S be a finite set which contains more than one element and let S be called the symbol class. We shall use i, j, k, m, n, p, q, r, s, t as integer variables, that is variables ranging over g.

12.02. DEFINITION. A { right } { left } ray is a subset R of g such that { R = [i I p < i] } { R = [i I i 5 p] } for some p E g. An interval is a subset I of g such

that I = [i p S i < q] for some p, q E g with p < q. If p E g, then { p, +-I, { - -, p } denote the rays [i p < i], [i I i :5 p]. If p, q E g with p < q, then 1p, q J denotes the interval [i j p =< i < q]. We also write { - co, + co } = g.

12.03. DEFINITION. A { bisequence } { right sequence) {left sequence } { block } is a

function on {g } { a right ray } { a left ray } { an interval } to S.

12.04. DEFINITION. We make the following definitions: (1) If A is a { bisequence } { right sequence } { left sequence), then the reverse of A, denoted A- or A, is the { bisequence } {left sequence l {right sequence l B

such that dmn B = -dmn A and B(-i) = A(i) (i E dmn A). (2) If A is a block, then the reverse of A, denoted Av or A, is the block B such that dmn B = dmn A and B(p + q - i) = A(i) (i (E dmn A) where p, q are the first, last elements of dmn A. (3) If A is a {bisequence } { right sequence } { left sequence) { block } and if n (E g, then the n-translate of A, denoted A", is the { bisequence } { right sequence }

{left sequence) {block} B such that dmn B = n + dmn A and B(n + i) _ A(i) (i E dmn A). (4) If A and B are { bisequences } { right sequences } { left sequences } (blocks),

then A is similar to B, this statement being denoted A- B, in case there exists

n E g such that A" = B. (5) If A is a right sequence or left sequence or block and if B is a bisequence

or right sequence or left sequence or block, then A is contained in B and B contains A in case A is a restriction of B or equivalently B is an extension of A; in the event that A is a { right sequence } { left sequence } { block) we may also say that A is a { right subsequence } {left subsequence } { subblock } of B. (6) If A is a { right sequence } { left sequence } { block } and if B is a bisequence

or right sequence or left sequence or block, then A appears in B in case A is similar to a {right subsequence } (left subsequence) { subblock } of B.

(7) If A is a block, then the length of A is the cardinal of dmn A. (8) If n > 0, then an n-block is a block of length n. 12.05. DEFINITION. Let A,, , A, , be a sequence of blocks such that each A. (n > 0) is contained in A"+1 . The union of Ao , A, , , denoted 102

112.111

SYMBOLIC DYNAMICS

103

Un a A. , is the bisequence or right sequence or left sequence or block A such

that dmn A = Un=0 dmn A. and each A (n > 0) is contained in A. Variations of this notation are obvious, for example, the union of a finite class of blocks. 12.06. DEFINITION. Let A be a {right sequence) { block). Then A denotes A-° where n is the least element of dmn A. Let A, B be blocks with lengths n, m. Then AB denotes the (n + m)-block C such that the initial n-subblock of C is A and the terminal m-subblock of C

is similar to B. Analogously AB denotes the (n + m)-block D such that the terminal m-subblock of D is B and the initial n-subblock of D is similar to A. A, The meaning of AIAZ is now clear where A, , , A. are blocks except that A, may be a left sequence (if p > 1) and An may be a right sequence.

Other uses of this "indexing-by-dot" notation are obvious. For example, if , A_, , Ao , A, , are blocks, then ( A_,AoA,A2A3 . . .) denotes a certain bisequence. If x is a bisequence, then we may write x = (. . . x(- 1)i(O)x(l) . . .). If, a, b, c E S, then a denotes a certain 1-block, ab denotes a certain 2-block, abe denotes a certain 3-block, etc. 12.07. DEFINITION. Let A, B be blocks with lengths n, m. Then AB denotes the (n + m)-block C such that the initial n-subblock of C is A and the terminal m-subblock of C is similar to B. Analogously AB denotes the (n + m)-block D such that the terminal m-subblock of D is B and the initial n-subblock of D is similar to A. Other uses of this "indexing-by-roof" notation are obvious. For example, if A, A, , , An are blocks, then AIAZ denotes a certain block which contains A, .

12.08. DEFINITION. The bisequence space is the set of all bisequences, that is, the set S. We consider the symbol class S to be provided with its discrete topology and we consider the bisequence space S9 to be provided with its product topology or equivalently its point-open topology. 12.09. STANDING NOTATION.

Let X denote the bisequence space S. Let

p : X X X --), (R be the function defined by p(x, y) = (1 + sup [n j x(i) = y(i)

for IiI
The shift transformation of X is the homeomorphism

Q:X° X defined by xv = (x(i + 1) 1 i E I) (x E X). The symbolic flow generated by S is the discrete flow on the bisequence space X generated by the shift trans-

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104

formation a of X and is denoted by (S, X, a-). We shall call S the symbol class of (S, X, a). 12.12. REMARK.

Let x E X and let a be the shift transformation of X. Then

xa _ (... x(_1)x(0)x(1) ...)a = (... x(0)x(1)x(2) ...). 12.13. REMARK. The symbolic flow generated by S { coincides If is isomorphic } with the { left } { right } functional transformation group over 9 to S.

12.14. REMARK. Let T be a set such that crd S = crd T. Then the symbolic flow generated by S is isomorphic to the symbolic flow generated by T. 12.15. DEFINITION.

Let n = crd S. By virtue of 12.14 the symbolic flow

generated by S may be called the symbolic flow on n symbols. 12.16. STANDING NOTATION. Let (S, X, u) denote the symbolic flow generated by S. Properties of invariance, recursion, etc., are relative to the shift

transformation.

Let x E X. Then: (1) x is periodic if and only if there exists p > 0 such that x(i + p) = x(i) 12.17. REMARK.

(iE9).

(2) If x is periodic, then the period of x is the least p > 0 such that x(i + p) _

x(i) (i E 9). 12.18. REMARK. Let x E X. Then the following statements are pairwise equivalent: (1) x is regularly almost periodic.

(2) If n > 0, then there exists p > 0 such that x(i + pj) = x(i) (I i 1 S n, i G 9). (3) If n E 9, then there exists p > 0 such that x(n + pj) = x(n) (j E 9). 12.19. REMARK. Let x E X. Then the following statements are equivalent: (1) x is isochronous.

(2) If n > 0, then there exist p > 0 and q E 9 such that x(i + pj + q) _

x(i)(IiI 5 n,jE9). 12.20. REMARK.

Let x E X. Then the following statements are pairwise

equivalent: (1) x is almost periodic.

(2) If n > 0, then there exists a syndetic subset E of 9 such that x(i + j) _

x(i)(Iil_<_n,jEE)

(3) If A is a subblock of x, then there exists k > 0 such that A appears in

every k-subblock of x.

(4) If n > 0, then there exists k > 0 such that every n-subblock of x appears in every k-subblock of x. 12.21. REMARK. Let x E X. Then the following statements are equivalent: (1) x is recurrent.

SYMBOLIC DYNAMICS

[12.301

105

(2) If n > 0, then there exists an extensive subset E of 9 such that x(i + j)

x(i)(Ii1
Let x E X. Then the following statements are equivalent: (1) x is transitive. (2) Every block appears in x. 12.22. REMARK.

Let x E X. Then the following statements are equivalent: (1) x is extensively transitive. 12.23. REMARK.

(2) If x = BA, then every block appears in both A and B. 12.24. REMARK. The following statements are valid: (1) The set of all periodic points is an invariant countable dense subset of X. (2) Q is regionally regularly almost periodic, regionally almost periodic and regionally recurrent. (3) The set of all recurrent points is an invariant residual G$ subset of X.

(4) The set of all extensively transitive points is an invariant residual Ga subset of X. (5) Q is mixing. (6) Q is expansive. 12.25. REMARK.

Let x, y E X. Then the following statements are equivalent:

(1) x and y are { positively } { negatively } { doubly } asymptotic.

(2) There exists n > 0 such that {x(i) = y(i) (i > n) } {x(i) = y(i) (i < n) } {x(i) = y(i) (I i 1 > n)}. 12.26. STANDING NOTATION.

Let S = [0, 11. Thus (S, X, v) denotes the

symbolic flow on 2 symbols.

12.27. DEFINITION. The dual of f01111, denoted 10'111'j, is 111101. Let A be a { bisequence } fright sequence } { left sequence) { block). The dual of A, denoted A', is the {bisequence } fright sequence } { left sequence } { block)

such that dmn A' = dmn A and A'(i) = A(i)' (i E dmn A). Let 3 denote the homeomorphism of X onto X defined by xS = x'(x (E X). 12.28. DEFINITION.

Define the sequence Q, , Q,

,

of blocks inductively

as follows:

(1) Q, = 0.

(2) If n > 0, then Define µ = QQ E X.

Q

12.29. DEFINITION.

Let A be a { bisequence } { right sequence left sequence }

{block} and let dmn A = {p, q}. The Q2-extension of A, denoted A*, is the { bisequence } fright sequence } { left sequence } { block }

such that dmn A* =

{2p, 2q + 1}, A*(2i) = A(i) (p
Let n _>_ 0. Then:

(1) Q is the initial 2''-subblock of

(p ? 0).

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106

[12.30]

(2) Qn = Qn if n is even. On = Qn if nis odd. (3) Qn = Qn+1

(4) Q* = Q. (5) Qn+3 = QnQ'Q'QaQnQnQaQ' Qn+3 =

(6) µ=µ. (7) µ I 1- 2', -1 } - Q. if n is even. µ { -2, -1 } N Q,' if n is odd. { - 2n, 2' - 11 = Q.Q. if n is even. { - 2n, 2n - 11 = if n is odd. (9) If n is even and if P. (i E g) are blocks such that JQn if µ(i) = 0

(8) µ

QnQn

P; = Qn

if µ(i) = 1 ,

then µ = (... P_j 0P1 ...). (10) If n is odd and if P. (i (E g) are blocks such that Q.

if µ(2) = 0 and i >- 0,

Qn

if µ(i) = 0 and i < 0,

n

if µ(i) = 1 and i > 0,

P; =

Qn

if µ(2) = 1

and

i < 0,

then µ = (... P_,P0P1 ... ). 12.31. REMARK. For n > 0 define 0(n) = E _oa; where n = E `_oa;2', , k - 1) and at = 1, and define 0(0) = 0. If n >= 0, then µ(n) = 0(n) (mod 2).

a; = 0 or 1 (i = 0, PROOF.

The statement is true for n = 0 or 1, that is, for n such that

0 -< n < 2" - 1 where p = 1. Let p > 1 and assume the statement is true for n such that 0 <--_ n 5_ 2" - 1. We show the statement is true for n such that 0 <--_ n < 2p+1 - 1. Let n be such that 2' - 1 < n < 2'+1 - 1. It is enough to show that µ(n) = 0(n) (mod 2). Since i(0) µ(2y+1 - 1) = Qa+l = QvQ,, ,

we have µ(n) = µ'(n - 2') 02 µ(n - 2') - 0(n - 2') 0, 0(n). Hence µ(n) _ 0(n) (mod 2). The proof is completed. 12.32. THEOREM. PROOF.

µ is almost periodic.

Let A be a subblock of µ. Choose a positive even integer n such

that A appears in Q,Qn . Let k = 2n+a and let B be a k-subblock of u. We show

SYMBOLIC DYNAMICS

112.41]

107

that A appears in B. By 12.30 (9), B contains a subblock similar to or By 12.30 (5), we have that B contains a subblock similar to Q.0. and Qn+3 . hence a subblock similar to A. The proof is completed. 12.33. DEFINITION.

Define M to be the orbit-closure of µ under a.

12.34. REMARK. The following statements are valid: (1) M is a minimal orbit-closure under a. (2) If A is a subblock of µ, then A, A', A* appear in µ.

(3) If x E M, then x, x', x* E M. 12.35. REMARK. The three unary operations µ yield exactly four different bisequences, namely, 12.36. DEFINITION. 12.37. REMARK.

applied successively to

Define v = µ* whence µ = v*.

In summary:

QV V = Q'Q,

V' = QQ'

are the only bisequences obtained from u by use of 12.38. REMARK. The following statements are valid:

(1) µ, µ', v, v' E M. (2) µ and µ' are separated. v and v' are separated. µ and v are negatively separated and positively asymptotic. µ' and v' are negatively separated and positively asymptotic. µ and v' are negatively asymptotic and positively separated. µ' and v are negatively asymptotic and positively separated. 12.39. THEOREM.

a I M is not {locally recurrent} {locally almost periodic}

{ isochronous } at any point of M.

12.40. REMARK. Let n = 2k > 0 and let A be the class of all orbit-closures under 0"° 1 M. Then crd A = n. Hence M is not totally minimal under a I M and v I M is not mixing.

12.41. REMARK. The set of all nonisochronous almost periodic points of X is dense in X. PROOF.

Let A be a block. Construct ( A

P; _

P_1PoP, ...) where

if µ(i) = 0,

(i E 4) A' if µ(a) = 1,

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108

[12.421

12.42. DEFINITION. Let x be a bisequence and let A be a block. An A-representation of x is a family (A; I i (=- g) of blocks such that A; = A or A'(i 5'!6 0), A(,

orA',0 E dmnA,, a n d

= (... A_1A(,A1 ).

Let n > 0 and let x be a bisequence which has a Q.-repreof x induced by (A; I i (=- g) is the unique Q"_1-representation (B; I i E g) of x such that Ao = B0B1 or B_1Bo . 12.43. DEFINITION.

sentation (A1 I i E g). The 12.44. LEMMA.

Let x E M and let n > 0. Then there exists exactly one

sentation of x.

PROOF. By 12.30(9 & 10) there exists a Qri representation of u and therefore

of ur' (i E 9). Since U:EI ua' is dense in M, it follows that x has at least one Qn representation.

For n > 0 let S. denote the statement that there exists at most one Qn representation of x. Clearly So is true. Let k > 0. Assume Sk is true. We show that Sk+1 is true. Let (A; I i E g) and (B; i (E 9) be Q,,,,-representations of x. In order to show that (A; I i E g) _ (B; i E g) it is enough to show that p = r where { p, q) = dmn A. and { r, 81 = dmn Bo . We may suppose that p 5 r p + 2k, q < s. Assume that p 5,4- r whence p < r. We seek a contradiction. If r then the Qk-representations of x induced by (A; I i (E g) and (B; i E g) are not identical which is contrary to Sk . Hence r = p + 2k. Each A; , B1 (i (E .4) is similar to Qk+1 = QkQk' or to Qx+1 = QkQk . Consideration of the common Qk-representation of x induced by (A; i E g) and (B; i (E g) now shows that x is periodic. This is a contradiction. The proof is completed. I

I

12.45. LEMMA. Let x E M and let A, B be left, right sequences such that x = AB or AB. Then:

(1) IfA-QorQ',then B.QorQ'. (2) IfBNQorQ',then A PROOF.

QorQ'.

Let x E M and let P be a left sequence such that x = PQ. By 12.34,

it is enough to show that P = Q or Q'. Let n > 0. It is enough to show that x if - 2, -1 } - Q" or Q, since the end elements of { Q" } { Qn } are { 0's } { 1's } when n is even. To show this it is enough to prove that if (A; i E g) is the Q,, representation of x, then 0 is the least element of dmn Ao . Let (A; I i E g) be the Q,, representation of x and let (B; i E g) be the Qn representation of u = QQ. Now 0 is the least element of dmn Bo . Choose y E M such that for of positive integers we have some sequence n1 , n2 , lim xor"` = lim uQ"' = Y. I

I

It follows that if 0 is not the least element of dmn Ao , then y has two different Q,, representations. The proof is completed. 12.46. DEFINITION. Let x E M, let n > 0 and let (A; I i (E g) be the Q.-representation of x. Then (A; J.i E g) is {left} {right} indexed provided that {A0 = Bi,B1} (A0 = B_1Ba} where (B; I i E I) is the Q"_1-representation of X.

SYMBOLIC DYNAMICS

[12.49)

12.47. DEFINITION.

a, = 0 or 1 (n = 0, 1,

Let a = (a0 , a, ,

109

) be a dyadic sequence, that is,

). Define the sequence F,, , F,

,

of blocks inductively

as follows: (1) F,, = ao

(2) If n > 0, then if a,+, = 0, F-11 =

FnF if

1.

Define F. = Un=o F, . Let a be a dyadic sequence. Then: (1) F. !., Q. or Qn (n > 0) in the notation of 12.28. (2) Every subblock of { F. } { µ } appears in { j } {F,,}. F. (3) If a contains infinitely many 0's and infinitely many 1's, then Fa is a bisequence such that Fa E M. 12.48. REMARK.

(4) If a contains only finitely many { 0's } { 1's 1, then { F. '-' Q or O' l { F, or Q'}.

(5) If a = (0, a, ,

) and if b = (1, a,

,

), then F. = Fb

Q

.

12.49. THEOREM. Let N be the set of all bisequences x such that x = F. or P.P. or F,Fa or P.P. or FaF, for some dyadic sequence a. Then N = M.

That N C M follows immediately from 12.48. We show M C N. Let x E M. Define the dyadic sequence a = (a0 , a, , PROOF.

)

as follows: ao = x(00).

a; = S 0T if the Q,-representation of x is {right} indexed (i > 0). We adopt the notation Fo , F, ,

of 12.47. For n > 0 let (A;

5) be the

Q,- representation, of x.

We first show that F. = Aa (n 0). Clearly F,, = Ao . Let k 0. Assume F,. = Ao . We Show F", = Ao+'. If a,.+, = 0, then Ao+' = AOAI = AoAo, _ F,F, = Fk+, . If at,, = 1, then Ao+' = Ak,Ao = Ao'Ao = FaPR = F.+, . This completes the proof that F. = Ao (n 0). We now have that F. is contained in x since F. = Un=0 F, = U.=0 Ao' and Un=o Aa is contained in x.

Case I. The sequence a contains infinitely many 0's and 1's. Since F. is a bisequence, x = F. and x E N. Case II. The sequence a contains only finitely many 0's. Then F. is a left sequence such that F. '' Q or Q'. Now x = F,P for some right sequence P. By 12.45, P Q or Q'. Therefore P P. or Pa" x = P.P. or F,FF , and x E N. Case III. The sequence a contains only finitely many 1's. Then F. is a right sequence such that F. '' Q or Q'. Now x = PF, for some left sequence P. By

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110

[12.49]

P. or F' , x = P.P. or P.P. , and x E N.

12.45, P - Q or Q'. Therefore P The proof is completed. 12.50. LEMMA.

Let x E M such that x does not belong to the union of the

orbits of µ, µ', v, v' and let n > 0. Then there exists m > 0 such that if (A; l i E 4) is the Qom,-representation of x, then { -n, n C dmn Ao PROOF.

Cf. the proof of 12.49.

12.51. DEFINITION.

Define r : X -* X by 0

if x(i - 1) 0 x(i)'

1

if x(i - 1) = x(i),

(xr) (i) =

(x E X, i E 9)

Define

n=I
=m, H

Mr.

It follows from 12.31 that n(0) = 1, (0) = 0, and

12.52. REMARK.

{O

if k is odd,

(i, j, k E 9; i 0 0; i = j2'; j odd).

12.53. NOTATION. Let x, y E X. The statement that x and y are { negatively } { positively } { doubly } asymptotic is denoted by { x J, y } { x T y } { x 2 y 1.

12.54. REMARK.

Let x, y E X. Then:

(1) r is an exactly 2-to-1 continuous-open mapping of X onto X.

(2) xr = x'r. xrr-' = [x, x'] = [x, x6]. (3) xor = xro. (4) If x is recursive, then xr is recursive. (5) If x ,} y, then xr . yr.

If x T y, then xr T yr. (6) µr = n = Ij'r, nr ' = [A, µ']. yr = r = v'r, r7r ' = [v, v']. (7) n(0) = 1.

(0) =0.

n(i) _ r(i) (i 0 0). (8)

and hence neither n nor r is regularly almost periodic.

(9) n, r E H. (10) H is a minimal orbit-closure under o. 12.55. THEOREM.

The following statements are valid:

(1) n and r are isochronous.

SYMBOLIC DYNAMICS

112.581

111

(2) The set of all regularly almost periodic points of H coincides with the comple-

ment in H of the union of the orbits of I and . (3) The orbits of -1 and r are the only nonseparated orbits in H. (4) If co is a homeomorphism of H onto H such that oSo = (pa, then cp = v" for some integer n. (5) The only nondegenerate traces under a I H are [i7a", o ] (n E J). PROOF. Let x E M such that x does not belong to the union of the orbits of p, p', v, V. It is enough to show that y = x7r is regularly almost periodic. Let n > 0. By 12.50 there exists m > 0 such that if (A; I i E J) is the Qm representation of x and if dmn Ao = 1p, q } , then { -n, n j C l p + 1, q } . Let (A; I i E g be the Q,n representation of x and let dmn A U = { p, q 1. If j E 5, then either

x (i + 2"'j) = x(i) (i E {p, q}) or x(i + 2"`j) = x(i)' (i E {p, q}). Hence y(i + 2'"j) = y(i) (i E {p + 1, q}, j E g and y(i + 2"j) = y(i) (I i I n,

j E 9). The proof is completed.

The following statements are valid: (1) The only nonseparated orbits in M are the obvious ones, namely, the orbits 12.56. THEOREM.

of tandv, z' and v',pandv',p'andv. (2) If ,p is a homeomorphism of M onto M such that ocp = coo, then rp = 0"S' for some integers n, m (m = 0, 1). (3) Let E be the union of the orbits of p, p', v, V. Then the traces under o I M are:

(x E M - E),

[x, x']

[po", p'v", vv",

v'Q"]

(n E g)

12.57. DEFINITION. Let C be an oriented circle (1-sphere) of circumference 1 and let C be provided with its natural topology. Let a be an irrational number such that 0 < a < 1. Let A be a closed interval in C of length a and let { a } { a' }

be the { first } { last } endpoint of A. Let A - = A - a+ and let A' = A - a-. Let p : C C be the rotation of C such that a -p = a+. For p E C define xy xp E X as follows:

12.58. REMARK.

0

if pp' CA-,

I1

if pp` Er A-,

0

if pp' E A+,

L1

if pp' $ A+,

(iEg) (iE.c)

Let p, q E C. Then:

(1) The following statements are pairwise equivalent: (I) xp = xp . (II) pp' a(III) pp" 0 a+

(nEJ). (n (=- g).

112

TOPOLOGICAL DYNAMICS

(2) If n E 9 and if pp ` = a , then: (1) xp(i) = x+(i) (i E 9; i (II) xy(n) = 0, xn(n) = 1,

[12.581

n, n + 1).

xD(n + 1) = 1, x+(n + 1) = 0.

(III) xy I xn (3) If n E 9 and if pp' = a+, then: (I) xn(i) = x+(i) (i E 9;; i 34 n - 1, n). (II) xp(n - 1) = 0, x9(n) = 1, x ,(n - 1) = 1, xp(n + 1) = 0. (III) xD S xD (4) [xp , xD ] (l [xo , xo ] _ 0 if and only if p 5,1 q. 12.59. DEFINITION. For D C C define D* = [xn I p E D] U [x; I p E D]. Define F = C*. Define 5 to be the class of all subsets E of r such that:

(1) If p E C such that x., = xD E E, then D* C E for some open interval D in C which contains p. (2-3) If p E C such that xD 94- xD and { xn E E) { xD E E}, then D* C E for some nondegenerate closed interval D in C with { initial) { terminal } endpoint P-

E C. Then J(-) lim;-m p; = p} 12.60. NOTATION. Let p, p, , P2 , I(+) lim;-m p; = p} means that if D is a nondegenerate closed interval in C with { initial } { terminal } endpoint p, then p; E D for all sufficiently large i. The symbol xD denotes x, or xn

.

12.61. REMARK. Let p, p, , p2 ,

E C. Then:

(1)

If lim p; = p and if xD = x9 , then lim x,-; = xa in X, lim x,, = xD in X.

(2)

(4)

If (-) lim p; = p and if xD xn , then lim xa, = xa = lim xD in X. If (+) lim p; = p and if xn 54 xp , then lim x9i = xy = lim xD; in X. If lim xp*, = xD in X, then lim p; = p.

(5)

If lim xp; = xD in X and if xr = x.- 54 xa , then (-) lim p; = p.

(6)

If lim xD; = x, in X and if xa 5,6 xa = xa* , then (+) lim p. = p.

(3)

12.62. REMARK. The topology of r induced by the topology of X coincides with 3.

SYMBOLIC DYNAMICS

112.641

12.63. THEOREM.

113

The following statements are valid:

(1) r is a totally minimal orbit-closure under a. (2) a r is locally almost periodic. (3) r contains only one pair of nonseparated orbits, namely, the orbits of xa -

and xa + and x.- - I xa - . (4) If rp is a homeomorphism of r onto r such that (pa = app, then cp = a' I r for some integer n.

(1) and (2) Cf. Hedlund [4]. (3) Obvious.

PROOF.

(4) Use 10.10. 12.64. NOTES AND REFERENCES.

References to the literature on symbolic dynamics up to 1940 can be found in the papers of Morse and Hedlund [3, 4]. (12.28) The bisequence µ was originally defined by Morse (cf. Morse [1]) and used by him to study the behavior of the geodesics on surfaces of negative curvature. (12.31) The equality of µ and 0 was noted by G. D. Birkhoff ([1], vol. 1, p. 691).

(12.47) The constructive method described for obtaining all the bisequences in the minimal set M is due to S. Kakutani (Personal communication). (12.61) The bisequence +7 was defined by Garcia and Hedlund [1] without reference to the bisequence µ and the equivalence of their definition with (12.52) was observed by J. C. Oxtoby. (12.57) The bisequence defined is Sturmian (cf. Hedlund [4]).

13. GEODESIC FLOWS OF MANIFOLDS OF CONSTANT NEGATIVE CURVATURE 13.01. THE HYPERBOLIC PLANE. Let U be the unit circle z2 = x2 + y2 = 1 of the complex z-plane e and let M be its interior. The hyperbolic plane 1t is the two-dimensional analytic Riemannian manifold defined by assigning to M the differential metric

ds

(1)

2 _ 4(dx2 + dy2)

4 dz dz

(1 - x2 - y2)2 - (1 - z2)2

If (p is an are of Class D' in Bi, the hyperbolic length of rp, or h-length of cp, denoted 2,((p), is (2)

£h(cP)

- Lump 2l-x2 + yydt. 2

Since in (1) we have gl, = g22 and g12 = 0 = g2, , angle in 91t coincides with euclidean angle. Let E be a measurable subset of M. The hyperbolic area of E, or h-area of E,

denoted by %(E), is

ah( - E (1 -4x2dx -dyy2)2_

(3)

13.02. GEODESICS IN THE HYPERBOLIC PLANE. Let 91Z* be the two-dimensional analytic Riemannian manifold defined by assigning the differential metric dx2 + dy2 ds2 = (4) y2

to the space M* = [z I z E e, .4 (z) > 0]. Let 8 denote the complex sphere. The transformation

iz + i -Z + 1 zG8)

(5)

is an analytic homeomorphism of 8 onto S. If µ denotes the restriction of (5) to M, then µ is an analytic isometry of J1l onto 91Z*. The following three sets coincide:

(a) The set of all geodesics parameterized by arclength in M*. (0) The set of all curves ((x(s), y(s)) I s E (R) of Class C2 in

by arclength in M* such that

y(s) z(s) - 2x(s)y(s) = 0

(s E (R),

y(s) y(s) - 2(y(s))2 + (y(s))2 = 0

(s E (R).

114

parameterized

GEODESIC FLOWS

113.051

115

(y) The union of the following four sets of curves: [((a tanh (s + c) + b, a sech (s + c)) 18 E 6i) I a, b, c E 61 & a > 01, [((a tanh (-8 + c) + b, a sech (-s + c)) I s E 61) I a, b, c E 61 & a > 0], [((a,

e8+b)

I s E (R) I a, b E a],

[((a,e 8+b) IsEE (R) Ia,bE 61].

It follows that 3tz* is complete (in the sense of Hopf and Rinow [2]) and the range of any geodesic in M* is the intersection of M with a circle in S which is orthogonal to the x-axis. Thus M is complete and the range of a geodesic in M is the intersection of M with a circle in e which is orthogonal to U. 13.03. HYPERBOLIC LINES, RAYS AND LINE SEGMENTS. Let C be a circle in S which is orthogonal to U. The set C (\ 9ii is called a hyperbolic line or h-line. Any are of C which together with its endpoints p and q lies in nt is called a hyperbolic line segment or h-line segment and is said to join p and q. Given two different points p and q of M, there exists a unique h-line segment joining p and q. Let L be an h-line and let p E L. Either of the two components of L - p together with p is called a hyperbolic ray or bray of which p is called the initial

point.

Let L be an h-line. Then L = C n nt, where C is a circle in S orthogonal to U. The two points in which C meets U will be called the points at infinity of L. Given different points u and v of U, there exists a unique h-line with u and v as its points at infinity. Let R be an h-ray. The set R has just one limit point on U and this point is called the point at infinity of R. Given p E 9TZ and u E U, there exists a unique h-ray with p as initial point and with u as point at infinity. 13.04. HYPERBOLIC DISTANCE. Let p and q be different points of 1l and let S be the unique h-line segment with endpoints p and q. All geodesic arcs

in 91z with range S have the same h-length and this h-length, denoted by D(p, q), will be called the hyperbolic distance or h-distance between p and q. The h-distance between p and q is the greatest lower bound of the h-lengths of curves of Class D' in it and joining p and q (cf. Hopf and Rinow [2]). 13.05. ISOMETRIES OF THE HYPERBOLIC PLANE. Let r be an analytic isometry of NTt onto M. Then r is a conformal (directly or indirectly) analytic homeomorphism of M onto M and r admits a unique extension r* to M U U such that r* is a homeomorphism of M U U onto M U U. Let r* have distinct fixed points u, v E U. If h denotes the hyperbolic line with points at infinity u and v, then hr = h and either all points of h are fixed under r or no point of h has this property. In the latter case it is said that r is an isometry of M onto M1 with

axis h and h is called an axis with endpoints u and v. If r is an isometry of YTt onto nz with axis h and r advances points of h toward v, v is the positive fixed point

of r and u is the negative fixed point of r. The points u and v are the only fixed points of r* in M U U. If V is any neighborhood in c of { u } { v } and A is any subset of M U U such that u, v Er A, there exists an integer N > 0 such that

In < -N, n E J} In > N, n E 9} implies A(r*)" C V.

TOPOLOGICAL DYNAMICS

116

[13.051

To a large extent, consideration of isometries of 91L onto 911 with axes will suffice for later developments, but the collection of all isometries of 911 onto 911 admits a complete and simple analysis which we develop briefly. Let E+ denote the group of linear fractional transformations

((az+c)/(cz+a) IzES) where a, c E e with ad - cc = 1. Let y E E+. Then Ua = U, My = M, o is a directly conformal analytic homeomorphism of 8 onto S and any transformation with these properties is an element of V. It is easily verified that the restriction of a to M is an analytic isometry of M onto M.

Let a = ((az + c)/(cz + a) I z E 8) E Z+ and suppose or is not the identity. Since a + a is real, y must be either hyperbolic with fixed points on U, parabolic with fixed point on U, or elliptic with fixed points inverse to U.

Let a E E+, let a be hyperbolic, and let u, v E U be the fixed points of y. converges to Let p E S with p 5 u, v. Then the sequence p, pa 1, pv 2, converges to the one of the fixed points of a and the sequence p, py, py2, other fixed point of y. The first of these points will be called the negative fixed point of y, and the other will be called the positive fixed point of a. Let u be the negative fixed point of a, let u+ be the positive fixed point of a and let A be any

subset of S such that u , u+ ($ A. Then if V is a neighborhood of {u } {u+} there exists a positive integer N such that { n < - N, n E 411 n > N, n E 9 } implies Ay" C V. Any circular are of S with endpoints u and u+ is invariant under a. The axis of a is the hyperbolic line h with points at infinity u , u+, and these points will also be called the endpoints of the axis. Let T = a I M. Then T is an isometry of 9Th onto 911 with { negative positive } fixed point { u } { u+ } and with axis h.

Let or E E+, let T be parabolic, and let u E U be the fixed point of a. If A is any subset of S such that u EF A, and V is any neighborhood of u, then there

exists an integer N > 0 such that I n I > N, n E 9, implies Aa° C V. If C is any circle which is tangent to U at u then Ca = C. Let LT E E+ and let a be elliptic. Then y has two fixed points in S which are

inverse with respect to U. There exists a disjoint class of circles covering S except for the fixed points of a such that each member of this class is invariant under y and the fixed points of a are inverse with respect to each member of this class.

Let E- denote the set of transformations ((az + c)/(cz + d) z E 5) where I

a, c E G with ad - cc = 1. Let y E E-. Then Ua = U, Ma = M, a is an inversely conformal analytic homeomorphism of S onto S and any transforma-

tion with these properties is a member of Z-. The restriction of a to M is an analytic isometry of 911 onto M.

Let a- E E-. Then, either there exists a circle C C S orthogonal to U such that p E C implies pa = p, or a has exactly two fixed points u, v and u, v E U. In the first case y is an inversion in C. In the second case, let T be the inversion in the circle D which passes through u and v and which is orthogonal to U. Let

[13.06]

GEODESIC FLOWS

117

Q1 = or. Then al is directly conformal, Ual = U, Mo-1 = M, a, cannot be the identity mapping, a-, (E V and uo-1 = u, vo1 = v. Thus Q1 is a hyperbolic trans-

formation with fixed points u and v. We have a- = r lv1 = rQ1 = Qtr and we conclude that a is the product of a hyperbolic transformation and an inversion in a circle orthogonal to U which passes through the fixed points of o. The transformation a is called a paddle motion. The { negative } { positive } fixed point of a- is defined to be the_{ negative } { positive) fixed point of a-, . If A is any subset of S such that u, v (Er A and V is any neighborhood of the { negative } I positive }

fixed point of v, there exists an integer N > 0 such that In < -N, n E a} In > N, n E g } implies AQ° C V. The axis of or is the axis of a, . Let r = or I M. Then r is an isometry of T onto B with axis h and with { negative } { positive } fixed point that of a-. Let E = 1- U V. Then is a group of conformal analytic homeomorphisms of S onto S and any conformal analytic homeomorphism r of S onto S such that

Ur = U and Mr = M is an element of E. Let T CM and let or = r I M. Then -a is an analytic isometry of DR onto M, and conversely, if a- is an analytic isometry

of J1l onto 3, then there exists a unique element r E E such that a = r I M. This extension of a will be denoted by 6. 13.06. HYPERBOLIC CIRCLES AND EQUIDISTANT CURVES. Since the differential metric 13.01(1) is invariant under rotations about the origin, the locus of points of Uff at constant h-distance from the origin is a circle with center at the origin 0. If p is any point of on, there exists a C E such that Oo- = p; thus the locus of points at constant h-distance r from p is again a circle C containing p in its interior, though unless p is at the origin the euclidean center of C will

not coincide with p. We call C the hyperbolic circle or h-circle with center p and with radius r. It is invariant under the group of elliptic transformations which have p and its inverse in U as fixed points. Given an h-line L which does not pass through the origin 0, there is a unique

h-line passing through 0 and orthogonal to L. By application of a suitable transformation belonging to we see that the same is true for any h-line L and point p E n, p not on L. Let q be the intersection of L and the h-line through p orthogonal to L. The h-line segment pq is the hyperbolic perpendicular from p to L and D(p, q) is the h-distance from p to L. The h-distance from p to L is denoted D(p, L). The point q is the foot of the perpendicular from p to L. By consideration of the case where p is at the origin, it is evident that the h-distance

from p to L is less than the h-length of any h-line segment joining p to any point of L other than the foot of the perpendicular from p to L. Let L be an h-line with points at infinity u and v. Let C be a circle passing through u and v with C ;-, U and C not orthogonal to U. Let A = C (1 M. Then A is invariant under every hyperbolic transformation with u and v as fixed points. Let p, q E A. There exists a linear fractional transformation a such that ua = u, va = v, pa = q. But then My = M, Ua = U, a- E V, Aa = A, and a is a hyperbolic transformation with axis L. The hyperbolic perpendicular from p to L is transformed by or into the hyperbolic perpendicular from q to

TOPOLOGICAL DYNAMICS

118

[13.06]

L and thus all points of A are at the same h-distance d from L. We call d the h-distance from A to L. 13.07. HOROCYCLES.

Let C be a euclidean circle which is internally tangent

to U at u (E U and let H = C - u. Then H is invariant under any parabolic transformation in V which has u as fixed point and H is an orthogonal trajectory of the family of h-lines which have u as common point at infinity. We call H a horocycle and u its point at infinity.

Let H be a horocycle and let A be a circular arc such that A C H. We can parameterize A with euclidean arclength and thus define an are in with A as its range; the h-length of this arc will be called the h-length of A. Let LI and L2 be h-lines with common point at infinity u and let Hl and H, be horocycles with u as common point at infinity and such that H, is interior to H2 U u. Let { sI } { s2 } denote the h-length of the are of { Hl } { H2 } cut off by L1 and L2 and let s be the h-length of the h-line segment of L1 (or L2) cut off by

Hl and H2 . Then s2 = sle'.

(6)

To derive this formula we can assume that u is any point of U and, in particular, we can choose u = -I-1. Under the transformation 13.02(5), which is an analytic isometry of M onto !*, the image of { L, L2 } is the range of a geodesic { (x(t) = aI , y(t) = e`) t E R} { (x(t) = a2 , y(t) = e') t E R} and the image of {HI} I

I

{ H2 } is the range of the curve

{ (x(t) = t, y(t) = dI) I min (a, , a2) 5 t < max (a, , a,)) { (x(t) = t, y(t) = d2) I min (a,

,

a2) < t < max (a, , a2) } with d, > d2

We then have a,

s; = I f d; I dt

d, I

a.

I aI - a2

i = 1, 2,

and d

s

=

dy

f,l y

= log (d1/d2)

which imply (6). 13.08. ASYMPTOTIC GEODESICS IN M. Let cp be a geodesic parametrized by

arclength in 1Z and let L be the range of gyp. The sequence {(p(-1), (p(-2),

}

} converges in a to a point {u (E U} {u+ E U}. The points u , u' are the points at infinity of L. The point {u } {u+} is called the {negative) { positive } point at infinity of (p. If { V- } { V' } is an open subset of G containing { u } { u+ } then there exists { s E at } { s+ E (R } such that { s < s } {,0(1), So(2),

{s > s+} implies {p(8) E V-} {cp(s) (E V+}. Let cp and t' be geodesics parameterized by arclength in 1 let {u-, u+}

GEODESIC FLOWS

[13.10]

119

{v , v+} be the negative, positive points at infinity of {,p} {¢} and let L = rng +'. Then (a) u = v and u+ 5-!s v+ implies lim D(,p(r), L) = 0,

r-.- m

lira D(,p(r), L) =+-.

r-.+m

(b) u 5 v and u+ = v+ implies lim D(,p(r), L) _ + -,

r-.-m

lira D(,p(r), L) = 0.

r-.+m

(c) u 0 v- and u+ 0 v+ implies lim D(,p(r), L) =+oo = lim D(V(r), L).

r-'- m

Let (p and ¢ be geodesics parameterized by arclength in 9Tt with different ranges. Then and ¢ are said to be { negatively } { positively } asymptotic provided there exists so E (R such that {lim,-_. D((p(s), &(s + so)) = 0} {lim,-+ D(,p(s),

4,(s + so)) = 01, or equivalently, provided there exists s1 E R such that {lim,._ D(,p(s + s1), ¢(s)) = 0} {lim,-+m D(.p(s + s1), k(s)) = 01. 13.09. THEOREM. Let rp and ,p be geodesics parameterized by arclength in Mt with different ranges. Then .p and ,l' are {negatively} {positively} asymptotic if and only if rp and 4, have the same { negative } { positive } point at infinity. PROOF. Let { u , u+ } { v , v+ } be the negative, positive points at infinity of {,p} {,'}. Suppose that u- F4- v-. Let d(s) = D(cp(s), rng,/). Then

lira d(s) = + o' . Let so E (R. Then D(,p(s), y'(s + so)) > d(s) and thus lim,._m D(,p(s), ,4(s + so))

+ co. It follows that 'p and 4, are not negatively asymptotic. Similarly, if u+ 5-1 v+ then 'p and ¢ are not positively asymptotic.

Now suppose u = v-. Let H be the horocycle with u as point at infinity and such that So(0) E H. Let V,(so) be the point in which H meets the range of V, and let t be the h-length of the are of H with endpoints ,p(0) and From (6) of 13.07 we have 'D(1p(s), 4'(s + so)) < tea,

s E R.

Thus lima-_0, D(,p(s), ¢(s + so)) = 0 and 'p and 4 are negatively asymptotic. Similarly, if u+ = v+ then 'p and y are positively asymptotic. The proof is completed.

13.10. THE GEODESICS FLOW OF M. Let p E M. A unitangent on 9Tt at p

is a unit contravariant vector at p. The unitangent space on 9Tt at p, denoted 0(m, p) is the set of all unitangents on 9Tt at p.

TOPOLOGICAL DYNAMICS

120

[13.101

The unitangent space on on, denoted X, is U, 3(911, p). Let the transformation µ of X onto 911 be defined as follows. If x E X and x is a unitangent at p E t, then xµ = p. The transformation µ is the projection of X onto W. Let x1 , x2 E X, let pi = x1µ, P2 = x2µ and let &(x1 , x2) denote the absolute value of the angle between x2 and the unitangent obtained by parallel displacement of x1 to p2 along the unique geodesic segment joining pl and p2 . For r E (R+, define a, _ [(x1 , x2) I x1 , x2 E X, h(x1µ , x2µ) + 6(x1 , x2) < r]. Define ti = [a, r E (R]. It is readily verified that `U is a uniformity-base of X. Let 9.1, be the uniformity generated by V. We provide X with this uniformity and I

assign to X the topology induced by `U.. Let G = I M and let g E G. Then g is an isometry of 911 onto 911, g defines a homeomorphism dg of X onto X and the set dG = [dg I g E G] is a homeo-

morphism group of X which is universally transitive, and dG is uniformly equicontinuous relative to 9.1.. If a is an open index of X, there exists a nonvacuous open subset E of X such that for (x, y) E X we have (x, y) E a if and only if x dg, y dg E E for some g e G. Let y be the transformation of X X (R onto X defined as follows. Let x E X

and let s E a. Let be a geodesic parameterized by arclength in 911 such that x = rp(s(,) is the tangent vector to (p at cp(so). Let y = ip(s + s(,). We define (x, s)y = y. Then g _ (X, (R, y) is a transformation group of X which is called the geodesic flow of M.

Let p be a geodesic parameterized by arclength in 91t. The set of tangent vectors to (p at all elements of (R is an orbit under 9 and is denoted by 0,, . 13.11. THEOREM. Let p and 4, be geodesics parameterized by arclength in 91Z with different ranges. Then 0, and O,. are {negatively} {positively} asymptotic if and only if (p and 4, have the same {negative} {positive} point at infinity. PROOF.

The necessity follows from 13.09 and the fact that the projection

µ : X ---> 911 of X onto 911 is uniformly continuous.

To prove the sufficiency, let p and ¢ have the same positive point at infinity u. (Proof of the other case is similar.) We may suppose there exists a horocycle H(0) with u as point at infinity and such that X0(0), ¢(0) E H(0). Let H(s)

(s (=- (R) be the horocycle with u as point at infinity and such that cp(s), ,'(s) E H(s). Let v be the negative point at infinity of lp. Let a(s) (s E (R) be the hyperbolic transformation whose axis has u, v as endpoints and such that ,p(s)a(s) = cp(O). Then H(s)o(s) = H(0) and the unitangents ip(s) dc(s) _. (p(0), J'(s) du(s) are unitangents at (p(0), l(s)v(s) E H(0) which are internally orthogonal to H(0). Since or(s) is an isometry of 911 onto 911, D(lp(0), ¢(s)Q(s))

h(s) = e8h(0), where h(s) is the arclength of the are of H(s) with endpoints cp(s), ¢(s). The conclusion follows.

It is now clear that if cp and 4i are geodesics parameterized by arclength in it with different ranges such that 0, and 0,, are neither negatively nor positively asymptotic, then there exists a geodesic 0 parameterized by arclength in 911 such that Oe is negatively asymptotic to 0, and positively asymptotic to 0,,, .

GEODESIC FLOWS

113.171

121

13.12. THE HOROCYCLE FLOW. For x E X let cp. be the geodesic parameterized by arclength in on such that x = ;7s(O), let u -(x) be the negative point at

infinity of (p. and let % : (R -* 91Z be the analytic curve parameterized by arclength

in 9IZ such that rng i , is a horocycle with u as point at infinity, such that %(0) = xµ, and such that (jz(0), x) is positively oriented. Define K : X X CR -p X

such that x E X, s E (R implies that (x, s)K is the unitangent at qs(s) which is externally orthogonal to mg % . The continuous flow (X, (R, K) is called the horocycle flow of 31t and denoted by 3C. 13.13. LIMIT SET OF A SUBGROUP OF Z. Let 9 denote a subgroup of Ti. If p, q E M, then p1l 0 U = qS2 (1 U. The limit set of S2, denoted A(S2), is the set pSt (1 U where p E M. Clearly A(Q) is closed and invariant under Q.

13.14. LEMMA. Let E be a finite subset of U, let u E A(S2) and let A be an arc of U with midpoint u. Then there exists w E 0 such that Ew n (U - A) consists of at most one point. PROOF. Let (3 be the least angle formed at the origin 0 by pairs of h-rays with initial point 0 and with points at infinity distinct points of E. Since E is finite, 6 > 0. Let p E M and let a(p) be the angle subtended by A at p by h-rays with initial point p and points at infinity the endpoints of A. Then lima- a(p) = 2a. There exists a sequence w1 , w2 , of elements of 0 such that lim, Ow = u. Let N be a positive integer such that a(OWN) > 27r - 0.

Since w is conformal, (U - A)wN1 can contain at most one point of E. It follows

that EWN n (U - A) consists of at most one point. The proof is completed. 13.15. THEOREM. If crd A(S2) is finite, then crd A(S2) = 0, 1 or 2. If crd A(0) is not finite, then A(S2) is self-dense and either A(S2) = U or A(S2) is nowhere dense on U.

PROOF. We suppose that crd A(12) > 2. Let a, b, c be different points of A(U). Let u E A(S2) and let A be an are of U with midpoint u. Since w E S2 implies that aw, bw, cw E A(2), it follows from 13.14 that there exist at least two points of A(S2) in A. Thus A(S2) is self-dense.

To complete the proof it is sufficient to show that if A(S2) contains an are of U, then A(S2) = U. Assuming that the are A C A(2), we let a be its midpoint

and let S E R. There exists co E 9 such that the angle subtended by A at Ow by hyperbolic rays exceeds 2a - S. But then the angle subtended by at 0 by hyperbolic rays also exceeds 27r - S. Since Aw-1 C A(12) and S can be chosen arbitrarily small, we infer that A(S2) = U. The proof is completed. Aw-1

13.16. THEOREM. PROOF.

Let u E A(12) and let crd uSt > 2. Then uSt = A(U).

Use 13.14.

13.17. THEOREM.

Let crd A(12) > 2. Then 0 is transitive on A(S2).

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[13.171

PROOF. Let a, b E A(Q) and let A, B be open arcs of U with midpoints a, b. By 13.15, A(ct) is self-dense, and thus there exist different points u, v such that u, v E A(U) n A. According to 13.14 there exists w E 0 such that either uw E B or vw E B. Since uw, vw E A(SZ), the proof of the theorem is completed.

13.18. THEOREM.

Let crd A(S7) > 2. Then there exists at most one point of

A(Q) which is not transitive under ct. PROOF. Let u, v E A(Q) and suppose that neither u nor v is transitive under Q. It follows from 13.16 that crd uI = 1 = crd vct, or, equivalently, that utl = u, va = v, and thus, if L denotes the h-line with points at infinity u, v, then Lw = L

for every w E Q. Let p E L. It is evident that U n pcl can contain at most the two points u and v, and thus crd A(Q) <- 2, contrary to hypothesis. The proof is completed.

13.19. REMARK. Let Q E E and let there exist a closed interval A of U such that Ao- C int A. Then v has an axis with the positive fixed point of or in int A

and the negative fixed point of a in int (U - A). 13.20. THEOREM. PROOF.

Let crd A(Q) > 2. Then some element of ct has an axis.

Suppose that no element of cl has an axis. Let A be a closed interval

of U such that A z U and let A* = U - int A. It follows from 13.19 that if w E ct, then A*w (1 A 0 0. We show that if u E A(11) and B is a closed interval of U such that u B, then there exists w E 0 such that Bw n B = 0. Let B be such an interval and let C be a closed interval of U such that u E int C and B n C = 0. Let D be a closed interval of U such that C C int D and D (1 B = 0. Let a be a positive number less than either of the two angles subtended by h-rays at the origin 0 by the two intervals which constitute D - C. There exists E 12 such that the angle which C' = U - C subtends by h-rays at Or is less than a. But then the angle which C'r ' subtends by h-rays at 0 is less than a. Let C* = U - int C. Then the angle which C*1 ' subtends by h-rays at 0 is also less than a and since C C C CD Bw

n B = 0.

Now let u and v be distinct points of A(Sl). Let A, B be open disjoint intervals of A(Q) containing u, v respectively. It has been shown that there exists w, E ( such that A'w, C A and w2 C SZ such that B'wz C B. But then A'wlwz C B C int A', contrary to the assumption that no element of 0 has an axis. The proof is completed. 13.21. LEMMA.

Let crd A(ct) > 2. Then there exist infinitely many distinct

axes of transformations of Sl and the set of endpoints of these axes is dense in A(Sl). PROOF. It follows from 13.20 that there exists w E SZ such that co has an axis. Let L be the axis of w and let u, v be the endpoints of L. Let a E U. Then La is an axis of v 'wa, which is an element of St, and the endpoints of Lo are uo, vv. Now use 13.18.

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[13.24]

13.22. DEFINITION.

The subgroup 2 of

123

is mobile provided that no point

of U is fixed under all elements of R. 13.23. LEMMA. Let 0 be a mobile subgroup of E and let crd A(S2) > 2. Then there exists a pair of axes of transformations of a such that these axes have no endpoints in common. PROOF.

Let L, with endpoints a and b, be an axis of co E Q. Not all of the

infinitely many distinct axes of transformations of 2 have a as common endpoint.

For if this were the case, since some element a E 0 moves a, there would be infinitely many distinct axes with av as endpoint, of which some one would not have a as endpoint. Thus there exists an axis L, of w, E 12 which does not have a as endpoint. Similarly, there exists an axis L2 of W2 E 52 which does not have b as endpoint.

If the statement of the lemma is not true, L, must have b as one of its endpoints, L2 must have a as one of its endpoints, and L, and L2 must have a common endpoint c. But co, moves L2 into an axis which has no common endpoint with L. The proof of the lemma is completed. Let &2 be a mobile subgroup of E with crd A(S2) > 2. Then (1) A(S2) is minimal under Q. 13.24. THEOREM.

(2) If A and B are open arcs of U such that A Cl A(R) Fd 0 0 B Cl A(S2), then there exists co E SZ such that co has an axis L with endpoints a, b such that

aEAandbEB.

(3) If 4 is the set of all geodesics

PROOF.

(1) Use 13.16:

(2) We can assume that A Cl B = 0. Let u E A Cl A(12) and let v E B n A(12). It follows from 13.23 that there exist co, , co, E S2 such that co, has an axis L, , co,

has axis L2 , and L, and L2 have no endpoint in common. Since, if a- E 0 has axis L and w E 2, then Lw is the axis of w-lvw, we can assume, in view of (1), that one endpoint of L, is in A. But then some power of co, transforms L. into an axis L. of w, such that both endpoints of L. are in A. Similarly there exists wb E S2 with axis Lb such that both endpoints of Lb are in B.

Let A' = U - A and let B' = U - B. There exists an integer n such that A'wa C A, A'wa C A, B'wn C B andB'wb " C B. Define co = wbwb. Then Aw = Awbwa C Bwa C A and Bw-1 = Bwa nC Awb" C B. It follows that co is a transformation with the desired properties. (3) Use (2).

(4) Let A, B, C, D be open arcs of U such that each intersects A(12). Choose COI E 9 such that w, has axis L, with endpoints a E A and d E D. There exists an integer n such that F = (B (1 A(SU))w, Cl D - 0. Choose w2 E 0 such that

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[13.24]

w2 has axis L2 with endpoints f E F and c E C. There exists an integer m such 0. that (A (1 A(S2))wiw2 ( \ C 5;6 0. Then also (B rl A(S2))wiw2 h D The proof is completed. 13.25. GEODESIC PARTITION FLOWS OF W. Let 12 be a subgroup of E and let G = S2 I M. Then G is a group of isometrics of W. We observe that dG is a uni-

formly continuous homeomorphism group of X and 9 is dG motion preserving. Define the partition XG = [x dG I x E X] and let XG be provided with its partition uniformity as defined in 2.34. We observe that the partition XG is starindexed and thus, by 2.36, the projection of X onto XG is uniformly continuous and uniformly open. Let 9° = [X0 , R, ya], called the geodesic partition flow induced by G, be the partition flow on X G induced by 9. For x E X let (Pz be

the unique geodesic parameterized by arclength in M such that x(0) = x. We remark that if x E p E XG , then p is periodic under 9 if and only if mg (p= is the axis of some member of G. 13.26. DEFINITION. A subgroup 0 of E is {limit-partial} [ limit-entire } provided that {crd A(S2) = X and A(S2) - U} {A(S2) = U}. We observe that if 12 is limit-partial, then A(12) is a Cantor discontinuum. 13.27. THEOREM.

Let 2 be a limit-entire, mobile subgroup of E and let

G=12IM.Then (1) The set of all 90-periodic points of XG is dense in XG (2) 90 is regionally transitive. PROOF.

Use 13.24.

COROLLARY. Let a be a limit-entire, mobile subgroup of E and let G = E2 I M. Then there exists a point of XG which is transitive under the geodesic partition flow 9o. PROOF. It is easily proved that the space X of unitangents on f1Z is a second-

countable, locally compact, Hausdorff space, and it follows that XG is also second-countable. Let 0 be the projection of X onto XG . Let x E X, let A be a compact neighborhood of x in X and let { 63n I n (E 9+ } be a base for the open

sets in Xo . Since the geodesic partition 9 is transitive there exists x1 E int A such that the orbit of x10 under $ meets B. There exists a compact neighborhood

N1 of x1 in X with N1 C A and such that y E N1 implies that the orbit of yO under 9° meets 631 . There exists x2 E int N1 such that the orbit of x20 under $o meets 632 and thus there exists a compact neighborhood N2 of x2 in X with N2 C N1 and such that y E N2 implies that the orbit of yO under $o meets 632 . Proceeding inductively, we define a sequence N1 , N2

,

of compact neighbor-

and y E N implies that the orbit of y0 under $ meets 63 . But then n =1 Ni 3-` 0 and x* E n:=1 Ni implies that hoods in X such that N1 D N2 D

x*0 is a transitive point of XG under $ The proof is completed. 13.28. DEFINITION.

Let (X, T, 7r) be a transformation group and let T = 6t

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[13.331

125

or t. The point x E X is

{ positively } { negatively } transient under T provided that, C being any compact subset of X, there exists to E T such that It > to }

{ t < t,,) implies xt ($ C. The orbit 4) under T is { positively } { negatively } transient

provided there exists x E 4) such that x is { positively } { negatively } transient. 13.29. THEOREM. Let 0 be a limit partial subgroup of E and let G = 0 1 M. Let (p be a geodesic parameterized by arclength in f such that the {negative } {positive } point at infinity of cp does not belong to A(&1). Let 0 be the projection of X onto XG . Then OO B is a {negatively) {positively) transient orbit under 9G . PROOF. Let a be a compact subset of XG . There exists a compact subset A of X such that A 0 D a. Let the { negative } { positive } point at infinity of p be u and suppose u ($ A(U). There exists a neighborhood V of u in g such that V () U,EG Ag = 0. There exists ro E R such that r E R with {r < r(,} Jr > ro} implies cp(r) E V. The conclusion follows.

13.30. THEOREM.

Let 12 be a limit-partial, mobile subgroup of

and let

G = 0 1 M. Let X* _ Uweo 0, where 4) is the set of all geodesics parameterized by arclength in Jtt such that their points at infinity belong to A(S2). Let X*G be the trace of XG on X*. Let 9*G be the restriction to Xo of 9G . Then: (1) The set of all 9a-periodic points of Xa is dense in X*G

(2) 9 is transitive. PROOF.

Use 13.24.

13.31. HOROCYCLE PARTITION FLOWS.

Let 52 be a subgroup of E+ and let

G = S2 I M. We observe that dG is a uniformly equicontinuous homeomorphism

group of X and the horocycle flow 3C is dG-orbit preserving. Define the partition XG = [x dG I x E X] of X and let XG be provided with its partition uniformity. Let 3CG = (XG , (R, KG), called the horocycle partition flow induced by G, be the partition flow on XG induced by 3C. 13.32. DEFINITION.

Let r E CR. Let x E X and let vx E 2 be the elliptic

transformation such that (xµ)v= = xµ and such that do-, rotates x through the angle r in the positive sense. The transformation (x do-, I x (E X) is an analytic homeomorphism of X onto X which we call a rotor and denote by p'.

Let ri : at --* )1t be an analytic curve parameterized by arclength in M such that mg q is a horocycle. Let f (s) = h('1(0), q(s)) (s E R). Let s E R and let (p be a geodesic parameterized by arclength in ) such that ,1(0) = p(O) and q(s) = p(r) for some r E W. Let a(s) (s E (R+) be the smallest positive angle from (r) to the unitangent externally normal to rng , at n(s). Let a(s) = -a(s) (s E (R-). Then the function a : s -> a(s) (s E Gt, s -- 0) is analytic 13.33. REMARK.

with lim,,,-m a(s) = 0, the function f : s -* f (s) (s E (R) is analytic with lim i .I-. f (s) _ + , and s E cR with s 0 0 implies p*pa(a)yf(s)pa(s) = B

(7) where { y' } {)c' }

is the s-transition of the { geodesic } { horocycle } flow in X.

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[13.341

Let S2 be a limit-entire, mobile subgroup of E+ and let

13.34. THEOREM.

G = 12 1 M. Then the horocycle partition flow induced by G is transitive. PROOF. Let 0 be the projection of X onto XG . Let (t and (B be open subsets of XG and let A = (tB-', B = (BB-'. There exists an open set C C Ap", an open set D C B with D saturated with respect to dG, and 5 E (R+ such that r E (R with I r I < S implies Cpr C Ap' and Dp' C B. By 13.27, there exists an orbit under 9G which intersects both CO and DO. Also by 13.27 the 9G periodic points of XG are dense in XG and thus there exists a 9G periodic point whose orbit

intersects both CO and DO. It follows that there exist arbitrarily large real numbers t such that Cy` n D P-4 0 and thus (cf. 13.33) we can choose s E (R 0. But then AK8 = Ap" pa(8) y r c 8)p a(8) D such that I a(s) I < S and Cyf(,' n D (Cyn') n D)p'"'> Cyr(8)pac8) and thus AK8 n B D Cyr(8)p'(') n 0, from which the conclusion follows. COROLLARY. Let S2 be a limit-entire, mobile subgroup of E+ and let G = 0 M. Then there exists a point of XG which is transitive under the horocycle partition flow.

PROOF.

The proof is similar to that of Corollary 13.27.

13.35. REMARK. Let Sl be a subgroup of E+ and let G = S2 I M. Let 0 be the projection of X onto X G and let {y'} { K' j { yG } { KG } be the s-transition of the {geodesic flow of )1t} {horocyele flow of l t} {geodesic partition flow in XG induced by G) { horocycle partition flow in X G induced by G I. Let x E X, let xO = xG a KG``, = xGK`y8 XK`ys and xGyG and let t, s E (R. Then xy'K" = GG If xG E X G is {periodic) {transitive} under the horocycle partition flow

3CG and r E CR, then EG7G is { periodic } [transitive) under 3CG. 13.36. LEMMA.

Let 12 be a limit-entire, mobile subgroup of E+ and let G = Sl M.

be the t-transition of the horocycle partition flow in XG induced by G. Let a C XG be the orbit of a periodic point of the geodesic partition flow aG and let

Let KG'

B = U,, 6, aKa . Then 63 = XG . PROOF. Let x AB-'. Let (p be the geodesic parameterized by arclength in M such that x = (O). The range of (p is the axis of some element of G and let u be the negative point at infinity of (p. Let B be the set of unitangents of X which are externally orthogonal to the horocycles with u as point at infinity. Then (B = BO. Since, by 13.24(1), uSt = U, the set B dG is dense in X and since (B = BO = (B dG)O it follows that (B is dense in XG . The proof is completed. 13.37. THEOREM.

Let S2 be a limit-entire, mobile subgroup of E+ and let

G = SZ I M. Let a E XG and let a be a periodic point of the geodesic partition flow Then a is a transitive point of the horocycle partition flow 3CG .

c7G .

PROOF. It follows from 13.34(Corollary) that we can choose b E XG such that b is transitive under the horocycle partition flow 3CG . Then (cf. 13.35)

s E (R implies that by' is transitive under 3CG . Let p E (R+ be such that aya = a and let 3) be an open subset of XG . There exists r E (R+ such that s E (R with

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[13.391

127

-2p -<- s =< 2p implies [byaK, I t E 61, t -< r] ( \ D 5-!5 0. There exists a neigh1

borhood 9 of b such that f E ff and s E R with -2p < s < 2p implies [f1' KG I t E (R, I t <- r] n 5)

and r2

0. According to 13.36 there exists r, E R with 0 <= r,

<--_ p

R such that ay'G Ka E Y. Thus there exists r, E R such that

ayGKpyG"KG (E 3) and consequently (cf. 13.07(6)) rsa'*i+ra

aKG

E C .

The conclusion of the theorem follows. 13.38. LEMMA. Let S2 be a limit-entire, mobile subgroup of E and let G = S2 I M. Let 12+ = SZ n V and let G+ = 12+ 1 M. Then S2+ is a limit-entire, mobile subgroup

of E and the geodesic partition flow 9G induced by G is regionally mixing if the geodesic partition flow !Ra+ induced by G+ is regionally mixing. PROOF. It is clear that S2+ is a subgroup of 2. If Q E 12, then v2 E 12+. It follows from 13.24 that if A and B are open arcs of U, there exists w E SZ such

that w has axis L with endpoints a, b such that a E A and b E B. But then w2 E 12+ and w2 has L as axis. We infer that S2+ is limit-entire and mobile.

Let 8 be the projection of X onto XG , let a, R be open sets in XG and let A = a8-', B- = (A8'. Let 8+ be the projection of X onto XG+ , let (t+ = (dO-1)8+ and let (B+ = ((38-1)8+. We assume that the geodesic partition flow ,a+ induced by G+ is regionally mixing and thus there exists s E R+ such that t E R, I t I > s implies d+yG+ n 63+ 0 0, where -Y'(;+ denotes the t-transition of the geodesic partition flow !g++ . But then t E (R., t I > s, implies ((t8-1)y` n ((P8-') 0 0, whence ay`G n (B 34 0. The proof is completed. I

13.39. THEOREM.

Let 12 be a limit-entire, mobile subgroup of 7, and let

G = 12 I M. Then the geodesic partition flow 9G induced by G is regionally mixing. PROOF.

Let SZ be a limit-entire, mobile subgroup of E and let G = SZ 1 M.

In view of 13.38 we can assume that 0 C V. Let a and (B be open subsets of XG . It follows from 13.27(1) that we can choose a E d and p E R+ such that ay'G = a. Let r E R be such that 0 =< r <--_ p.

From 13.37 we infer that ay'G is transitive under 3CG and thus there exists t(r) E (R such that ay'GKGc r E (B. We can choose S(r) E 6t with S(r) > 0 such

thatsEawithIs - rI <= S(r)implies [ayaKGItE(R,ItI < I t(r) 1](1(B o 0. A finite number of the intervals [s 1 s E 6t, I s - r I < S(r)] (r (E R) cover the interval [r I r E R, 0 <= r p] and thus there exists t. E 6t+ such that s E R implies [ay°KG I t E 61, t <- tj n 61 P6 0. Since a is open, there exists a E R+ such that [aKG t E R, t <- S] C a. Let so E R+ be such that Se' ° > to and let s E (R with s > so . Then I

1

I

1

63naya TO

[a7lc

Sly'G) _

ItEa,It1<51D

63n[ayaKGItER,It1
0.

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[13.39]

It remains to prove that there exists s, E at such that t < - s, , t E 61, implies a'Ya (\ 63 0 0. Let 0 be the projection of X onto XG and let A = aO-', B = 638-1.

The sets Ap' and Bp' are open and it follows from the first part of the proof

that there exists sl E 61' such that s > Si implies (Ap')y' (1 Bp' 0 0. But s E 61 and x E XG implies xp'y'p' = xy '. Thus s > s, implies Ay-' (1 B =

(Ap'y'p') (1 B = (Ap'y' n Bp')p' 0 0, or, equivalently, t E at, t < -sl implies Ay` n B 0 0, and thus (tya ( \ 63 0 0. The proof is completed. 13.40. THEOREM.

Let St be a limit-entire, mobile subgroup of 2;' and let

G = SZ 1 M. Then the horocycle partition flow induced by G is regionally mixing. PROOF. Let 0 be the projection of X onto XG . Let a and 63 be open subsets of XG and let A = dB-`, B = 639-1. There exists an open set C C Ap', an open set D C B with D saturated with respect to dG, and S E 6t with S > 0, such that r E 61 with I r I < S implies Cp' C Ap' and Dp' C B. By 13.39 the geodesic

partition flow ga is regionally mixing and we can choose so E 6t such that 1 81 > so implies I a(s) I < S and C'/" (1 D ; 0 (cf. 13.33). But then 1 s 1 > so implies Ap'pa(8)yfc8)p8(8) (1 Dpa 8) D (C'/ n D)pa(8> 0 0, from which AK' (1 B D

the conclusion of the theorem follows. 13.41. COMPLETE TWO-DIMENSIONAL RIEMANNIAN MANIFOLDS OF CONSTANT

Let 91 be a complete two-dimensional analytic Riemannian manifold of constant negative curvature -1. There exists (cf. H. Hopf [1]) a group G of isometries of AIL onto 1 such that nl is the universal covering manifold of 9'G with G the covering group. The two-dimensional Riemannian manifold MG obtained by partitioning 9 by G is isometric to R. NEGATIVE CURVATURE.

The group G has the property that p E ) implies the existence of a neighborhood W of p such that pg E W, g E G, only if g is the identity mapping. A group of isometrics of 1 with this property will be said to be discrete in M. Let F be a group of isometrics of M onto 91Z which is discrete in M. The two-dimensional Riemannian manifold obtained by partitioning JCL by F is then a complete two-dimensional analytic Riemannian manifold of constant negative curvature -1. Thus the problem of constructing the class of such manifolds is equivalent to the problem of constructing the class of groups of isometrics of 9tt which are discrete in M. 13.42. LEMMA.

Let SZ be a subgroup of

with crd A(12) > 2 and let G = 12 1 M

be discrete in M. Then Sl is mobile.

PROOF. We assume that 12 is not mobile and thus that there exists u E U such that uSl = u. Let wo E 9 be the identity mapping. Since G is discrete in f, w E 0, co 0 wo , implies that co is either parabolic, hyperbolic or a paddle motion. If all co E 0 other than co. were parabolic, the set A(11) would consist of [u], contrary to hypothesis. Thus there exists co, E Sl with axis Ll and L, must have u as one of its endpoints. If L, were the axis of every member of St other than co. ,

it would follow that A(12) = 2, which is not the case. Let co E 0 such that L,

[13.44]

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129

is not the axis of w. If w does not have an axis, then w is parabolic and Llw is an axis of w-'wiw E 0 with Llw 0 L, . Thus there exists w2 E Sl with axis L2 such that L, 0 L2. Since u12 = u, Ll and L2 must have u as common endpoint.

Let p1 E L, , let p2 E L2 and let S E W. There exists m E g such that 0 < D(plwi , L2) < S and thus there exists n E 9 such that the h-distance from p,w;w2 to some point of the h-line segment joining p2 and p2(02 is positive and less than S. Since a can be chosen arbitrarily small there exists a sequence {wk I k E 9', wk E Sl} such that p,w? 0 p,w* , i F& j, and such that the sequence {p,w,k* I k E g+} converges to a point p E L2 . But this implies that G is not discrete in Mt. The proof of the lemma is completed. 13.43. THEOREM. Let 12 be a subgroup of M, let G = 12 I M and let G be discrete in M. Let wo be the identity mapping of G. Then exactly one of the following statements is valid. (1) crd A(0) = 0;12 = [wo]. (2) crd A(12) = 1; there exists w E 12 such that w is parabolic with fixed point A(12) and 12 = { w" I n E .41. (3) crd A(12) = 2; there exists w E 12 such that w is hyperbolic with fixed points

A(12)and12 = {w"InE9}. (4) crd A(12) = 2; there exists w E 9 such that co is a paddle motion with fixed points A(12) and Sl = { co" I n E 4 1.

(5) crd A(12) > 2;12 is mobile and limit-partial. (6) crd A(12) > 2; Sl is mobile and limit-entire. PROOF. Since G is discrete in 91l, co E 12 with co F& coo implies that w is parabolic, hyperbolic or a paddle motion. Thus if A(12) = 0, it follows that 12 = [wo]. Suppose crd A(12) = 1. Let u = A(12). Then w E SZ with w 0 wo implies that

co is parabolic with fixed point u. Let p E 59 and let H be the horocycle with point at infinity u and such that p E H. Then p1l C H and since G is discrete in ni, there exists p, E $2 such that p' 0 p and h(p, , p) _<_ h(q, p) for all q E p12, q 0 p. Let p, = pw with co (E 12. Then w is parabolic with fixed point u and

12= {w"InE9}.

Suppose crd A(S2) = 2. Let A(12) = [u, v]. Since A(12) is invariant under 12 and G is discrete in 91Z, co E 12 implies that uw = u and vw = v. Let L be the h-line with points at infinity u and v and let p E L. Since G is discrete in MZ, there exists pi E p11, p 0 p, p E L such that h(p, , p) < h(q, p) for all q E p12, q 0 p. Let p, = pw with w E Q. Then 12 = { w" I n E 9) and w is either a hyperbolic transformation or a paddle motion with u and v as fixed points in either case.

If crd A(12) > 2, it follows from 13.42 that 12 is mobile. It follows from 13.15 that all possible cases have been considered. The proof is completed. 13.44. REmARc. Let 12 be a subgroup of X, let G = 12 I M and let G be discrete in M. Then G is countable and if p E M, co, , w2 E 0, pwl = pw2 then co, = w2 .

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[13.451

13.45. DEFINITION. Let 12 be a subgroup of Z, let G = 12 1 M and let G be ], where discrete in M. Let po be the origin 0 and let p0G = [p; I i = 0, 1, 2, h(p, p.)]. p; 0 pi provided i 0 j. For n E 9+ define R. = [p I p E t, h(p, The set R= (fin ,R is the fundamental region of G.

Let 0 be a subgroup of E, let G = 0 1 M, let G be discrete in 91t and let R be the fundamental region of G. Then: (1) R is a non-vacuous open subset of M. (2) R is h-convex in the sense that p, q E R implies that the h-line segment joining p and q lies in R. (3) If g, , g2 E G and Rg, n Rg2 0 0 then g, = 92 . (4) Corresponding to any compact subset A of on there exists a finite subset E of G such that A_C U,,E Rg. (5) a,,(R) = ah(R) 13.46. REMARK.

13.47. THEOREM.

Let 12 be a subgroup of Z, let G = 12 1 M, let G be discrete

in n and let R be the fundamental region of G. If R is of finite h-area, then crd A(0) > 2 and 0 is mobile and limit-entire. PROOF. It follows from 13.43 that if crd A(0) 5 2 then R is not of finite h-area. Thus we can assume that crd A(0) > 2 and by 13.42 12 is mobile. We show that 12 is limit-entire. Let 5 E W. There exists r E 61, 0 < r < 1, such that if C, denotes the circle with center 0 and euclidean radius r, then any h-convex subset of 59 which is exterior to C, has euclidean diameter less than 5. There exists a finite subset E of G such that C, and its interior are contained in U,EE Rg. Let u E U and let W be a euclidean neighborhood of u of diameter less than 5 and exterior to C, . Then W (\ f is of infinite h-area and there must exist g* E G, g* ($ E, such that Rg* n W 0 0. But then Rg* is exterior to C, , the euclidean diameter of Rg* is less than 5 and Og* is within euclidean distance 25 of u. It follows that u E A(0) and thus U C A(12) and 12 is limit-entire.

13.48. GEODESIC FLOWS OF TWO-DIMENSIONAL MANIFOLDS OF CONSTANT

Let t be a complete two-dimensional analytic Riemannian manifold of constant curvature -1. Let p E 91. A unitangent on 9L at p is a unit contravariant vector at p. The unitangent space on t at p, denoted 5(9l, p) is the set of all unitangents on Yt at p. The unitangent space on 91, denoted X, is U,Ea 3(gt, p). Let x E X and let NEGATIVE CURVATURE.

x be a unitangent at p E X. Let r E 6t+ and let A, (x) be the set of all unitangents

on 9Z at p and forming an angle less than r with x. Let U,(x) be the set of all unitangents obtained from A, by parallel transport along all geodesic segments of length less than or equal to r and with initial point p. For r E 6t+ define a, = [(x, , x2) I x, , x2 E X, x2 E U.(xl)] Define `U = [a, I r E 6t+]. It is readily verified that U is a uniformity base. Let 9t be the uniformity generated by `U. We provide X with this uniformity and assign to X the topology induced by 91. Let y be the transformation of X X 6t onto X defined as follows. Let x E X

GEODESIC FLOWS

[13.52]

131

and let s E a. Let p be a geodesic parameterized by arclength in 91 such that x = (so) is the tangent vector to ip at p(so). Let y = cp(s + s(,). We define (x, s)y = y. Then G = (X, (R, -y) is a transformation group on X which is called the geodesic flow of 91. 13.49. THEOREM. Let 91 be a complete two-dimensional analytic Riemannian manifold of constant curvature -1 and of finite area. Then the geodesic flow of

91 is regionally transitive, regionally mixing, and the periodic orbits of the geodesic flow of 9Z are dense in the space of unitangents on 9Z. PROOF.

Use 13.27, 13.39 and 13.47.

13.50. CONSTRUCTION OF TWO-DIMENSIONAL MANIFOLDS OF CONSTANT NEGA-

As indicated in 13.41, the problem of construction of twodimensional Riemannian manifolds of constant negative curvature -1 is equivalent to the problem of construction of groups of isometries of M1 which are discrete in M. The problem can be completely solved by geometric methods TIVE CURVATURE.

involving the construction of fundamental regions (cf. Fricke-Klein [1], Koebe [1] and Lobell [1]). These manifolds include compact orientable manifolds of genus

at least 2 and compact non-orientable of every topological type other than the projective plane and Klein bottle. 13.51. n-DIMENSIONAL MANIFOLDS OF CONSTANT NEGATIVE

CURVATURE,

n > 2. A large number of the results of this section can be extended to manifolds of constant negative curvature of dimension exceeding 2 and the proofs have been so designed that these extensions obtain with scarcely any modifications of the proofs given for the case of dimension 2. In particular, the extensions

of the results concerning the density of the periodic geodesics and regional transitivity of the geodesic flow are valid. The extension of the concept of the horocycle flow is not immediately obvious, but mixing properties can be attained for higher dimensional manifolds (cf. E. Hopf [3]). The construction and classification of manifolds of constant negative curvature and dimension exceeding 2 is largely an unsolved problem. Compact manifolds of constant negative curvature and of dimension 3 have been constructed by Lobell [2] and Salenius [1], while non-compact manifolds of finite volume are known to exist, but these examples appear to represent only a small number of the possibilities. 13.52. NOTES AND REFERENCES.

(13.01) This model of the hyperbolic plane is commonly associated with Poincare due to his extensive use of it in the development of the theory of automorphic functions, although it appears to have been known earlier to Beltrami (cf. Beltrami [1]). (13.02) Cf. Bianchi [1], p. 584. (13.11) Cf. E. Hopf [3], p. 268.

(13.13) It is usually assumed in the definition and analysis of limit sets that G = 0 1 M is properly discontinuous (cf. L. R. Ford [1]).

132

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[13.52]

(13.27, 13.34, 13.39) Cf. Hedlund [2] for references to the literature. (13.41) The expression discrete in M replaces the more commonly used expression properly discontinuous. The phrase discrete and without fixed points in T would, perhaps, be more appropriate. A group of isometries of can be topologized in various ways. It can be considered as a set of mappings of t

onto Yl and assigned the compact-open topology. It can be considered as a set of mappings of X onto X and assigned the compact-open topology. It can be considered as the restriction of a subgroup of 7, and thus be topologized by defining a base a for the neighborhoods of the identity of E as follows: let e E 6t+,

let

[as +c a,cES;ad -cc= 1;1a-11 <e,IcI <eT cz -+d and let (B = UeER+ B(e). Then G is discrete in M, as defined in 13.41, if and only if no element of G other than the identity has a fixed point in 1, and G is discrete in each of the stated topologies.

14. CYLINDER FLOWS AND A PLANAR FLOW 14.01. STANDING NOTATION. Throughout this section Y denotes a topological space, aR denotes the set of real numbers with the natural topology, F(Y) denotes

the set of all continuous functions on Y to at and H(Y) denotes the set of all homeomorphisms of Y onto Y.

Let X = Y X aR and let v denote the projection of X onto aR defined by (y, r)v = r (y E Y, r E (R). The subset A of X is {bounded 14.02. DEFINITION.

above) { bounded below } provided AP is { bounded above) { bounded below}, and

A is bounded provided A is bounded both above and below. 14.03. DEFINITION.

Let 0 E H(Y) and let f E F(Y). Let X = Y X aR

and let So be the homeomorphism of X onto X defined by (y, r)cp = (yO, r + f (y))

(y E Y, r E a). The homeomorphism p will be denoted q(Y, f, 0) and called the cylinder homeomorphism determined by Y, f and 0. 14.04. DEFINITION. Let X = Y X aR, let s E aR and let X -* X be defined by (y, r)0. = (y, r + s) (y E Y, r E (R). The homeomorphismk. of

X onto X will be called the translation of X by s.

14.05. REMARK. Let X = Y X aR, let s E aR and let ¢, be the translation of X by s. Let f E F(Y), let 0 E H(Y) and let (p = po(Y, F, 0). Let x E X, let fi(x) be the orbit of x under c and let r(x) = c1(x). Then (1) (2) (3)

x4,.,P.

'(x)4,. _ (k(x ¢.) .

r(x)V,. = r(x4,.)

14.06. DEFINITION. Let f E F(Y), let 0 E H(Y) and let (p = cp(Y, f, 0). Let { A- } { A+ } { SZ- } { Sl+ } denote the set of a1 lx E X = Y X aR such that {the

negative semiorbit of x under (p is bounded below) {the negative semiorbit of x under o is bounded above } { the positive semiorbit of x under 'p is bounded below the positive semiorbit of x under (p is bounded above 1. Let { a } { a+ } { w } { w+ } denote the set of all x E X such that

{ lim xcpnv = - co } { lim WP = + - I I lim xco v = - c } { lim x5onv n-.-m

n-.-m

-++m

Let B+ = A+ (\

n- +m

W. Let B- = A- (1 Q. The collection of sets [A-, a, a+, w, w+, A-', A+', fl-', St+', B+, B-] will be denoted by D(,p).

=+ A+,

}

.

Q-, SZ+,

14.07. REMARK. Let 0 E H(Y), let Y be a minimal orbit-closure under 0, let f E F(Y) and let e = cp(Y, f, 0). Let s E aR and let 4,, be the translation of X by s. Let D E D (cp) . Then D is invariant under p, D is invariant under and if D 76 0 then D is dense in X. 133

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134

14.08. LEMMA.

[14.081

Let Y be a compact metric space, let 0 E H(Y), let f E F(Y),

let (p = (p(Y, f, 0) and let v be the projection of X = Y X (R onto (R. For each i E 9+ let there exist x; E X and n; E 9+ such that { x,(p''v - x,v > i } { x,(P "v - x, v < Then (1)

{A+ 0 0} {A- 0 0}.

(2)

{12-00} {12'00}.

PROOF. It is sufficient to prove the first reading. We assume that corresponding to i E 9+ there exists x; E X and n, E 9+

such that x,(p"v - x,v > i. Corresponding to i E 9+ let p, , q, E 9+ be so chosen (x,p'v - x,(p"v). that 0 <- p, < q, _< n; and x,(pQ'v - x,(p"'v = Then x,p"`v - x,(p"v > i and k E 4, p, _5 k <- q, implies x,(p?'v < x,(pkv xi(po'v. Since Y is compact and f is continuous on Y there exists L E (R+ such that y E Y implies I yf 1 <- L and thus x E X implies I x(pv - xv 1 5 L. It follows that i E 4+ implies x,(p"v - x,(p"'v < (q, - p,)L and thus q, - p, > i/L, whence

lim,-.+W (q; - p;) = +-.

i E 9+. Then x*v = 0 (i E 9+) and - s, = xgoQ1+kv - x,(pQ'v" 0, k = 0, - 1, ... p p, - q, . Since Y is compact, the sequence x* , x2 , - - contains a subsequence Let s; = x,(pQ'v and let x* = x,(pa.+kv

x*(pkv =

converging to a point x* E X and n E g- implies x*(p'v < 0. Thus x* E A+ and (1) is proved. Let t; = x,(pD'v and let x; = x,(p9i '_,, , i E 9+. Then x;v = 0 (i E 9+) and .+kY kV = x, - t = x. v.+kv - x. v.V 0, k = 0, 1, ... xi kV = x, v. q, - p, . The sequence x; , x2 , contains a subsequence converging to a point x' E X and n E 9+ implies x'(p'v > 0. Thus x' and (2) is proved. 14.09. LEMMA.

Let Y be a compact metric space, let 0 E H(Y), let f E F(Y),

let (p = (p(Y, f, 0) and let v be the projection of X = Y X (R onto (R. For each i E 9+

let there exist x; E X and m; , n; E 9+, with 0 < m; < n; , such that {x,co"v x,v > i} {x,(p"1v - x,v < -i) and {x,(p"1v - x{SO'`V > i} {x,(p"'V - x,(p''v < -i).

Then {B+ 0 0}{B- 0 0}. PROOF.

It is sufficient to prove the first reading.

We assume that for each i E 9+ there exists x, E X and m; , n, E 9+ such

that 0 < m, < n, and x,co"v - x,v > i, x,(p"P - x,(p"v > i. Let p, E 9+ be such that 0 < p, <--_ n, and x,(p"'v = sup09 ,,, x,(p' . Then x,(p"v - x,v > i and x,(p9"v - x,(p"v > i, i E 4+, in consequence of which lim,.+m p, = +

and lim,-+m (n, - p,) = + -. Let s, = x,(p"v, let 4,_ be the translation of X by -s, and let x* = x,(p"','_s,, i E 9+. Then x*v = 0 and x*(pkv = x,(P,,i x,SoD'+kv

- p, + 1,

-

- x,(p?'v, i E 9+, k E 9, and consequently x*(pkv <- 0, k = - - -

, n, - p, . Since Y is compact, the sequence x* , x2 ,

-

-p: ,

contains

a subsequence converging to a point x* E X and n E 9 implies x*(p'v <- 0. Thus x* E B+, and the proof is completed. 14.10. LEMMA.

Let 0 E H(Y), let f E F(Y), let X = Y X (R and let

(p = (p(Y, f, 0). Let x = (y, r) E X, let r(x) be the orbit-closure of x under (p and

CYLINDER FLOWS AND A PLANAR FLOW

[14.111

let there exist s E (R, s 5 0, such that g+]

135

(y, r + s) E r(x). Then [xq; I n E

= [(y, r + ns) I n E g+] C r(x) PROOF. Let 4, E r(x) and let n E g+. Since r(x) is closed and invariant

under cp, it follows that r(4,) C r(x). With the aid of 14.05(3), it follows by induction that r(x+p;) C r(x), whence x>G; E r(x) and [x¢; I n E g+] C r(x). 14.11. THEOREM. Let 0 E H(Y), let Y be a compact minimal orbit-closure under 0, let f E F(Y) and let rp = cp(Y, f, 0). Then the following statements are

equivalent:

(1) There exists x E X such that at least one of the semiorbits of x under cp is bounded.

(2) x E X implies the existence of gt E F(Y) such that the orbit-closure of x under -p is g=

.

(3) There exists g C F(Y) such that f (y) = g(y0) - g(y) (y E Y). ,n=o f (yo,) is bounded on Y X g+. (5) (p is pointwise almost periodic. (4)

PROOF. Assume (1). We prove (2). Let xo E X and let the positive semiorbit of X. under rp be bounded. That is, there exist a, b C (R such that if v denotes

the projection of X onto (R and n E g+, then a < x0So"v < b. Let w(xo) be the w-limiting set of xo under (p. Then w(xo)v is contained in the closed interval (a, b), w(xo) is a compact invariant nonvacuous set and w(xo) contains a minimal set M. Let x* _ (y*, r*) E M. Since M is invariant under gyp, we have [x*0' I n C g+]

[(y*0", r* + En-o f (y*o)) I n E g*] C M and since My C (a, b), it follows that n E g+ implies a < r* + E;=o' f (y*0D) < b. Since Y is a compact minimal orbit-closure under 0, it follows that [y*0' I n E g+] is dense in Y and thus, if R(y) denotes the set [(y, r) I r C at], it follows that y E Y implies M (1 R(y) ; 0. For each y E Y, the set M n R(y) consists of exactly one point. For if (y, r) E

M and (y, r + s) E M, with s 0 0, it follows from 14.10 that [(y, r + ns) I n E 4+] C M, which is not possible, since M is bounded.

Let g : Y - R be defined by (y, g(y)) E M, y E Y. Then g(y) is uniquely defined for each y E Y, g(y) E (R, and g is a bounded function on Y. Since g = M = M = g, it follows that g is continuous and thus g EE F(Y). Let x = (yo , r) E X. Then (y(, , g(yo)) E M, the orbit-closure of (yo , g(yo)) under cp is M, x C is the orbit-closure of x under (p. and Let gs : Y -> (R be defined by gz(y) = r - g(y(,) + g(y), y E Y. Then g= C F(Y) and gs = , which is the orbit-closure of x under 0. Thus (2) is true for the case under consideration. The similar proof that the existence of a bounded negative semiorbit implies (2) will be omitted. Assume (2). We prove (3). Let x E X and let g E F(Y) be the orbit-closure of x under gyp. Then (y0, g(y0)) = (y, g(y)),p = (y0, g(y) + f (y)), y C Y, and thus f (y) = g(yO) - g(y), y E Y, which implies (3).

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[14.11]

Assume (3). We prove (4). Let g E F(Y) exist such that f (y) = g(y0) g(y), y E Y. Let y E Y and let n E 9'. Then n-1

n-1

V-0

.f(ye°) _

D-0

{ g(ye'+')

- g(ye) } = g(yen) - g(y)-

Since Y is compact, g is bounded on Y and there exists b E 61 such that y E Y implies I g(y) I< b. Thus y E Y and n E 9+ implies n-1

E f(yep)

v-0

< I g(yen) - g(y) I < 2b,

which implies (4). Clearly (4) implies that the positive semiorbit of any point x E X is bounded and thus (4) implies (1). Assume (2). We prove (5). Let xo = (yo , ro) E X and let U be a neighborhood of xo . Let go E F(Y) be the orbit-closure of xo under rp. Since go is continuous

on Y, there exists a neighborhood V of yo in Y such that y E V implies (y, go(y)) E U. Since Y is a compact minimal orbit-closure under 0, there exists a syndetic subset A of .4 such that a E A implies yoO" E V. But then a E A implies that x0(p' = (yo0 go(yoe°)) E U and thus x0 is almost periodic under (p, which implies (5). Clearly (5) implies (1).

The proof of the theorem is completed.

14.12. REMARK. Let Y be a compact metric space, let 0 E H(Y) and let Y be an almost periodic minimal orbit-closure under 0. Let y E Y. By 4.47 there exists a unique group structure of Y which makes Y a topological group such that (Y, ay), where ay : g -* Y is defined by iav = y(p', i E 5, is a compactification of g and thus (cf. Halmos [1]) there exists a unique normalized regular Haar measure µ on Y. 14.13. THEOREM. Let Y be a compact, connected, locally connected, metric space, let 0 E H(Y), let Y be an almost periodic minimal orbit-closure under 0, let f E F(Y), let p = co(Y, f, 0) and let A be the normalized Haar measure on Y.

Then the following statements are equivalent: (1) The discrete flow generated by cp is transitive. (2) The discrete flow generated by So is point extensively transitive.

(3-4) There exists x1 E X such that that the {positive} {negative} semiorbit of x1 is not bounded below and there exists x2 E X such that the { positive } { negative } semiorbit of x2 is not bounded above.

(5-6) There exists x E X such that neither semiorbit of x is bounded { below } {above}.

(7) f y f (y) dµ(y) = 0 and E'.---O' f (y0') is not bounded on Y X g.

It follows from 9.23 that (1) and (2) are equivalent. Clearly (2) implies (3), (4), (5) and (6). PROOF.

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[14.131

137

Assume (3). We prove (1). Let x E SZ-'. If x E w , then x E W. If 0. Similarly, R" 0 0 implies

x E S2-' - w , it follows from 14.08 that S2+

Let U and V be open subsets of X = Y X a. Let Y. be an open nonvacuous

subset of Y and I. an open bounded nonvacuous interval of (R such that Y X I C U. Let Y2 be an open nonvacuous subset of Y and I, an open bounded nonvacuous interval of (R such that Y2 X I, C V. Since Y is an almost periodic minimal orbit-closure under 0, there exists an open nonvacuous set Y, C Y2

and a syndetic subset A of 1 such that a E A implies Y,0 C Y. . Thus there exists M E g+ such that n E g implies that for some integer j, 1 <= j 5 M and Y,Bn+'

C V..

Since 12+ 0 0 0 9-, it follows from 14.07 that each of these sets is dense in X. Let x+ = (y+ , r+) E (Y, X I,) n S2+ and let x_ = (y_ , r_) E (Y, X I,) n 12-

such that x+ and x_ lie in a connected subset C of Y, X I, . We observe that x+ E 2-' and x_ E SZ+, for otherwise it would follow from 14.11 that U-1 = 0, contrary to hypothesis. Let v be the projection of X = Y X (R onto at. Since Y is compact, f is bounded on Y and there exists a E (R+ such that x E X implies xv - xcpv I < a. Let

and let c E R with c > b + Ma. Then x E X with xv > c b = sup [r I r E implies xpv > b, i = 1, 2, , M. There exists L E g+ such that n E g+ implies that at least one of the points x_lp"+',

,

x_cp"+L satisfies the condition xv > c. For otherwise there exist

sequences n1 , n2 ,

and L1, L2 ,

of positive integers with lim;-+. L; = + OD

and such that i E g+ implies that x_rp'v < c, j = n; + 1, , n; + L; . Since x_ E S2 there exists d E (R such that n E g+ implies x_(p'v > d. The sequence (X-(P n`+' i (=- g+) contains a subsequence converging to a point xo of X. The positive semiorbit of xa is bounded and it would follow from 14.11 that (3) is not valid, contrary to hypothesis. I

Let e = inf [r I r E I,.]. Since x+ E S2-' there exists t E g+ such that x+(p`+`v < e,

i = 1, 2,

,

L + M. Of the points

at least one,

x_,p`+',

< e, k = j + 1, j + M. There exists an integer p with t + j + 1 < p 5 t + j + M such

x_,p`+', is such that x_(p'+'v > c. Thus x_cp°+kv > b and

x+(P'+kv

that Y,B' C Y. . Consider the connected set Co' C Y X R. It contains x_cp' and x+cp'. Since x_rp'v > b and x+cp'v < e, it follows that Ccp' must meet Y. X Iw

and thus U. We infer that U 1 1 U%E6 Vl' 00, which implies (1). The proof that (4) implies (1) is similar and will be omitted.

Assume (5). We prove (3). Let x E A-' (l 12'. By 14.08, A+ 0 0 0 S2-. Let x* E 12-. Then x* E S2+'. For otherwise it follows from 14.11 that A-1 = 0, contrary to hypothesis. This proves (3). Similarly, (6) implies (4). Assume (6). We prove (7). It is known (cf. Oxtoby [3]) that

lim - E f(y6') = f f(y) dA(y) = lim 1 f(y9-D), -+w n y.0 Y n-+m n y=1

y E Y.

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[14.131

Thus, if f Y f (y) dµ(y) > 0, then w+ = X = a- and if f Y f(y) dµ(y) < 0, then w- = X = at In either case (6) is not valid and we infer that f f (y) dµ(y) = 0. Since x = (y, r) E X and n E J+ implies x,pn = (yO', ED-o f (yB")), it follows from (6) that E (y0P) is not bounded on I' X g+. The proof that (6) implies (7) is completed. Assume (7). We prove (5). Since f y f (y) dµ(y) = 0, it follows that

f

Y

a=0

f(yep)

dµ(y) = 0,

nE

g+,

and thus, corresponding ton E y+ there exists yn E Y such that EpI .f (ynO") = 0. Let xn = (y , 0). Then xv = 0 and xnlpnv = 0. Corresponding to m E 9+ there , n(m), satisfy exists n(m) E y+ such that not all the points xn(,n) ', j = 0, 1, the condition I xv I 5 m. For otherwise there would exist a bounded orbit under

gyp, and from 14.11 it would follow that En o f (yO') is bounded on Y X 0. If B+ 0 and contrary to hypothesis. By 14.09, either B+ 5 0 or Bx E B+, it follows from 14.11 that x E A-' f T' and thus (5) is valid. If B- F-4 0 and x E B-, it follows from 14.11 that x E A+' (l Sl+' and thus (6) is valid. But it has been shown that (6) implies (5), and thus, in either case, (5) is valid. The proof of the theorem is completed. 14.14. EXAMPLES OF CYLINDER FLOWS. It follows from 14.11 that it is easy to construct nontrivial examples of cylinder homeomorphisms which are point-

wise almost periodic. Using the notation of 14.11, we choose g E F(Y) and define f E F(Y) by f (y) = g(yO) - g(y), y E Y. Then (p = cp(Y, f, 0) is pointwise almost periodic.

It is more difficult to construct examples of transitive cylinder flows. The following method yields such examples. Let Y be a compact connected separable abelian (additive) topological group. Then Y is monothetic (cf. Halmos and Samelson [1]). Let y* be a generator of Y;

that is, Y = [ny* I n E .4]-. Let 0 : Y - Y be defined by yO = y + y*, y E

Y.

Then Y is an almost periodic minimal orbit-closure under 0. Let C be the unit circle zz = 1 of the complex plane Z. Let X : Y -p C be a character of Y such that x(y*) = e'' 0 1. Since Y is connected, ,li/a is irrational. Let 0 < nl < n2 < . . . be a sequence of integers such that E.'-, I x(nky*) is convergent. Since 0/7r is irrational, such a sequence 1 I = Ek=, I e'n`# - 1 exists. Let (an I n E 4) be defined by:

-

I

J an = l

0 unless n E [nk I k E I+] or n E [-nk I k E g+],

l an = a_.,. = I

e'nko

-1

I,

k c g+

The series E±m an x(ny) is absolutely and uniformly convergent on Y. Le f : Y --* Z be defined by f (y) = E±.' anx(ny), y E Y. Then f is continuous on Y and since x(-y) = x(y), y E Y, it follows that f (y) E lR, y E Y. By the

CYLINDER FLOWS AND A PLANAR FLOW

[14.18]

139

orthogonality property of characters, n E 5, n 5;6 0, implies f Y X(ny) dµ(y) = 0, where µ is the normalized Haar measure on Y, and thus f y f (y) dµ (y) = 0. Now suppose that there exists g E F(Y) such that

y E Y.

.f(y) = g(y + y*) - g(y),

(A)

Let b = f r g(y)X(ny) dµ(y), n E y. Then bkbk < -. But a simple - 1), n E 4. Thus computation shows that an = bn(e;nfl

"nk =

I

e" - 1

a.,

k E 9+,

1,

We infer that there cannot exist g E F(Y) such that

and hence E±m bkbk (A) is valid.

Let Y be also locally connected and let p = cp(Y, f, 0). It follows from 14.11 that En-o f (y0D) is not bounded on Y X g+ and thus, by 14.13, the discrete flow generated by 'p is transitive. 14.15. REMARK. The dyadic tree is a dendrite whose endpoints form a Cantor

discontinuum and whose branch points are all of order three. There exists a homeomorphism 'p of the dyadic tree X onto X such that: (1) p is regularly almost periodic on X. (2) cp is periodic at every cut point of X. (3) The set of all endpoints of X is a minimal orbit-closure under 'p. 14.16. REMARK. The remainder of this section is devoted to the construction of a compact, connected plane set which is minimal under a homeomorphism and which is locally connected at some points and not locally connected at other points. 14.17. REMARK.

Let f be a continuous real-valued function on a dense

subset of a real interval such that the closure of (the graph of) f in the plane is compact, connected and locally connected. Then f is uniformly continuous. (Since f is compact, it is enough to show that the relation f is single-valued. This may be done by use of the arcwise connectedness theorem.) 14.18. DEFINITION.

Let A. = [2k7r I k E J], let Xo = '31 - Ao and let

fo : Xo -* (R be defined by: fo(x) = sin 7r2

IX

-7r

x G 7r,

x 54 0,

J iII

fo(x+2ir) = fo(x),

xEX0.

For n E 1, n 0 0, let An = [2k7r + n I k E 5], let X. = (R - A. and let

(

fn : Xn -* (R be defined by f n(x)= f .(x - n), x E Xn .

Let X = (R-

UnE9 A. =

I

InES

f(x) = 10 +

Xn and let f : X ---> (R be defined by:

2-'n'fn(x),

x E X.

140

TOPOLOGICAL DYNAMICS

[14.191

14.19. REMARK. We adopt the notation of 14.18. The following statements are valid: (1) f is continuous on X.

(2) If I is any open interval of (R, then f is not uniformly continuous on

X(lI.

(3) f is 5-chained for every 8 E (R+ 14.20. LEMMA. We adopt the notation of 14.18. Let N = [(x, f (x)) I x E X]. We consider N as a subspace of the product space (R X R. Let ¢ : N N be defined by (x, f (x))+/' = (x + 1, f (x + 1)), x E X. Then ¢ and -' are uniformly continuous homeomorphisms of N onto N. PROOF. It is clear that ¢ is a one-to-one transformation of N onto N. That ¢ and ¢-' are continuous on N follows from the continuity of f on X. Thus ¢ is a homeomorphism of N onto N. We prove that ¢ is uniformly continuous on N; the proof that -' is uniformly continuous is similar. Suppose that k is not uniformly continuous on N. Then there exists e E 6i.+ such that corresponding to 8 E (R+ there exist x, x' E X such that I x - x' I < 8, I f (x) - f (x') I < 8 and I f (x + 1) - f (x' + 1) I > e. Let (Sn n E 9+) be a sequence of positive real numbers such that 81 > 82 > ... and limn-+m 8n = 0. Then there exists a sequence of pairs ((xn , xn) I n E 9+) such that x , xn E X, I X. - xn I < B. , I f (xn) - f (X.) i < Sn , f (xn + 1) - f (x.' + 1) I > e, for

all nE9+ We can assume that lim-+m xn = x = lim., x; , where x E R. Since lim infra-+m I f (xn + 1) - f (xn + 1) I > e it follows that x + 1 Er X and there

exist k, m E 9 such that x + 1 = 2k7r + m. If n E 9+, then e < I f(xn + 1) - f(xn + 1)

I=IE

2-'

[fn(xn

+ 1) - fn(xn + 1)]

There exists M E 9+ such that M > m and 12-"'[fv(xn + 1) - fn(xn + 1)]

< e/3

and

E 2-'ni[f9(xn + 1) - fn(xn + 1)] M+1

<e/3,

nE9+

E 2 -'P' [fp(Xn + 1) - f,(xn + 1)] > e/3, -M

n E 9+

Thus M

Since p E 9, p F-4 m, implies lim I fn(xn +

1)

- fn(xn + 1) 1 = 0,

CYLINDER FLOWS AND A PLANAR FLOW

(14.23]

there exists P E 9+ such that n E 9+, n > P. implies UU

provided n > P, n E

12-Iml[ fm(xn + 1)

{{=fm-1(xn), n E 9+, we have

f m(x'. + 1)] 1 > e/4. Since fm(xn + 1) 2-Im1[fm-1(xn)

141

- m-1(xn)] I > e/4

4+.

Since n E 9+ implies +m I

E 2- l" [fa(xn) - fn(x:) ]

f(xn) - f(x;) I =

< un ,

there exists Q E 9+, Q > m, such that Q`

2-Ivl[ f (xn)

- fn(xn)]

< Sn + e/10

(n E 9+).

Since p E 9, p 0 m - 1, implies lim n-++m

I

,(xn) - f y(xn) I = 0,

there exists S E 9+ such that n > S implies 12-1--11[fm-1(xn)

- fm-1(x.')] I < Sn + e/9.

Let t E 9+ with t > P such that 12-Im-11[fm-1(xt)

- f'n-1(x!)]

I

<.E/8,

whence I m-11 2-lm-11 > tl-Im-ll I fm-1(xt) 4 = 8. I > fm-1(xt) g 2-1-1

From this contradiction we infer the validity of the lemma.

14.21. DEFINITION. We adopt the notation of 14.18. Let M be the graph of r = f (0), 0 E X, in polar coordinates. Let tp : M -+ M be defined by (0, f (0))tp =

(0 + 1, f (0 + 1)), 0 E X. 14.22. THEOREM. The transformation p : M M is a uniformly continuous homeomorphism of M onto M, tp 1 is uniformly continuous on M, tp and p-' are pointwise almost periodic, and M is a minimal orbit-closure under (o. PROOF.

It is obvious that c is a homeomorphism of M onto M, that tp and

are pointwise almost periodic and that M is a minimal orbit-closure under tp. The uniform continuity of tp and tp 1 follows from 14.20. (P-1

14.23. REMARK.

Let X be a compact metric space, let Y be a nonvacuous

subset of X and let (p be a pointwise almost periodic homeomorphism of Y onto Y such that (p and tp 1 are uniformly continuous on Y and such that Y is a minimal orbit-closure under (p. Then there exists a homeomorphism of Y onto Y such that I Y = (p and 7 is a minimal orbit-closure under gyp.

TOPOLOGICAL DYNAMICS

142

[14.24]

14.24. THEOREM. Let M be the plane set defined in 14.21. Then M is compact, connected, and locally connected at some points but not locally connected at other points. There exists a homeomorphism 77 of M onto M such that k is a minimal orbit-closure under ,. PROOF.

Clearly M is locally connected at each of the points (0, f (0)),

0 E X. To complete the proof, use 14.17, 14.19(3), 14.20, 14.22, 14.23. 14.25. NOTES AND REFERENCES.

(14.01) Cylinder homeomorphisms with Y a circle were considered by A. S. Besicovitch [1, 2], who constructed transitive models of cylinder flows. (14.15) Cf. Zippin [1], pp. 196-197 and Gottschalk [4]. (14.16-14.24) This example of a minimal set is due to F. B. Jones (Personal communication).

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INDEX Admissible orbit pairs, 10.38 Admissible subsets of T, 3.12 Almost periodic, 3.38 Asymptotic, 10.21 Asymptotic geodesics, 13.08 Averaging process, 4.75 Axiom

continuity, 1.01 homomorphism, 1.01 identity, 1.01 Axis of a hyperbolic transformation, 13.05 Baire subset, 9.11 Base, 3.29 Bisequence, 12.03 Bisequence space, 12.08 Block, 12.03 Block, n-, 12.04 Bounded, 14.02 Bounded set of mappings, 11.11 Cantor-manifold, 2.16 Center, 7.17 Centralizer of A in G, 9.31 Compactification of a topological group, 4.47 Compactive, 8.11 Composition, 4.74 Contained in, 12.04 Cylinder homeomorphism, 14.03 Decomposition, 1.37 Discrete, 3.15 Discrete in M, 13.41 Dual, 12.27 Dyadic tree, 14.15 Envelope x-, 2.08 Equicontinuous set of mappings, 11.07 Expansive, 10.31 Extension left S-, 1.43 right S-, 1.43 Or, 12.29 Extensive in T, 3.37 Fixed point, 3.06, 10.33 positive, 13.05 negative, 13.05 Flow, 1.30 geodesic, 13.10

geodesic partition, 13.25 horocycle, 13.12 horocycle partition, 13.31 Fundamental region, 13.45 Generative topological group, 6.01 Group motion preserving, 9.28 orbit preserving, 9.30 phase, 1.01 transition, 1.08 Homeomorphism motion preserving, 9.26 orbit preserving, 9.29 periodic, 10.33 pointwise periodic, 10.33 Homeomorphism group, 1.14 discrete, 1.14 topological, 1.12 total, 1.14 Horocycle, 13.07 Hyperbolic area, 13.01 circle, 13.06 distance, 13.05 length, 13.01 line, 13.03 plane, 13.01 ray, 13.03 Indexed, left, 12.46 Indexed, right, 12.46 Intrinsic property, 1.03 Invariant set, 1.22 Isochronous, 3.38 weakly, 3.38 Isometry with axis, 13.05 Isomorphism topological, 1.02 uniform, 1.02 Length (of a block), 12.04 Limit-entire, 13.26 Limit-partial, 13.26 Limit point P-, 6.33 a-, 10.15 co-, 10.15 149

150

INDEX

Limit set, 13.13 P-, 6.33 a-, 10.15 w-, 10.15

Locally bounded set of mappings, 11.30 Mean, 4.81 a-, 4.78

Minimal orbit-closure, 2.11

S-,2.11 totally, 2.27 Mixing, regionally, 9.02 Mobile, 13.22 Motion-projection, 1.08 Motion-space, 1.08 Motion x-, 1.08 Nonseparated, 10.02, 10.38

Normalizer, 9.31 Orbit, 1.26, 10.12 Orbit-closure, 1.26 minimal, 2.11 Paddle motion, 13.05 Partition, 1.27 Period of a homeomorphism, 10.33 of a point, 10.33 of T, 3.04 of T at x, 3.02 Periodic, 3.06 almost, 3.38 at x, 3.06 pointwise, 3.06 regularly almost, 3.38 weakly almost, 3.38 Periodic point, under homeomorphism, 10.33 Permutation of a set, 11.15 Topological group monothetic, 4.49 solenoidal, 4.49 Phase group, 1.01 projection, 1.01 space, 1.01

Point at infinity of a horocycle, 13.07 of an h-ray, 13.03 Points at infinity of an h-line, 13.03 Projection motion, 1.08 phase, 1.01 transition, 1.08

Property intrinsic topological, 1.03 intrinsic uniform, 1.03 Ray left, 12.02 right, 12.02 Recurrent, 3.38, 10.18 Recursive, 3.13 uniformly, 3.42, 3.44, 3.47, 3.50 weakly, 3.42, 3.44, 3.47, 3.50 Regularly almost periodic, 3.38 Replete in T, 3.37 Representation, A-, 12.42 Restriction, 1.32 Reverse, 12.04 Rotor, 13.32 Saturation, 1.34 Semigroup, 2.05 Semiorbit, 10.12 Separated, 10.02, 10.38 Sequence left, 12.03

right, 12.03 Shift transformation, 12.11 Similar, 12.04 Space product, 1.49 Star, (1-, 1.34 Star-closed, 1.35 Star-indexed, 2.35 Star-open, 1.35 Subblock, 12.04 Subgroup restriction, 1.32 Subsequence left, 12.04 right, 12.04 Subspace restriction, 1.32 Symbol class, 12.01 Symbolic flow, 12.11 Syndetic, 2.02, 2.03 Topology

bilateral compact-index, of a set of permutations, 11.34

bilateral compact-open, of a set of permutations, 11.44 bilateral space-index, mutations, 11.16

of a set of per-

compact-index, of a set of permutations, 11.34

inverse compact-index, of a set of permutations, 11.34

inverse compact-open, of a set of permutations, 11.44

INDEX

inverse space-index, of a set of permutations, 11.16 point-index, of a set of mappings, 11.24 space-index, of a set of mappings, 11.01 space-index, of a set of permutations, 11.16 Total power, 4.74 Total homeomorphism group, 1.14 Totally minimal, 2.27 Totally noncompactive, 8.12 Trace, 2.40 Transformation group, 1.01 bilateral, of T, 1.51 discrete, 1.14 effective, 1.09 equicontinuous, 1.52 equicontinuous at x, 1.52 functional, over (X, T, 7r) to Y, 1.68 ,p-inverse partition, 1.42 left, of T, 1.51 left functional, over T to Y, 1.63 left, of T/S induced by T, 1.56 left, of S induced by T under gyp, 1.58 left uniform functional, over T to Y, 1.62 partition, 1.39 (D-orbit partition, 1.40 ''-orbit-closure partition, 1.41 topological, 1.01 right, of T, 1.51 right functional, over T to Y, 1.63 right, of S induced by 1' under p, 1.58 right, of T\S induced by T, 1.56 right uniform functional, over T to Y, 1.62

uniform functional, over (X, T, a) to Y, 1.66

151

uniformly continuous, 4.36 uniformly equicontinuous, 1.52 Transformation subgroup, 1.32 Transient, 13.28 Transition group, 1.08 projection, 1.08 t-, 1.08 Transitive, 9.02 Translate, n-, 12.04 Uniform convergence at a point, 9.34 Uniformity

bilateral compact-index, of a set of permutations, 11.34 bilateral space-index, of a set of permutations, 11.16 compact-index, of a set of mappings, 11.24

compact-index, of a set of permutations, 11.34

inverse compact-index, of a set of permutations, 11.34 inverse

space-index,

of

a

set

of

per-

mutations, 11.16 partition, 2.34 point-index, of a set of mappings, 11.24 space-index, of a set of mappings, 11.01 space-index, of a set of permutations, 11.16 Uniformly equicontinuous set of mappings, 11.08

Union of a sequence of blocks, 12.05 Unitangent, 13.10, 13.48 space, 13.10, 13.48 Universally transitive, 9.02 Weakly recursive, 3.13, 3.42, 3.44, 3.47, 3.50

Colloquium Publications 1.

2.

31. 32.

4.

H. S. White, Linear Systems of Curves on Algebraic Surfaces; F. S. Woods, Forms of Non-Euclidean Space; E. B. Van Vleck, Selected Topics in the Theory of Divergent Series and of Continued Fractions; 1905, xii, 187 pp. $3.00 E. H. Moore, Introduction to a Form of General Analysis; M. Mason, Selected Topics in the Theory of Boundary Value Problems of Differential Equations; E. J. Wilczynski, Projective Differential Geometry; 1910, x, 222 pp. out of print G. A. Bliss, Fundamental Existence Theorems, 1913; reprinted, 1934, out of print ii, 107 pp. E. Kasner, Differential-Geometric Aspects of Dynamics, 1913; reprinted, 1947, ii, 117 pp. 2.50 L. E. Dickson, On Invariants and the Theory of Numbers; W. F. Osgood, Topics in the Theory of Functions of Several Complex Variables; 1914, xii, 230 pp.

out of print

5,. G. C. Evans, Functionals and their Applications. Selected Topics, Including Integral Equations, 1918, xii, 136 pp. out of print 52.

6. 7. 8.

0. Veblen, Analysis Situs, 1922; 2d ed., 1931; reprinted, 1951, x, 194 pp. 3.35 G. C. Evans, The Logarithmic Potential. Discontinuous Dirichlet and Neumann Problems, 1927, viii, 150 pp. out of print E. T. Bell, Algebraic Arithmetic, 1927, iv, 180 pp. out of print L. P. Eisenhart, Non-Riemannian Geometry, 1927; reprinted, 1949, 2.70 viii, 184 pp.

9. 10.

G. D. Birkhoff, Dynamical Systems, 1927; reprinted, 1952, viii, 295 pp. 4.60 A. B. Coble, Algebraic Geometry and Theta Functions, 1929; reprinted, 4.00 1947, viii, 282 pp.

11.

D. Jackson, The Theory of Approximation, 1930; reprinted, 1951,

12. 13.

3.35 out of print R. L. Moore, Foundations of Point Set Theory, 1932, viii, 486 pp. out of print viii, 178 pp. S. Lefschetz, Topology, 1930, x, 410 pp.

14.

J. F. Ritt, Differential Equations from the Algebraic Standpoint, 1932; reprinted, 1947, x, 172 pp. 3.00

15.

M. H. Stone, Linear Transformations in Hilbert Space and their Applications to Analysis, 1932; reprinted, 1951, viii, 622 pp.

8.00

16.

G. A. Bliss, Algebraic Functions, 1933; reprinted, 1947, x, 218 pp.

out of print J. H.-M. Wedderburn, Lectures on Matrices, 1934; reprinted, 1949, x, 205 pp. 3.35 18. M. Morse, The Calculus of Variations in the Large, 1934; reprinted, 1947, x, 368 pp. 5.35 19. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, 1934; reprinted, 1954, viii, 184 pp. + portrait plate 4.00 20. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 1935, x, 382 pp. out of print 21. J. M. Thomas, Differential Systems, 1937, x, 118 pp. out of print 22. C. N. Moore, Summable Series and Convergence Factors, 1938, vi, 105 pp. out of print 23. G. Szego, Orthogonal Polynomials, 1939; reprinted, 1948, x, 403 pp. out of print 24. A. A. Albert, Structure of Algebras, 1939; reprinted, 1952, xii, 210 pp. 4.00 25. G. Birkhoff, Lattice Theory, 1940; enlarged and completely rev. ed., 1948, xiv, 283 pp. 6.00 26. N. Levinson, Gap and Density Theorems, 1940, viii, 246 pp. 4.00 27. S. Lefschetz, Algebraic Topology, 1942; reprinted, 1948, vi, 393 pp. out of print 28. G. T. Whyburn, Analytic Topology, 1942; reprinted, 1948, x, 280 pp. out of print 29. A. Weil, Foundations of Algebraic Geometry, 1946, xx, 288 pp. 5.50 30. T. Rado, Length and Area, 1948, vi, 572 pp. 6.75 31. E. Hille, Functional Analysis and Semi-Groups, 1948, xii, 528 pp. out of print 32. R. L. Wilder, Topology of Manifolds, 1949, x, 402 pp. 7.00 33. J. F. Ritt, Differential Algebra, 1950, viii, 184 pp. 4.40 34. J. L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions, 1950, viii, 384 pp. 6.00 35. A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht 17.

36.

Functions, with a chapter on The Region of Values of the Derivative of a Schlicht Function by Arthur Grad, 1950, xvi, 311 pp. 6.00 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, 1955, viii, 151 pp. 5.10 AMERICAN MATHEMATICAL SOCIETY Providence, R. I., 80 Waterman Street