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I. 31.5.9.5...95.’ t. in. xiii 1....- «slittiv... 1A.! . .3138 at. .75.. . 2%.... .: \ 3 1...... .3 pt... .100...ui\:4! \rvxf .2. .0: a. r ..1nv .. 59!. .0 ’1. 0.4!! 2. 9923. F \O1I12.\:.Ielc.2..‘d( . \. .. .5 5.... ..$1D;‘\... . lip... r. .5 fv. I. Z...m r messs I 2 ()0. l LIBRARY Michlgan fataie nivers t LJ’ “ML... This is to certify that the dissertation entitled Minimality of Flows and Almost Periodicity of Points Under Various Constructions in Topological Dynamics presented by Al ica Mi l ler has been accepted towards fulfillment of the requirements for Ph.D. degree inMaihemaiicL WM 8 W V V Major professor Date May 3. 2001 MSUiJ an Affirmative Action/Equal Opportunity Institution 042771 PLACE IN RETURN Box to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIFIC/DateDuepGS-pJS MINIMALITY OF FLOWS AND ALMOST PERIODICITY OF POINTS UNDER VARIOUS CONSTRUCTIONS IN TOPOLOGICAL DYNAMICS By Alica Miller A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2001 ABSTRACT MINIMALITY OF FLOWS AND ALMOST PERIODICITY OF POINTS UNDER VARIOUS CONSTRUCTIONS IN TOPOLOGICAL DYNAMICS By Alica Miller In Chapter 1 we deal with the minimality of the flows obtained by various con- structions from compact minimal abelian flows. We first find a criterion for minimality of “syndetic” restrictions of compact minimal abelian flows in terms of eigenvalues (and a criterion for total minimality of compact minimal abelian flows). Using the criterion for minimality of restrictions we give a new proof of a classical theorem of W. Parry about minimality of group-extensions of compact minimal abelian flows. Then we prove a criterion for minimality of products of two compact minimal abelian flows, one of which is almost periodic, in terms of eigenvalues. For each of the cri- teria we give several applications. We also introduce the notion of SK groups, and use it to generalize some statements which relate total minimality, weak mixing and triviality of the structure group, as well as to improve various conditions which imply non—total-minimality of compact minimal abelian flows. In Chapter 2 we deal with the question whether almost periodicity of a point in a flow transfers to the appropriate points in the flows obtained by various constructions Alica Miller applied to the original flow, like restrictions, subflows, factors, extensions, products, etc. The most difficult is the case when we have a morphism f : X —> y, an almost periodic point y in y, and a point a: E f"1(y). In general a: is not necessarily almost periodic, but several conditions are known under which that happens. They fall into either “compact” or “noncompact” conditions, depending on whether X and y are assumed to be compact or not. In “noncompact” conditions other assumptions are restrictive. We find a criterion for lifting of almost periodicity of y, which generalizes both “compact” and “noncompact” statements at the same time. ACKNOWLEDGEMENT I would like to thank my advisor Clifford Weil for all his help and support over the years, my committee members Ronald Fintushel, Jay Kurtz, Joel Shapiro and William Sledd for showing interest in my work and being nice to me and my dear friend Joe Auslander for his encouragement. Finally my thanks to my parents and my husband for their love, encouragement and support. iv Contents 0.1 Introduction ................................ 1 0.2 Notations and preliminaries ....................... 3 0.2.1 General topology; topological groups .............. 3 0.2.2 Flows ............................... 5 0.2.3 x-envelopes; orbit-closures .................... 6 0.2.4 Almost periodicity, proximality, distality, weak mixing . . . . 7 1 Minimality of restrictions, group-extensions and products of com- pact minimal abelian flows 10 1.1 The notion of an X—enveloped subgroup ................ 10 1.2 A criterion for minimality of restrictions and a criterion for total min— imality ................................... 12 1.3 A new proof of a theorem of Parry ................... 15 1.4 Minimality of group-extensions ..................... 17 1.5 Minimality of a product of two compact minimal abelian flows, one of which is almost periodic ......................... 19 1.6 SK groups ................................ 21 1.7 1.8 Total minimality of X in terms of the structure group I‘ (X ) ..... Compact minimal abelian flows that are not totally minimal ..... Almost periodicity of a point under various constructions 2.1 The notion of a skew-morphism of flows ................ 2.2 Almost periodicity of a point under various constructions ....... 2.3 The notion of a skew-morphism good over a point with respect to orbit-closures ............................... 2.4 Examples of skew—morphisms good over a point with respect to orbit- closures .................................. 2.5 A criterion for lifting of almost periodicity of a point ......... 2.6 Applications of the criterion for lifting of almost periodicity of a point Bibliography .......................................................... vi 24 27 27 28 30 32 34 35 65 0.1 Introduction The thesis consists of two chapters. We will first roughly describe the content of each of the two chapters; then we will give a more precise description of each section of each of the chapters. In Chapter 1 we deal with the minimality of the flows obtained by various con- structions from compact minimal abelian flows. We first find a criterion for minimality of “syndetic” restrictions of compact minimal abelian flows in terms of eigenvalues (and a criterion for total minimality of compact minimal abelian flows). Using the criterion for minimality of restrictions we give a new proof of a classical theorem of W. Parry about minimality of group—extensions of compact minimal abelian flows. Using Party’s theorem we prove a criterion for minimality of a product of two com- pact minimal abelian flows, one of which is almost periodic, in terms of eigenvalues. For each of the criteria we give several applications. Among other things, we intro— duce the notion of SK groups, well adjusted to work with the criteria, prove some properties of this notion, and then use it to generalize some statements which re— late total minimality, weak mixing and triviality of the structure group, as well as to improve various conditions which imply non-total-minimality of compact minimal abelian flows. In Chapter 2 we deal with the question whether almost periodicity of a point in a flow transfers to the appropriate points in the flows obtained by various constructions applied to the original flow, like restrictions, subflows, factors, extensions, products, etc. The most difficult is the case when we have a morphism f : X ——> )7, an almost periodic point y in y, and a point a: E f‘1(y). In general :1: is not necessarily almost periodic, but several conditions are known under which that happens. They fall into either “compact” or “noncompact” conditions, depending on whether X and y are assumed to be compact or not. In “noncompact” conditions other assumptions are restrictive. We find a criterion for almost periodicity of :c, which generalizes both “compact” and “noncompact” statements at the same time. The questions investigated in Chapter 1 and Chapter 2 are naturally related to each other since, in the case of compact flows, almost periodicity of a point can be expressed via minimality. In particular, in a compact minimal flow every point is almost periodic. We now give a more precise description of each section of each chapter. First let us mention that in Section 0.2 we give the notation and the terminology we use throughout the thesis, as well as some relevant basic facts. We first give a description of each section of Chapter 1. In Section 1.1 we introduce the notions of the X - envelope of a syndetic subgroup S of T and of an X - enveloped subgroup of T. We prove some properties related to these notions and give some examples which illustrate them. Using these notions, in Section 1.2 we formulate and prove two results, which we name as a criterion for minimality of restrictions and a criterion for total mini- mality. The first criterion gives necessary and sufficient conditions for a restriction X5 = (S, X), S syndetic, of a compact minimal abelian flow X : (T,X) to be minimal. One of the conditions is in terms of eigenvalues, the other one in terms of X -envelopes. The second criterion gives necessary and sufficient conditions for a 2 compact minimal abelian flow X 2: (T,X) to be totally minimal. We apply these criteria to the cases T = Z and T = R. The statement we deduce in the case T = IR is a result of J. Egawa. In Section 1.3 we investigate skew-extensions of compact minimal Z—flows. In the context of a classical theorem of Parry we apply the criterion for minimality of restrictions to an appropriately defined flow and conclude that the condition coming from the criterion is equivalent to the well-known Parry’s condition. In that way we get a new proof of this theorem. In Section 1.4 we give a new proof of a more general theorem of Parry about minimality of group—extensions. In the proof we use the criterion for minimality of restrictions. In Section 1.5 we formulate a criterion for minimality of a product of two com- pact minimal abelian flows X and )7, one of which is almost periodic, in terms of eigenvalues, and prove it using Party’s theorem. Applying this criterion we deduce a new proof of a characterization of minimality of X x y in terms of common factors. In Section 1.6 we introduce the notion of SK groups, give examples and prove some properties of this notion. These properties will then be used in Sections 1.7 and 1.8. In Section 1.7 we investigate the relation between weak mixing and total minimal- ity in the context of SK acting groups (using the criteria developed in Section 1.2). We generalize a statement of N. Markley (about the equivalence of weak mixing and total minimality) and a statement of W. Gottschalk (about total minimality of X in terms of the structure group I‘( X )) by extending the class of acting groups for which 3 their statements remain valid. In Section 1.8 we use the criteria developed in Section 1.2 to investigate some conditions which, when imposed on compact minimal abelian flows, necessarily imply non-total-minimality. In that way we generalize and give a shorter proof of a result of H. Chu (in which for example we avoid the use of Pontryagin’s duality theory), as well as several other results. Now we give a description of each section of Chapter 2. Morphisms of flows with not necessarily the same acting group were not seriously considered in the literature since it seems that they don’t give anything more than “standard” morphisms of flows with the same acting group. In fact there is only one paper in which these “new” morphisms of flows were considered; namely [26]. There in total three propositions involving this notion were proved and no example was given. In Section 2.1 we call these morphisms “skew-morphisms” and give several natural situations where they appear. We use them in a systematic manner in the rest of this chapter. In Section 2.2 we give facts about almost-periodicity of a point in various con— structions. Some facts are stated for skew-morphisms instead of morphisms and, in some instances, it is illustrated that, using skew-morphisms, we sometimes get sim- pler and more natural proofs, as well as new statements. Some statements show that it is much easier to deal with almost periodicity of a point in the case of compact flows (since in that case there is a natural connection between almost periodicity of a point and minimality of flows). Our goal is a theorem which unifies various known statements about lifting of 4 almost periodicity of a point in both the compact, as well as not necessarily compact, case. The first important statement about lifting was given by R. Ellis in [12] for compact flows. (Applications of this statement to free abelian topological groups are given in [11].) Later Markley and others obtained some statements for not necessarily compact flows. In [27] Markley said that his results “differ from other results of this genre in that we do not assume that either space is compact.” But his other assump- tions were quite restrictive and were later relaxed a little bit by S. H. A. Kutaibi, F. Rhodes and others. Some other related results appeared later, like for instance a theorem of V. Pestov [30]. In order to extract what is essential in all these state- ments, in Section 2.3 we introduce the notion of a skew-morphism good over a point with respect to orbit-closures and give some natural examples. In Section 2.4 we give several more complicated examples of skew—morphisms good over a point with respect to orbit closures. In Section 2.5 we formulate and prove a criterion for lifting of almost periodicity of a point, which works for not necessarily compact flows. In Section 2.6 we show that various other statements about lifting of almost peri- odicity of a point (both “compact” and “non-compact”) are corollaries of our criterion. As corollaries we get results of Ellis, Markley, Kutaibi-Rhodes, Pestov. 0.2 Notations and preliminaries 0.2.1 General topology; topological groups 0.2.1. If X is a set, we denote its cardinality by |X|. All topological spaces are assumed to be Hausdorff. If X, Y are topological spaces, then Homeo(X) denotes the group of homeomorphisms of X, and C (X ,Y) denotes the set of continuous maps from X to Y. The map (x,y) t—> a: (resp. (x,y) v—> y) from X X Y to X is denoted by prl (resp. prg). If T is a topological group, Td denotes the group T equipped with the discrete topology. 0.2.2. Let X ,Y be topological spaces, f : X —> Y a continuous map. Then the map 9 : X —> Gr(f), defined by g(.r) = (3:,f(:r)), is a homeomorphism. (Here Gr(f) = {($,f(:1:))|a: E X} is considered as a subspace of X X Y.) 0.2.3. 11‘ will denote the topological group of complex numbers of module 1. If T is an abelian group, the continuous homomorphisms X : T —-) T, are called continuous characters of T. The set of all continuous characters of T will be denoted by T. 0.2.4. Let T; and T2 be topological groups and let X E T372. Then for all t1 6 T1 and t2 6 T2, x(t1,t2) = x(t1, 1) - x(1,t2). If we denote by X1 the continuous character t1 r—> x(t1,1) of T1 and by X2 the continuous character t; H x(1,t2) of T2, we have X(t1,t2) = X1(t1)xz(t2). Whenever no confusion can arise, we will simply write X = X1X2. Similarly for products of any finite number of factors. 0.2.5. We use the theory of uniform spaces as it is presented in [40]. We call an en- tourage what is called a connector in [40], or an index in [2], [13],[22],[38]. Entourages 6 are denoted by small greek letters: a, B, etc. If X is a uniform space, a an entourage of X and a: E X, then a[:c] denotes the set of all y E X such that (:r,y) E a. On a compact topological space X there is exactly one uniform structure com— patible with the topology of X. The entourages of this uniform structure are all neighborhoods of the diagonal A in X x X. 0.2.6. Let T be a topological group. A subset A of T is syndetic if there exists a compact subset K of T such that T = K A. If S' is a syndetic subgroup of T, the quotient space T/S is compact. A subset A of T is discretely syndetic if it is a syndetic subset of T4. Lemma 0.2.7. Let h : T ——> T’ be a surjective group homomorphism. Then for every discretely syndetic subset S' of T’, h‘1(S') is discretely syndetic in T. Proof. There is a finite subset F’ = {b’1,,--- ,b;} of T’ such that T' = F'S'. For every b; E F' let b.- E T be such that h(b.) 2 b2. Let F 2 {b1,--- ,bn}. We claim that T = Fh‘l(S'). Indeed, for t E T, let h(t) = b’s’. Put 3 = b'lt. Then h(s) : h(b)'1h(t)= b’—lb’s’ = s’, so 3 E h‘1(S'). We havet = b-b‘lt E F-h"1(S"). [:1 Remark 0.2.8. Let h : T —> T’ be a surjective group homomorphism having the compact-covering property (i.e. for every compact K’ in T' there is a compact K in T such that h(K) = K’). Then if S’ is a syndetic subset of T', h"1(S') is a syndetic subset of T. This statement is from [26]. The proof is analogous to the proof of Lemma 0.2.7. Lemma 0.2.9 ([27]). Let T be a topological group, S a syndetic subset ofT, SI, - - ~ , 3,. subsets ofS such that S = UL1 5;, t1, - -- ,tn elements of T. Then the set UL1 t,S,- is syndetic. Proof. Let K be a compact subset of T such that T = KS. We have: (UL1 Kti—l) ’ (Ufl ItiSs') 3 U?=1Kt;“t.-S. = ULIKS.‘ = K(UL1 5,) 2 KS 2 T, and the set . —- — L Kt-_1 is com act. So the set L t,-S',- is syndetic. C] 3-1 a p i_l 0.2.10 ([4]). Let X and Y be topological spaces, f : X ——> Y a continuous map. We say that (X, f) is a covering of Y if for each point y E Y there is an open neighborhood V of y such that f "1( V) is a nonempty disjoint union of open subsets U,, i E I, of X, on which the restrictions f; : U,- —> V of f are homeomorphisms. An open neighborhood V of a point y E Y is called elementary if it satisfies the above condition. An open neighborhood U of a point :1: E X is called elementary if there is an elementary neighborhood V of the point y = f (x) such that U is one of the disjoint open subsets Ug, i E I, of X, whose union is equal to f‘1(V). A homeomorphism g : X —> X, a: H gr, is called a deck-transformation of the covering (X, f) if f (gm) = f (3:) for all x E X. The deck-transformations form a group A under composition (written as (g, g’) I~—> gg’). We say that A is transitive on the fiber f'1(y) of a point y E Y if for any two elements :12, x' E f—1 (y) there is an element 9 E A such that x' = gm. If (X, f) is a covering of Y, the fibers of f are discrete. Also f is a surjective local homeomorphism. In particular, f is open. (A continuous map f : X —-> Y is a local homeomorphism if for each a: E X there is a neighborhood U of a: such that f (U ) is a neighborhood of f(r) in Y and the map U —> f(U) which coincides with f on U is a homeomorphism.) 0.2.2 Flows 0.2.11. A triple X = (T, X,7r) consisting of a topological group T, a topological space X and a continuous action 7r : T X X —> X of T on X is called a flow on X. We write t.;2: or to: for rr(t,a:). We say that X is compact (respectively abelian), if X is compact (respectively if T is abelian). We say that X is trivial if [X] = 1. For x E X we denote by 7r’ : T -—> X the orbital map t I——> t.:r. For t E T we denote by 7r; E Homeo(X) the transition homeomorphism a: I—> ta. 0.2.12. When we have a Z-flow on X, X = (Z,X, 7r), then the transition homeomor- phism h := 7n completely defines the action: rr(n, 1‘) = h"(:r:). In that case we simply write X = (X, h) when no confusion can arise. 0.2.13. Every flow X5 = (S, X, 7r [XxS)s where S is a subgroup of T, will be called a restriction of the flow X = (T, X, 7r). Usually it is denoted simply by X5 = (S, X). If a subset Y of X is invariant under the action of T, then the canonical flow (T, Y) is a subfiow of X. If X = (T, X), y = (T, Y) are two flows with the same acting group T, then we define a T-flow on X X Y by t.(:1:,y) = (tz,ty), for t E T, :2: E X, y E Y. This flow is called the product of the flows X and y and denoted by X X y. 0.2.14. Let X = (T, X) and y = (T, Y) be flows. A map f : X ——> Y is a morphism of flows ifit is continuous and f(tx) = tf(a:) for all t E T and :1: E X. Iff is surjective, y is a factor of X, and X is an extension of )7. Endomorphisms, isomorphisms and automorphisms of flows are defined in a standard way. 9 0.2.15. Let X = (T,X) be a flow. A continuous function r] : X —> T is an eigen- function of X if there is a continuous character X E T such that r)(t.:r) = X(t)r)(r) for (t, 2:) E T X X. In that case X is an eigenvalue of X (the eigenvalue which corresponds to 1;) and 17 is an eigenfunction which corresponds to X. The following are equivalent: (i) X is trivial; (ii) 77 is constant on some Ta: (:1: E X); (iii) 7} is constant on every 3 (:1: E X). If X contains a point with dense orbit, then X is trivial iff r] is constant. 0.2.16. A flow X : (T,X) is minimal if the orbit T.:1: of every point :1: E X is dense in X. It is totally minimal if the flow X 5 is minimal for every syndetic (equivalently, closed syndetic) subgroup of T. If f : X —> y is a surjective morphism of flows, then if X is minimal (respectively totally minimal), y is minimal (respectively totally minimal). Two compact minimal abelian flows X = (T,X) and y = (T, Y) are disjoint if the product X X y is minimal. They are weakly disjoint if the product X X y has a point with dense orbit. 0.2.17. Every compact flow contains a minimal set. (The proof uses Zorn’s lemma ([2], [13]» [22], [38])-) 0.2.18. For :1: E X and U,V C X, the dwelling set D(U, V) (resp. D(:1:,V)) is the set of all t E T such that t.Ufl V 75 0 (resp. t.:2: E V). Lemma 0.2.19. Let X = (T, X,7r) be a flow, :1: E X. Then for every neighborhood V of :1: there are a neighborhood W of :1: and a neighborhood 0 of the unit element 6 E T such that 0D(:1:,W) C D(r, V). 10 Proof. Fix a neighborhood V of 1:. Since 7r : T X X —> X is continuous at (6,510), there is a neighborhood W of :1: and a neighborhood 0 of e such that 0W C V. We claim that then 0D(a:, W) C D(x, V). Indeed, let 0 E 0 and let t E D(r, W). Then ta: E W, hence 0(tx) E 0W, hence (ot)a: E V, i.e. at E D(zc, V). [:1 0.2.20. The Ellis semigroup E(X) of a flow X = (T, X,7r) is Cl{7r,|t E T} in XX (i.e. in Fp(X, X)) with the operation of composition. (Here Fp(X, X) denotes the set of all maps from X to itself, equipped with the topology of pointwise convergence.) If X is a compact flow, E (X ) is a compact Hausdorff right semitopological semigroup ([38, p.301]). 0.2.3 x-envelopes; orbit-closures 0.2.21. Let X = (T,X) be a flow. For :16 E X and S C T, the x-envelope of S , denoted by S”, is the set {t E T I to: E SE}. S” is a closed subset of T, it contains S, and $3.51 = E. If S is a syndetic normal subgroup of T, then S“ is a closed subgroup of T. ([22, 2.08—2.10]) 0.2.22. If S is a normal subgroup of T the following properties are easy to verify: (i) t3? = S.t.:1: for all t E T, :1: E X; (ii) If SE is a minimal subset of X under S, then: (1)15 = S—.:1:- if and only ift E S“; (2) (Vy E X) t.y E E if and only if ti; = fl. 0.2.23. Let X = (T,X) be a flow. It is easy to see ([22, 223]) that the following are equivalent: 11 (i) the set of orbit closures under T is a partition of X; (ii) (VxeX) (VyEX)yET.—;®T_.y=fi; (iii) every orbit closure under T is minimal under T. 0.2.24. Let X = (T, X) be a flow and S a syndetic normal subgroup of T. Then the set of orbit closures under S is a partition of X iff the set of orbit closures under T is a partition of X. In particular, if X is minimal, the set of orbit closures under S is a partition of X. ([22, 224]) We denote by 05 the set {Sax E X} and by R(Og) the relation (:13, y) E 12(05) <:> 3—9: ’53 on X. If X is compact minimal, R((’)3) is an equivalence relation which is open ([22, 2.30]) and closed ([22, 232]). 0.2.25. Let X = (T,X) be a minimal flow, S a syndetic normal subgroup of T, and K a compact subset of T such that T = KS. The following are easy to verify: (i) K.§ = X for every :1: E X; (ii) in particular, for every .27, y E X there is a k E K such that his? 2 Si; (and consequently lea: E 31]). 0.2.4 Almost periodicity, proximality, distality, weak mixing 0.2.26. Let X = (T,X) be a flow. (i) A point :1: E X is almost periodic (in X) if for every neighborhood U of :1: there is a syndetic subset A of T such that As: C U, i.e. the dwelling set D(r, U) is syndetic in T. A point :1: E X is discretely almost periodic if it is almost periodic in the flow Xd = (Td,X), where T; is the group T equipped with the discrete topology. Every 12 discretely almost periodic point is almost periodic. (ii) A point :1: E X is regularly almost periodic if for every neighborhood U of :1: there is a syndetic subgroup S of T such that 5:1: C U. Every regularly almost periodic point is almost periodic. (iii) A point :1: E X is locally almost periodic if for every neighborhood U of :1: there is a neighborhood V of :1: and a syndetic subset A of T such that AV C U. Every locally almost periodic point is almost periodic. A flow X is pointwise almost periodic (resp. pointwise regularly almost periodic; pointwise locally almost periodic) if every point :1: E X is almost periodic (resp. regu- larly almost periodic; locally almost periodic). (The adjective “pointwise” is omitted in the case of regularly and locally almost periodic flows.) A flow X = (T, X,7r) on a uniform space (X, LIX) is uniformly almost periodic (resp. equicontinuous) if for every 0 E LIX there exists a syndetic A C T such that As: C a[:1:] for every :1: E X (resp. if the family {71, | t E T} of transition homeomor- phisms is equicontinuous). In case of compact flows, uniformly almost periodic flows are the same as equicontinuous ones, and are simply called almost periodic flows. 0.2.27. Let X = (T,X) be a flow, :10 E X. The following are equivalent: (i) :1: is almost periodic in X; (ii) for every neighborhood U of :1: there is a compact subset K of T such that for every t E T, Kta: F) U 7&4 0; (iii) for every neighborhood U of a: there is a compact subset K of T such that Ta: c KU ([2],[13],[22],[38]). 13 0.2.28. Let X = (T, X) be a flow, at E X. If :1: has a compact neighborhood, then :1: is almost periodic iff T; is compact minimal. In particular, a point :1: in a compact flow X is almost periodic if and only iffl is minimal. ([2, p.11], [13, p.10], [38, IV(1.2)]) 0.2.29. Let X = (T,X) be a compact flow. The following are equivalent: (i) X is almost periodic; (ii) E (X ) is a compact topological group and the canonical map E (X ) X X —> X is a continuous action of E (X ) on X; (iii) E (X ) is a group and its elements are homeomorphisms of X. If in addition X is minimal, these conditions are equivalent to (iv) E(X) is a topological group. ([2, p.60], [13, p.25], [38, IV(3.34)]) 0.2.30. Let X = (T, X) be a compact minimal abelian flow. The following are equiv- alent: (i) X is almost periodic; (ii) for every :1: E X there is an abelian group structure on X with the unit element :1:, such that the orbital map 7r‘r : T —> X is a continuous group homomorphism; (iii) there is an element 6 E X such that there is an abelian group structure on X with the unit element 6, such that the orbital map 71'8 : T —> X is a continuous group homomorphism. ([36, Corollary 2.10], [38, IV(3.42)]) In particular, every nontrivial compact minimal abelian flow has a nontrivial eigen- value ([38, p.409]). 0.2.31. Let X be a compact flow. There is a smallest closed invariant equivalence relation on X, denoted S}, such that the quotient flow X / S; = (T, X / 5}) is almost 14 periodic ([2, p.125], [13, p.32] or [38, p.398]). The equivalence relation S; is called the equicontinuous structure relation of X. Then the Ellis semigroup E (X / SS.) is a compact topological group, called the structure group of X and denoted by I‘(X). 0.2.32. Let X = (T, X,7r) be a flow on a uniform space (X, LIX). (i) A pair (x,y) of points in X is proximal (and the points x,y are proximal to each other) if for every 01 E LIX there is a t E T such that (tx, ty) E a. The flow X is proximal if every pair of points in X is proximal. (ii) A pair (x,y) of points in X is regionally proximal (and the points x,y are regionally proximal to each other) if there is a point z E X such that for every neigh- borhood V of z and every neighborhood U1 ofx and U2 of y, D(Ul, V)flD(U2, V) 75 (b. The flow X is regionally proximal if every pair of points in X is regionally proximal. (iii) A pair (x,y) of points in X is distal (and the points x,y are distal to each other) if either x = y or (x, y) is not proximal. The flow X is distal if every pair of points in X is distal. A point x E X is distal if every pair (x,y), y E X, is distal. If X is compact, X is point-distal if there is a distal point x E X with dense orbit. 0.2.33. Let X be a compact flow. Then if X is almost periodic, it is distal ([2, p.65], [13, p.36], [38, IV(2.21)]). 0.2.34. If X = (T, X) is compact minimal abelian and x E X, then every pair (x, tx), t E T, is distal ([22, 10.07]). 0.2.35. Every nontrivial distal compact minimal flow has a nontrivial almost peri- odic factor ([2, p.104], [38, V(3.33)]). In particular, every nontrivial distal compact minimal abelian flow has a nontrivial eigenvalue ([2, p.105]). 15 0.2.36. Let X, y be distal compact minimal flows, f : X ——> y a morphism of flows. Then f is an open map ([2, p.98], [38, V(2.3)]). 0.2.37. Let X = (T, X,7r) be a flow on a uniform space (X,L(X). A pair (x,y) of points in X is proximal iff it belongs to the subset PX: H To aEux of X X X, which is called the proximal relation in X. This relation is reflexive, symmetric, invariant, but is not transitive nor closed in general ([2, p.66]). In case of compact flows, if Pg is closed, it is an equivalence relation ([2, p.88]). A pair (x, y) of points in X is regionally proximal iff it belongs to the subset Qx=flfi aeux of X X X, which is called the regionally proximal relation in X ([38, p.283]). Every proximal pair in a compact flow is regionally proximal. The relatioan is reflexive, symmetric, invariant and closed, but not necessarily an equivalence relation ([38, p.401]). If X is compact, S} is the smallest closed invariant equivalence relation on X which contains Q1 ([38, p.399]). In case of compact minimal abelian flows, Q x is an equivalence relation, and hence in that case Q3» 2 S}([2, p.130] or [38, p.404]). 0.2.38. A compact flow X is proximally equicontinuous if P4» = 6231 ([38, V(1.7)2]). In that case P31 is closed, hence Px (i.e. Qx) is an equivalence relation. So P3: = Qx = S}. All compact equicontinuous (i.e. almost periodic) and all compact proximal flows are proximally equicontinuous. Also all compact (pointwise) locally almost periodic flows are proximally equicontinuous ([38, p.364]). 16 0.2.39. A flow X = (T,X) is weakly mixing if for any open subsets U, U’, V, V’ of X there is a t E T such that at the same time tU H V # (b and tU' H V’ 7t 0, i.e. D(U,V) fl D(U’, V') # (b ([38, p.273]). If X is compact minimal abelian, then X is weakly mixing iff S; = X X X ([33, p.279], [2, p.133] or [38, V(1.19)]). Intuitively speaking, this says that weakly mixing compact minimal abelian flows are opposite to the almost periodic flows: the equicontinuous structure relation is the whole X X X, i.e. maximal possible, which is opposite to the case of almost periodic flows where the equicontinuous structure relation is equal to the diagonal A x, i.e. minimal possible. The only almost periodic factor of a weakly mixing compact minimal abelian flow is the trivial flow. 0.2.40. If X is a weakly mixing compact flow, every eigenfunction of X is constant. If X is a compact minimal abelian flow, then if every eigenfunction of X is constant, X is weakly mixing ([38, p.409]). 0.2.41. If X is a nontrivial compact minimal abelian flow on a metric space X, then if X is weakly mixing, it is not point-distal ([38, p.408]). 17 Chapter 1 Minimality of restrictions, group-extensions and products of compact minimal abelian flows 1.1 The notion of an X -enveloped subgroup Proposition 1.1.1. Let X = (T, X) be a minimal abelian flow, and let S be a syndetic subgroup of T. Then S” = S” for every x,y from X. Proof. Let K be a compact subset of T such that T = K + S. Fix any x, y from X. There is a k in K such that —k.x = y' E S—y. Then x = k.y', y' E S—y. Let s E S”. We have: s.y E 3.Ty = s.S.y’ = 3.5. — k.x = s. — lag = —k.s.fi = —k.S.s.x = —k.S.x = S. — k.x = S.y' 2 S3. Hence 3 E S”. Thus 5” C S”. By symmetry S” C S”. Hence 18 S‘ = S”. E] Definition 1.1.2. Let X = (T, X) be a minimal abelian flow, and let S be a syndetic subgroup of T. The X-envelope (or simply envelope) of S, denoted by S", is the subset of T which is equal to S”, where x is any element of X. By the previous proposition the notion of the X -envelope of S is well defined. Definition 1.1.3. Let X be a minimal abelian flow. A syndetic subgroup S of T is called an X -enveloped (or simply enveloped) subgroup of T if S = 5“. Note that a subgroup S of T may be enveloped with respect to some flow X = (T,X), and at the same time not enveloped with respect to some other flow y = (T, Y). Proposition 1.1.4. Let X = (T, X) be a minimal abelian flow and let S be a syndetic subgroup ofT. Then S“ is an enveloped subgroup ofT and it is the smallest enveloped subgroup of T containing S. Proof. 5* is syndetic and (S*)” = 5". So S“ is an enveloped subgroup of T. Let E be an enveloped subgroup of T containing S. Then: E=E*={teT|t€E}3{teT|te§3}=S*. Thus an enveloped subgroup of T, containing S, contains S *. E] 19 Example 1.1.5. Let 0 7é 0 be a real number. Consider the compact minimal abelian flow X = (R,T,7r), defined by 71(t,z) = 62"“62, t E R, z E T. Let a E R, a 75 0. Consider the action of the subgroup Z01 of R on T, induced by 71. The orbit in this action of an element 2 E T has the form Za.z : {no.2 I n E Z} = {ezfimez I n E Z}. ,21rin019 2:11;}:5 1st case: 010 E Q. Suppose 00 = 1,9, (k,l) = 1. Then every 6 = e 1 is one of . . 1- 2m? 2m 1‘ ’ I...’ e the elements 1, e k. Hence 2m} ”1% Za.z = {2,e z,...,e 2}. So m = Za.z. To calculate (Z01)" it is enough to calculate (Za)z for any 2 E T, for example (Za)l. We have: 3 E (Za)l (i) 62“”.1 2 e2”‘""9 for some n E Z (i) ehwwm") = 1 (i) 0(fl—na) = q for somquZ<=>fi= $q+na®flEZa+Z%=Z%+—Z%=Zia. Thus we have Z0” 2 Zia, where 09 = %, (k,l) = 1. 2nd case: 010 E Q. Then {ezflmoz I n E Z} is dense in T for every 2 E T, so Za.z = '11. Then (zar = (Za)l = {s e R | em” 6 r} = R. More concretely, let 0 = 1. Then for example: (Z?,—)" = Z; = (Zfi): (Z\/2)i = R, etc. There are many subgroups of R that are enveloped, and many that are not. Example 1.1.6. Let X be an almost periodic compact minimal abelian flow with X non-connected. X can be written as a disjoint union X = Y U Z of two nonempty clopen sets Y and Z. Let a : (Y X Y) U (Z x Z). Since X is compact and 01 is 20 an open neighborhood of the diagonal A X, a is an entourage of the unique uniform structure on X. Claim: S := fl D(x,a[x]) xEX is a proper enveloped subgroup of T. Since X is almost periodic, S is a syndetic subset of T. Note that for y E Y, a[y] = Y, and for z E Z, a[z] = Z. So S = {t E T I (Vy E Y) t.y E Yand (VzEZ)t.zEZ}={tETIt.YCYandt.ZCZ}={tETIt.Y=Yand t.Z = Z}. It follows that t1,t2 E T implies t1 + t2 E S, and t E S implies —t E S. Thus S is a syndetic subgroup of T. Now consider any element of X, for example some y E Y. Since X is a compact minimal abelian flow, 3" = S” = {t E T I t.y E S_y} C {t E T I t.y E Y}. (The inclusion holds because S.y C Y and Y is closed.) Since this is true for any y E Y, we have 5* c ny{t e T | t.y e Y} = {t e T | (vy e Y) t.y e Y}. 116 If we do the same thing for every 2 E Z and combine the results, we get 5* C S. Thus 5" = S. To prove that S is proper, consider any y E Y. Since X is minimal and Z is open, there is a t E T such that t.y E Z. This t does not belong to S. The claim is proved. 21 1.2 A criterion for minimality of restrictions and a criterion for total minimality Proposition 1.2.1. Let X = (T,X) be a flow and S a normal syndetic subgroup of T. Let X E T be an eigenvalue of X. The following are equivalent: (i) ker (X) D S; (ii) ker (X) 3 S‘” for every x E X; (iii) ker(X) D S‘r for some x E X. Proof. (iii) => (i): clear, since S“ D S. (ii) => (iii): clear. (i) :> (ii): Fix any x E X. Let r] be an eigenfunction of X which corresponds to X. We have: 170$) = X(t)n(=v) (*) for all t E T, and, in particular, n(sx) = 17(x) for all s E S. If we let 2 = 17(x), we have r;(S.x) = {2}. By continuity 17(5) 2 {z} and consequently 17(S“.x) = {2}. Hence from (:1:), X(s) = 1 for all s E S”, i.e. ker(X) D S”. E] Corollary 1.2.2. Let X : (T, X) be a minimal abelian flow and S a syndetic sub- group of T. Let X E T be an eigenvalue of X. Then ker (X) D S if and only if ker(X) D S“. Cl Theorem 1.2.3 (criterion for minimality of restrictions). Let X = (T, X ,71) be a compact minimal abelian flow. Let S be a syndetic subgroup of T and let X5 = 22 (S,X). The following statements are equivalent: (i) X 5 is a minimal flow; (ii) X has no nontrivial eigenvalue whose kernel contains S; (n1) 5* = T. Proof. By Proposition 1.2.1, (ii) is equivalent with (ii ’) X has no nontrivial eigenvalue whose kernel contains S“. So we will prove the above theorem with (ii) replaced by (ii'). First we make some observations. Note that 05 2 {SE I x E X} is the same as 05- since '33? = S*.x. The equivalence relation R = R(05.), ((x,y) e R e m = S*.y) is open and closed by 0.2.24. Hence X/R is compact Hausdorff. We denote by pX : X —> X/R = X the quotient map and by pT : T —> T/ S "' the canonical homomorphism. The elements of X will be denoted by 5: = pX (m). The map 71 : T x X —> X is compatible with the relations (mod 5*) x R on T x X and R on X. Hence it induces a continuous map fr : T X X/( mod S“) X R —> X. Since mod 5" and R are both open, we may identify r x X/(mod 5*) x R with :1:/5* x X/R = :1:/5: x X. With this identification we have 7~r°(pr> (ii') : clear. (ii’) => (iii) : Suppose S’“ aé T. Define a continuous map f : X —-> T/S'“ by f = prl 095‘10px. Fort E Tlet translme : T/S“ ——> T/S“ be defined by translpT(t)(t’+ S“) = pT(t) + t' + S“. Then for every t E T f o 71, = translme 0f. (2) To prove (2), put f = prl 0 cf)“. For t1 E T we have f(tx) =t1+S*¢>f(tx)=t1+S* se(t1+szo)=t efimfieS—fi: SE<=> 11$: tfie (—t+tl)E= 3754:) ((—t+t1)a)"’ = e 4: ¢(—t+t1+S*,&) = a 4:) f(x) = -—t+t1+S* (:1 f(x) =(—t+S*)+(t1+S*)©(t+S*)+f(:c)=t1+S*. Thus (2) holds. Now let X : T/S* —> T be any nontrivial character of T/S“. For t E T let translxm : T —> T be defined by tranleIt)(z) = X(t)z. It is easy to see that for every t E T X 0 translpTU) = transl(XopT)(,) OX. (3) 24 Let r) = X o f : X —) T. Then 77 is a continuous function which satisfies 7} 0 7r, 2 translIxopTW) 077 for all t E T. (This follows from (2) and (3).) So 77 is an eigenfunction of X whose eigenvalue X 0 pp is nontrivial and whose kernel contains S *. [:1 Corollary 1.2.4 (criterion for total minimality). Let X = (T, X) be a compact minimal abelian flow. The following statements are equivalent: (i) X is a totally minimal flow; (ii) X has no nontrivial eigenvalue whose kernel is syndetic; (iii) T has no proper X -enveloped subgroup. Proof. By Corollary 1.2.2, (ii) is equivalent to {ii’} X has no nontrivial eigenvalue whose kernel contains an enveloped subgroup of T. So we will prove this corollary with (ii) replaced by (ii’). (i) => (ii’) : clear from Theorem 1.2.3. (ii’) => (iii) : clear from Theorem 1.2.3. (iii) => (i) : Suppose (iii) holds. Let S be a syndetic subgroup of T. By assumption (iii), 3* = T (since S“ is enveloped). By Theorem 1.2.3, X5 is minimal. Cl As the first applications of these criteria, we investigate Z and R—flows. Corollary 1.2.5. Let X = (Z,X) be a compact minimal Z-flow. Then the following statements are equivalent: (i) X is totally minimal; 25 (ii) X has no eigenvalue X). = 62TH.) such that A E Q \ Z; (iii) X has no eigenvalue X(n) = z" with z 76 1 offinite order in T. Proof. (ii) and (iii) are clearly equivalent. We will show (i) (i) (ii). Every character of Z has the form XX 2 ehul'). We have ker(XX) = {n E Z I 62"“ =1} 2 {n E Z I An = k E Z}. Now by the criterion for total minimality, X is totally minimal iff X has no nontrivial eigenvalue XA such that {n E Z I An = k E Z} is syndetic in Z, iff X has no eigenvalue X), with A E Q \ Z. C] Remark 1.2.6. The direction (i) => (ii), i.e. (i) 2) (iii), of the Corollary 1.2.5 is well known; see for example [5], p.108. The opposite direction is probably also known, but the author could not find a reference. Corollary 1.2.7 ([9, Theorem 1]). Let X = (R,X) be a compact minimal R- flow. Let A(X) = {A E R I Xx = 821”.)‘(0 is an eigenvalue of X} and let M) =13 | A 6 M). n e Z\ {0}}. Let S = Z01, where a > 0 is a real number, and let X5 = (S,X). Then X5 is minimal ammomyugexmm Proof. By the criterion for minimality of reduced flows, X5 is not minimal iff X has an eigenvalue X» A 71 0, such that {t E R I At E Z} 3 Za, iff X has an eigenvalue )0, A 75 0, such that Zfi 3 Z01, iff X has an eigenvalue XX such that i = g, n E Z \ {0}, ingeMXI 26 1.3 A new proof of a theorem of Parry We will now investigate skew—extensions of compact minimal Z-flows. We will show that, in the special situation described in Theorem 1.3.1, the criterion for minimality of restrictions is equivalent to the well known Parry’s condition. The proof illustrates the way in which one can end-up with Parry’s condition after a sequence of natural steps, starting with the condition from the criterion. Theorem 1.3.1 ([2, p.72], [28, p.98], [38, II(8.22)]). Let G be a compact abelian topological group. Let y : (K?) be a compact minimal Z-flow, 1p : Y —+ G a continuous map, X = Y X G and let a E Homeo(X) be defined by 001.9) = My), «Rh/)9)- (1) Then the compact Z-fiow X = (X, o) is minimal ifl has no solution f E C(Y, T), 7 E C, with 7 751. Proof. Define a (compact abelian) flow Z = (Z X G,X) by (mm-3 = (pm(0"(x)),pr2(0”(x))g), (2) for n E Z, 9 E G, x E X. Writting x = (y,g’), we get from (1) and (2) fl (nag)-(y.g') = (7"(y).H¢(T""(y)) - 9’9). i=1 27 fl (—n,g).(y.g') = (r‘"(y),H (COND2). If we put 9 = 1 in (COND3), we get 17(0(y,g’))= 77(y.g'), y E Y, 9’ E G- Hence n(0"(y,g’)) = n(y.g’), n E Z, y E Y, 9’ E G- (3) Then we replace (y, g’) by 0""l(y,g’) = (n—1,1).(y, g’) in (COND3) and get (COND2) using (3). So (COND2) (:1 (COND3). Also (COND3) :> (COND4) <=> (COND4’) => (COND5). Now if we put f (y) = 11(y, 1) we get (COND5) => (COND6). Also (COND6) (i) the negation of the condition from the statement of the proposition. Conversely, suppose that (COND6) holds and define n(y,g') = 7(9’)f(y)- (4) We will show that these 77, 7 satisfy (COND3). We have n((1,g)-(y.g’)) = n(r(y).r(y)g’g) = 7(r(y’))r(g’)7(g)f(r(y)) = (from = 7(9)n(y,g’)- Thus (COND6) => (COND3) (i) (CONDl). Now since S is a syndetic subgroup of Z X G, by the criterion for total minimality of reduced flows and (at), X is not minimal iff the condition (CONDI) holds. But, as 29 we have just shown, (CONDl) is equivalent to the negation of Parry’s condition from the statement of the proposition. This completes the proof. El Remark 1.3.2. We could omit the proof of this version of Parry’s theorem since we are giving in Section 1.4 a proof of a more general theorem of Parry (from which this one can be deduced), also by applying the criterion for minimality of restrictions. But we decided to keep this proof as well, since it illustrates how the (“natural”) condition from the criterion for minimality of restrictions can be transformed, in a complicated concrete situation, to a condition which looks misterious and for which it is not clear where it is coming from. So we may say that the criterion for minimality of restrictions also sheds some light on Parry’s theorem. Remark 1.3.3. Some related types of skew—extensions are discussed in [19] and [17]. 1.4 Minimality of group-extensions In this section we give a new proof of a more general theorem of Parry about mini- mality of group-extensions, using the criterion for minimality of restrictions. Definition 1.4.1 ([2], [28], [38]). Let X = (T,X) and y = (T, Y) be compact flows, K a compact topological group. An extension p : X —> y is called a K-extension if the following conditions are satisfied: (i) there is a continuous action K on X which commutes with the action of T on X ; (ii) the fibers of p are precisely the K—orbits in X. 30 If in addition {K acts effectively} {K acts freely} {every character of K is an eigen- value of (K,X)}, we say that p is an {efiective} {free} {simple} K-extension. If we don’t want to specify the group, we say group-extension instead of K- extension. Example 1.4.2. Let y = (T, Y) be a compact flow, K a compact topological group. Put X = Y X K. Let t.(y,k) : (ty,k) and k.(y,k’) = (y,kk’). Let p : X —> Y be defined by p(x,y) = y. Clearly (X,p) is a free K-extension of y. Let X E K. Put fx(yak) = x(k). Then we have fx(k-(yik')) = fx(y.kk') = X(kk') = X(k)x(k') = X(k)fx(yik')- So (X, p) is a simple free K-extension of y. In what follows, if 7 is an eigenvalue of some flow, we denote by f.7 an eigenfunction of 7. Theorem 1.4.3 ([28]). Let X = (T,X) and y = (T, Y) be compact Abelian flows, y minimal, K a compact Abelian topological group. Suppose that (X, p) is a simple free K -extension of y. Then X is minimal ifl the functional equation f(t _P__($)) ___f7(t-$) has no solution f, f,, with f E C(Y, T) and 7 E K \ {1}. Proof. Define a (compact Abelian) flow Z = (T X K,X) by (t,k).x = t(kx) = k(tx), 31 for t E T, k E K, x E X. This flow is minimal. ( Indeed, let x E X and let U be an open subset of X. Since p(U) is open in Y and y is minimal, there is a t E T such that tp(x) E p(U), i.e. p(tx) E p(U). Then tx E p‘1(p(U)) = 'LeJU Kx’. Hence there is a k E K such that k(tx) E U.) Let S = T X {1}. Since (t, 1).x = tx, we may identify flows Z5 and X. So X is not minimal iff Z5 is not minimal. By the criterion for minimality of restrictions of compact minimal Abelian flows and 0.2.4, Z5 is not minimal iff the following condition holds: (:1:) r;((t,k).x) = 7(k)17(x) , t E T, k E K, x E X, has a solution r] E C(X, T), 7 E K, with 7 7‘— 1. It remains to show that the condition (:1:) is equivalent with the negation of the condition (1). Suppose that (at) holds. Define f E C (Y, T) by f(P($)) = f7($)/77($)i x E X ' ' ' h(k”) _ ‘YIkIf (37} _ full') (This is well defined Since by (:1:) Wm) _ 7(k)ri(x) — "(3, .) Since n(t.x) = 17(x), for t E T, x E X (which follows from (:1:) for k = 0), we easily get (1). Conversely, suppose that the negation of the condition (1) holds. Define 17 E C(X,T) by Then we have: n((t,k).x) = 7706”) = —f(](’°,:,), = ————’(fk(’,£”(:t;” = 7a) fortET,xEX,kEK. Cl Remark 1.4.4 ([23]). Fix 7 e I?\{ 1}. In the context of Theorem 1.4.3 the following are equivalent: (i) there is an eigenfunction f, of 7 such that the equation ( 1) has a solution f, f,, with f E C(Y, T); (ii) for every eigenfunction f,’, of 7, the equation (1) has a solution f’, 1,, with f’ E C (Y, T). (Indeed, (ii) :> (i) is clear. Conversely, suppose that (i) holds. If ff, is any other eigenfunction of 7, then (%)(gx) = (gig-Xx) for all g E G, x E X, so If can be written as h o p for some h E C(Y, K). Therefore f(t-p(:r)) = f3,(t$)h(19(t$)) = f;(tx)h(tp(x)) f(p(2=)) f1,(:v)h(p($)) ff,(9=)h(p(x)) ’ so the equation (1) has a solution f’ = '5, f4.) Remark 1.4.5. Let X = (T,X) be a simple free K —extension of y = (T, Y), where X, y are compact Abelian flows, and y minimal. For every 7 E K fix an eigenfunction f,’, of 7 (for the flow (K ,X )) Then the flow X is minimal iff the equation (1) has a solution f, f; with f e C(Y,r) and 7 e 1?\ {1}. (Indeed, the direction 4: is clear. The direction => follows from Remark 1.4.4.) 33 1.5 Minimality of a product of two compact min- imal abelian flows, one of which is almost pe- riodic In this section we use Parry’s theorem to prove a criterion for minimality of a product of two compact minimal abelian flows X and y, one of which is almost periodic, in terms of eigenvalues. We also give some applications of this criterion. Theorem 1.5.1 (criterion for minimality of products). Let X = (T, X, 71), y : (T, Y, p) be compact minimal abelian flows and suppose that y is almost periodic. Then the product X X y is minimal if and only if X and 32 have no nontrivial common eigenvalue. Proof. Fix any 6 E Y. Since y is almost periodic, there is a compact abelian group structure on Y such that e is the identity element and the orbital map p“3 : T ——> Y, t I—> te, is a continuous group homomorphism (0.2.30). Denote the group operation on Y by *. We have t(y :1: y’) = ty :1: y’, for t E T, y,y’ E Y. Define an action of the group You X X Yby y.(x,y’) = (x,yaky’) and amapp : X X Y ——> X by p(x,y) = x. In this way (X X y, p) becomes an Y-extension of X. If for every 7 E Y we define f1, : X X Y —> T by f;(x, y) = 7(y), we can conclude (as in Example 1.4.2) that X X y is a simple free Y—extension of X. Now by Remark 1.4.5, X X )7 is minimal iff the functional equation 34 has no solution f E C(X,T) with 7 E Y, 7 75 1. Since te*y = ty for t E T, y E Y, X X y is not minimal iff the functional equation f(tiv) = 7(te)f(x) (2) has a solution f E C(X,T) with 7 E Y, 7 76 1. We show that this condition is equivalent with X, y having a nontrivial common eigenvalue. A First note that for every 7 e Y, 7 o as is an eigenvalue of 31 (since 7(ty) = 7(te*y) = 7(te)7(y)). Now if (2) has a solution f e C(X,r) with 7 e Y, 7 ,1 1, then 7 o p" is a common eigenvalue of X and y, which is at 1 (since (7 o pe)(T) is dense in Y). Conversely, suppose that 6 e if, i at 1, is a common eigenvalue of X and 31. Then there is a 7 e C(Y, 11) such that 7(ty) = 6(t)7(y) and we can choose 7 so that 7(e) = 1. Then 7(te) = 6(1) and 7(te =1: t’e) = 7((t + t’)e) = 6(t + t’) = 7(te)7(t’e). It follows that 7 e Y, 7 a 1. Also there is a f e C(X, '1‘) such that f(tx) = 6(1) f(x). Hence f(tx) = 7(te)f(x), i.e. (2) has a solution f E C(X, T) with 7 E Y, 7 751. [:1 Remark 1.5.2. A measure-theoretic analogue of Theorem 4.1 was proved in [29]: let X and y be metric compact abelian flows which support closed ergodic invariant measures. Then X and y are weakly disjoint iff they have no nontrivial common eigenvalue. Remark 1.5.3. (a) The above theorem can also be proved using the criterion for minimality of restrictions instead of Parry’s theorem. We would consider the flow Z = (T X Y,X X Y), defined by (t,y).(x,y’) = (tx,ty’ at y) and its restriction Z5, where S = T X {e}. (Here * and e would be the same as in the above proof.) (b) Here is one more way to prove the easy direction (=>) of Theorem 1.5.1. 35 Suppose that X and y have a nontrivial common eigenvalue X. Let f : X —> T, g : Y —> T be the corresponding eigenfunctions. Then the function f g : X X Y ——> T, defined by fg(x,y) : f(x)g(y), is nonconstant and invariant. Hence X X y is not minimal. Example 1.5.4. Consider almost periodic compact minimal flows X = (IR, T, 71) and )7 = (R, T, p), defined by 71(t,z) = e2"“"z and p(t, z) = ezflmz, where a,fi E R. The eigenvalues of X (resp. y) are all Xm : t t—> 62""0‘ (resp. Xng : t +—> e”"‘"”’), n E Z. Hence, by Theorem 1.5.1, X X y is minimal iff oz and 6 are linearly independent over Q. Similarly the eigenvalues of the restriction Xz (resp. ya) of X (resp. y) are all Xm : k 1—> 62""0" (resp. Xng : k H 62“"3"), n E Z. Hence, by Theorem 1.5.1, XZ X ya is minimal iff a, )8 and 1 are linearly independent over Z. Corollary 1.5.5 ([2, p.161]). Let X, y be compact minimal abelian flows, and suppose that X is almost periodic. Then the product X X y is minimal if and only if X and 37 have no nontrivial common factor. Proof. (=>) Let Z be a nontrivial common factor. Then Z X Z is minimal, as a factor of a minimal flow X X 3). Hence Z is trivial, a contradiction. (<:) Let X and y have no nontrivial common factor. Suppose that X and y are not disjoint. Then by Theorem 1.5.1, they have a nontrivial common eigenvalue X. Let f : X —+ T and g : Y ——> T be the corresponding eigenfunctions. We may assume that there are points x0 E X and yo E Y such that f(xo) = 1, g(yo) : 1. Then f(Txo) = g(Tyo) = X(T). If X(T) is finite, then by continuity of f, f(TTE—S) = X(T) 36 and similarly g(Ty—6) = X(T). If X(T) is infinite (so dense in T), then f(m) = g(Ty?) = T, since the sets f(Tx—o) and g(TiiJ) are compact and contain a dense subset of T. In both cases, minimality of X and 3) implies f (X ) = g(Y). Hence the subflow on f(X) = g(Y) of the flow (T, T), (t,z) t—-> X(t)z, is a nontrivial common factor of X and y, a contradiction. 1:] Remark 1.5.6. A different proof of the statement of this corollary is given in [2], p.161. Also note that, conversely, Theorem 1.5.1 can be deduced from this corollary. Remark 1.5.7. If we don’t assume that either of the flows X, y is almost periodic, it is possible to construct two nondisjoint compact minimal abelian flows with no nontrivial common factor. A complicated example was given in [20]. The analogous problem with eigenvalues is trivial: take any weakly mixing flow X and put 32 = X. They are nondisjoint, but have no nontrivial common eigenvalue. Ergodic analogues of these questions are discussed in [39]. Remark 1.5.8. Let X and y be compact minimal abelian flows. It is known that if X is distal and y is weakly mixing, the product X X y is minimal; i.e. X and y are disjoint ([2, p.163], [15, Theorem II.3], [38, IV(2.39)1], [41, VI.2.18]). The first proof of this fact was given by Furstenberg ([15]) who showed that a group extension of a flow disjoint from all weakly mixing compact minimal flows is itself disjoint from all weakly mixing compact minimal flows if it is minimal. Recall that a compact minimal abelian flow 3) is weakly mixing iff y has no nontrivial eigenvalue ([33]). So it is natural to ask whether, more generally, X and y are necessarily disjoint if we assume that X is distal and X and 3) have no nontrivial 37 common eigenvalue. If we want to construct a counterexample, y must be non- weakly-mixing (the statement above) and also at least one of the spaces X, Y must be non—metric (this can be deduced from [2], p.161). We know of no such counterexample; i.e. we know of no example of two non-disjoint compact minimal abelian flows X and y, with X distal, and X and y having no nontrivial common eigenvalue. 1.6 SK groups In this section we introduce the notion of SK groups, which will be used in Sec- tions 1.7 and 1.8. For example, Proposition 1.6.6 below will play a role in the proof of Proposition 1.8.5(ii). The motivation for introducing SK groups comes from the criterion for minimality of restrictions and the criterion for total minimality. Definition 1.6.1. A topological group T is said to be SK, if the kernel of every continuous character X E T is a syndetic subgroup of T. Remark 1.6.2. The name “SK” means “syndetic kernels.” Example 1.6.3. (i) R. (ii) Every compact group. (iii) Every abelian minimally almost periodic group; in particular, every abelian extremely amenable group, see [23, 23.32] for examples of such groups. (Recall that an abelian topological group T is called minimally almost periodic if it has no nontrivial continuous characters. A topological group T is called extremely amenable if every T-flow on a compact space has a fixed point. It is easy to see that every abelian extremely amenable group is minimally almost periodic.) 38 Example 1.6.3(iii) shows that there are non-LCA SK groups. Also, not all LCA groups are SK, for example Z, Rd, Td, R X Rd, etc. Proposition 1.6.4. A finite product of SK groups is an SK group. Proof. Let T1,...,T,, be SK groups and let T = T1 x x T... Let x e T. For each (x1,...,xn) E T we have X(x1,...,x,,) = X(x1,0,...,0)... X(0,0,...,x,,). For i = 1,2,...,n denote by X,- the continuous character x,- +—> X(0,...,x,-,...,0) of T,-. So we have X(x1,...,x,,) : X1(x1)...X,,(x,,), where Xi E T,- (i = 1,2,...,n). Let S.- = ker(Xg), and T,- = S,- + Kg, where K, is a compact subset of T,- (i = 1,2, ....,n) Since ker(X) D S] X X S", and 5'1 X X 5,, is syndetic (K = K1 X X Kn is compact and 31 X X S, + K = T), ker(X) is also syndetic. Corollary 1.6.5. Every connected LCA group is SK. Proof. By [23, 9.14], connected LCA groups have the form R" X C, where n 2 0 and C is a compact connected abelian group. Since R and C are SK, the corollary follows from Proposition 1.6.4. [I Proposition 1.6.6. Let T be a topological group, S a subgroup of T. Let X E T be such that: (i) X(S) = T, and (ii) ker(XI5) is syndetic in S. Then ker(X) is syndetic in T. Proof. Let S’ = ker(X I 5). We have S = S’ + K for some compact subset K of S. For each t E T, let 3, be an element of S such that X(St) = X(t)'1. Then t+st+5’ C ker(X) (Indeed, X(t+st+5’) = X(t)-X(st)-X(5’) = X(t)-X(t)"l ° {1} = 39 {1}.) Thus ker(X) D U (t+st+S’). Now ker(X) + K D U (t+s¢+S’) + K = tET tET U (t + 3, + S’ + K) = U (t + S) = T. So ker (X) is syndetic in T. C] tET teT Corollary 1.6.7. Let T be a topological group which contains a connected SK sub- group S. Let X E T be such that S ¢ ker (X). Then ker (X) is syndetic in T. Proof. Conditions (i) and (ii) of the previous proposition are satisfied. D Corollary 1.6.8. Let T be an LCA group, To its connected component of identity. Let X E T be such that To ¢ ker (X). Then ker (X) is syndetic in T. Proof. To is connected and it is SK by Corollary 1.6.5. So the statement follows from Corollary 1.6.7. E] Example 1.6.9. Let T = R X Td. Then To = R X {0}. If c is the trivial character of R, then ker(c - raw) 2 R X {0} and this is not a syndetic subgroup of T. For any other character X E R, ker(X - id’r) is a syndetic subgroup of T by Corollary 1.6.8. (For notation X1 - X2 see 0.2.4.) Remark 1.6.10. Abelian SK groups in a natural way generalize minimally almost periodic groups (which are never LCA unless trivial), but also contain connected LCA groups. The fact that in recent years it has become clear that extremely amenable (and minimally almost periodic) groups are not “exotic” ([32]), can give some impor- tance to SK groups. In connection with this, let us mention that it is not known if minimally almost periodic groups are extremely amenable ([31]) even in the case of monothetic groups. It is proved in [18, Theorem 3.3], that an example of a polish minimally almost periodic group, which is not extremely amenable, would solve in 40 the negative the old problem from combinatorial number theory and harmonic anal— ysis, asking if the set S — S, where S is a syndetic subset of Z, is big enough to be a neighborhood of 0 in the Bohr topology on Z. (Recall that the Bohr topology on an abelian topological group T is the weakest topology on T in which all originally continuous characters of T remain continuous.) Let us also mention that an abelian topological group T is extremely amenable iff every compact minimal T-flow is trivial, iff the universal compact minimal T-flow MT 2 (T, MT) is trivial. (Recall that for every topological group T, the universal compact minimal T-flow is defined as a compact minimal T—flow MT 2 (T, MT) such that for every compact minimal T-flow X = (T,X) there exists a morphism of flows of MT onto X. It is well-known that MT exists and is unique, see [2, p.115-117], [13, p.61-62], [38, IV(3.27)], and also [37, Appendix]. It is shown in [37] that MT is not 3-transitive.) Obviously, if a topological group T admits at least one compact minimal non-totally-minimal flow, then M T is not totally minimal. 1.7 Total minimality of X in terms of the structure group I‘(X) Remark 1.7.1. Combining 0.2.40 and 0.2.15, for compact minimal abelian flows we have: X is weakly mixing iff every eigenvalue of X is trivial. (Another characterization of weakly mixing compact minimal abelian flows was recently given in [3].) Proposition 1.7.2. Let X = (T,X) be a compact minimal abelian flow and suppose 41 that T is an SK group. Then X is totally minimal ifl X is weakly mixing. Proof. From 1.7.1 and the criterion for total minimality we conclude that if a compact minimal abelian flow is weakly mixing, it is totally minimal. Suppose now that X is totally minimal. By the criterion for total minimality, X has no nontrivial eigen- value whose kernel is syndetic. Since T is SK, this implies that X has no nontrivial eigenvalue at all. By Remark 1.7.1 X is weakly mixing. [:1 Remark 1.7.3. Let X be a compact minimal abelian flow. The direction “X weakly mixing implies X totally minimal” is true without T being SK (see [24, p.480] for another proof). According to [24, p.480], the equivalence “X is weakly mixing iff X is totally minimal” was first proved by N. Markley for T = R. Proposition 1.7.2 extends this result since R is SK. Remark 1.7.4. If T is not SK, Proposition 1.7.2 is not true in general. Consider for example a compact minimal (almost periodic) Z-flow X = (Z,T,7r), defined by 71(n, z) = e2"‘9"z, for n E Z and z E T, where (9 E R is irrational. This flow is totally minimal. (Follows from 1.8.1 below, but it is also easy to check directly.) However this flow is not weakly mixing. (Follows easily from the definition of weak mixing.) More generally, any nontrivial compact minimal Z-flow X on a connected space X, which satisfies S} # X X X, is totally minimal but not weakly mixing. (Total minimality follows from 1.8.1 below. Weak mixing follows from 0.2.39.) Corollary 1.7.5. Let X = (T,X) be a compact minimal abelian flow and suppose that T is SK. Then X is totally minimal if and only if the structure group F(X) is trivial. 42 Proof. For compact minimal abelian flows, X is weakly mixing if and only if S} = X x X (0.2.39). Hence (by Proposition 1.7.2) X is totally minimal iff S; = X X X. Finally, since X is minimal, S} = X X X iff I‘(X) is trivial. (Indeed, if S} = X X X, then clearly P(X) is trivial. Conversely, if F(X) is trivial, E(X/Sf-r) = {idX/5ft}. So all elements of T fix every element of X / S}. This means that for every equivalence class C C X of the relation S3,, t.C C C for all t E T. Thus C is a (closed) invariant subset of X under T. Since X is minimal, there is only one equivalence class; i.e. F(X) is trivial.) El Remark 1.7.6. Thus the class of abelian topological groups for which the total minimality of every compact minimal abelian flow X is equivalent with the triviality of F (X ), includes abelian SK groups. (It would be interesting to characterize this class.) In the case that T is a connected LCA group, the previous corollary was stated by Gottschalk ([21, p.56]). Since, by Corollary 1.6.5, connected LCA groups are SK, we have a larger class of acting groups for which the statement holds. 1.8 Compact minimal abelian flows that are not totally minimal 1.8.1. Although total minimality is a strong condition, there are many examples of totally minimal flows. For example, the following statement holds ([22, 228]): every minimal flow X = (T,X), with T discrete and X connected, is totally minimal. Indeed, if S is a syndetic normal subgroup of T, then T = F S 2 SF for some 43 finite set F c T. Let x e X. Then (using 0.2.21, 0.2.23 and 0.2.24) X = T3 = FS.x = FEST; = U LS}, where F1 is some subset of F and the union is disjoint. tEF1 Since X is connected, it cannot be a finite union of > 1 disjoint closed sets. So X 2 ST. Since this holds for any x E X, X is totally minimal. 1.8.2. In [16, p.36] a family of examples of minimal Z—flows on T" (n 2 1 any integer) is given. By 1.8.1, these flows are necessarily totally minimal. (See also [38, III(1.18)— III(1.20)] and [14].) 1.8.3. There exists a minimal continuous R—flow on T2, with no nontrivial continuous eigenvalue ([25]). By 1.7.1(ii) and (iv), this flow is necessarily totally minimal. 1.8.4. Note that by 1.8.1 and Corollary 1.2.5, every nontrivial eigenvalue X ,\ = 62“.”) of a compact minimal Z—flow on a connected space X, satisfies /\ E Q. (But not every such flow has a nontrivial eigenvalue. However, it is proved in [14, Theorem 5.1] that every minimal Z—flow on T”, X = (T2,h), such that the homeomorphism h is not homotopic to the identity transformation, has a nontrivial eigenvalue.) We will now give some conditions on compact minimal abelian flows which necessarily imply non—total—minimality. Proposition 1.8.5. Let X = (T, X,7r) be a compact minimal abelian flow. Then in each of the following situations X is not totally minimal: (i) X almost periodic, X non-connected; (ii) X almost periodic, T contains a connected SK subgroup which acts nontrivially on X; 44 (iii) X proximally equicontinuous, T contains a connected SK subgroup which acts nontrivially on X; (iv) X/Sfp non-connected; (v) X distal, X totally disconnected, IXI > 1; (vi) X point-distal, T SK, X metric, IXI > 1; (vii) X regularly almost perodic at at least one point, T contains a connected SK subgroup which acts nontrivially on X, X metric. Proof. (i) Follows from Example 1.1.6 and the criterion for total minimality. (ii) Let S be a connected SK subgroup of T and a E X, and suppose that IS.aI > 1. By 0.2.30, X has a compact abelian group structure such that a is the identity element and the orbital map 71“ : T ——> X is a continuous group homomorphism and rr°(T) 2 TE 2 X. Since ”STE is a nontrivial closed connected subgroup of X, there is a surjective continuous character of 3:, f0 : m —-> T. Let f be a continuous character of X which extends f0. Define X E T by X = f 0 7r“. Since 7r°(S) = S.a is a nontrivial connected subgroup of X, which is dense in —S.—a, X(S) = f (71“(S )) is a nontrivial connected subgroup of T. Hence X(S) = T. By Corollary 1.6.7, ker (X) is a syndetic subgroup of T. Since f(ker (X).a) = (f o rr")(ker(X)) = {1}, we have f(ker(X).a) = {1}. Ift E ker(X)“, then X(t) = f(t.a) E f(ker(X).a) : {1}. Hence t E ker (X). Thus ker(X)“I C ker(X). Hence ker (X)* = ker (X)a = ker (X). So ker (X) is a proper enveloped subgroup of T. Now by the criterion for total minimality ((i) 4:) (iii)) X is not totally minimal. (iii) Let S be a connected SK subgroup of T and a E X, and suppose that 45 IS.aI > 1. Since X is proximally equicontinuous, Pg 2 S; (0.2.38). Consider the almost periodic flow (T, X / 3}) This flow satisfies the conditions of (ii). Indeed, if b 79 a is an element of S.a, then a and b are distal (0.2.34). Hence the images 51. and bof a and b in X/S} are distinct points, and b E Sii. Now by (ii), (T, X/Sfp) is not totally minimal. Consequently X is not totally minimal. (iv) The flow (T, X / S3.) is almost periodic with a non-connected phase space. By (i), (T, X / S3,) is not totally minimal. Consequently X is not totally minimal. (v) Since X is distal and [X] > 1, S} # X X X (0.2.35). The canonical map 1p : X —+ X / S3, is not only closed, but also open (0.2.36). Since X is totally disconnected, its image X / S ff under a continuous clopen map is totally disconnected. In particular, X / S} (having more than one element) is not connected. By (iv), X is not totally minimal. (vi) Follows from Proposition 1.7.2 and 0.2.41. (vii) By [22, 5.24] X is locally almost periodic. Hence it is proximally equicontin- uous. Now by (iii) X is not totally minimal. C] Remark 1.8.6. (i) The statement 1.8.5(ii) was first proved in the case T = R by E. E. Floyd (see [22, 4.55 and 487]). It was generalized by H. Chu to non-totally— disconnected LCA groups with the connected component of the identity acting non- trivially on X ([6]) We extend the class of acting groups for which the statement is true. Also our proof is simpler than that in [6] (no need for Pontryagin’s duality theory). 46 (ii) We could finish the proof of 1.8.5(ii) in a different way, by showing that X is a nontrivial eigenvalue of X whose kernel is a syndetic subgroup of T. For this purpose it remains to show that f(t.x) = X(t)f(x) for all (t, x) E T X X. Denote the operation in X by * . We have (t1 + t2).a = 71°(t1 + t2) = 71"(t1) =1: rr“(t2) 2 ho * t2.a for any t1,t2 E T. For any x E X there is a net tXa —> x. Hence for any t E T, t.a * x = t.a =1: (limtXa) = lim(t.a * tXa) = lim (t + tX).a = limt.(t,\.a) = t.(limt)‘.a) = t.x. So f(t.x) = f(t.a * x) = f(t.a)f(x) = X(t)f(x) for any (t,x) E T X X. Now we use the criterion for total minimality ((i) 4:) (ii)). (iii) The statement 1.8.5(iii) for X locally almost periodic (hence proximally equicontinuous) and T non—totally-disconnected LCA group with the connected com- ponent of identity acting nontrivially on X, was proved in ([7, p.380]). We extend the class of acting groups and replace ”locally almost periodic” by a weaker condition ”proximally equicontinuous”. The part of the proof in which the statement (iii) is reduced to the statement (ii) follows [7]. The proofs of (ii) are different. (iv) The statement 1.8.5(v) was proved in [24, 3.2] as an application of a cri- terion for weak mixing that was formulated and proved there. Although 1.8.5(v) implies 1.8.5(iv), we stated both of them since the proof of 1.8.5(v) reduces to the proof of 1.8.5(iv). (v) A complete characterization of flows which satisfy 1.8.5(vii), with T = R, in terms of their eigenvalues, is given in [10, Theorem 2]. Remark 1.8.7. Note that proximal compact minimal abelian flows are trivial [38, IV(2.18)], in particular totally minimal. 47 Chapter 2 Almost periodicity of a point under various constructions 2.1 The notion of a skew-morphism of flows Definition 2.1.1. Let X = (T, X), y = (T’, Y) be two flows. A pair of maps (h, f), where h : T —> T’ is a continuous group homomorphism and f : X —1 Y is a continuous map, is called a skew-morphism of flows if f(tiv) = h(1)f(il7) for alltE T and all x E X. We write (h,f) : X —> y. A skew-morphism (h, f) is called a skew-isomorphism if h is an isomorphism of topological groups and f is a homeomorphism. Example 2.1.2. Let X = (T,X), y = (T, Y) be two flows with the same acting group T and let f : X —> Y be a morphism of flows. Then (idT,f) : X -—> y is a 48 skew—morphism. Also if X; = (Td, X), then (idT, idX) : X; ——+ X is a skew-morphism (but not necessarily a skew-isomorphism). Example 2.1.3. Let X = (T,X) be a flow, f : X -—> T be an eigenfunction of X and X E T the corresponding eigenvalue. Let T = (T,T) be the flow defined by the action of the unit circle T on itself by multiplication. Then (f,X) : X ——> T is a skew-morphism. Example 2.1.4. Let X = (T,X), y = (T’,Y) be two flows, (h,f) : X —> y a skew-morphism, y E Y, x E f‘l(y). Since f(Tx) C T’y, we have f(TE) C T—’y. Let f1 : T; —> T—’y be the restriction of f to these sets. Let X’ = (T, fl) and y’ = (T’,T’y) be the canonical flows. Then (h, f1) : X’ —> y’ is a skew-morphism of flows. Example 2.1.5. Let X = (T, X,7r) be a flow, S a normal subgroup of T, x E X, t E T. Consider the canonical flows y = (S, SE) and Z = (S, m) Notice that .517: :15. Let h = Int, : S —> S, h(s) =tst-1, and let f = 71th —> X, 71¢(x) = tx. Then (h, f) = (Inthrrt) : )7 -—> Z is a skew-isomorphism of flows. In T is abelian, Int, 2 id5, so we have a skew-isomorphism (id5, 71¢) : ST —> Stx. Example 2.1.6. Let X = (T, X, 71) be a compact minimal abelian flow, S a syndetic subgroup of T. The orbit-closures under S form a partition of X. Let R be the equivalence relation on X defined in that way, X = X/R, px : X —> X/R the canonical map. For x E X denote by x the element pX(x) of X. Let S‘ be the X- envelope of S, p1 : T —) T/S* the canonical homomorphism. The function fr : T/ S " X X/R —+ X/R, given by ir(t + S‘,:i:) = t2, defines a flow X = (T/S*,X/R,ir) (follows 49 ~ from the proof of the criterion for minimality of restrictions). Then (pp, p x) : X —-> X is a skew-morphism of flows. Proposition 2.1.7. Let X = (T,X), )1 = (T’,Y) be two flows, (h,f) : X —+ y a skew-morphism. (i) Ifh is surjective, then f(X) is an invariant subset of y (and hence (T’,f(X)) is a subflow of 32). (ii) If X is minimal and f is surjective, then y is minimal. (iii) IfX is totally minimal, h, f are both surjective and h has the compact-covering property, then y is totally minimal. Proof. (i) and (ii) are easy. (iii) Fix a syndetic subset S’ of T’ and an element y E Y. By Remark 0.2.8, S = h‘1(S’) is a syndetic subset of T. Let x E f’1(y). Then S; = X. Hence: S—’y_ = 11(5),, = h(S)f(x) = f(Sx) 3 f(fi) = f(X) = Y. So )2 is totally minimal. 13 2.2 Almost periodicity of a point under various constructions Proposition 2.2.1. Let X = (T,X) be a flow, x E X. Let Y be an invariant subset ofX which contains x and let )1 = (T, Y) be the subflow ofX on Y. Then x is almost periodic in X if and only if x is almost periodic in y. Proof. Follows from the definition. Cl 50 Remark 2.2.2. Let X = (T,X), y = (T’, Y) be two flows, (h,f) : X —> y a skew- isomorphism, x E X, y = f (x) Then x is almost periodic in X if and only if y is almost periodic in y. Proposition 2.2.3 ([2],[13],[22],[38](for morphisms)). Let X = (T,X), y = (T’, Y) be two flows, (h, f) : X —> y a skew-morphism with h surjective. Let x E X, y = f(x). Then ifx is almost periodic in X, y is almost periodic in y. Proof. Let V be a neighborhood of y and let U be a neighborhood of x such that f(U) C V. Let S be a syndetic subset ofT such that Sx C U. Then, from f(Sx) C V, h(S)y C V. Also T’ = h(T) = h(KS) = h(K)h(S). Since h(K) is compact, h(S) is syndetic. Thus y is almost periodic. [:1 Remark 2.2.4. The above proof is the same as the proof in case of morphisms. The next three propositions however illustrate how sometimes, using skew—morphisms, we can easily get simpler and more natural proofs of known statements, as well as new statements. Proposition 2.2.5 ([2, page 13]). Let X = (T, X, 71) be aflow, S a normal subgroup of T, X5 = (S, X) a restriction of X, x E X. Then ifx is almost periodic in X5, every tx, t E T, is almost periodic in X5. (In particular, if x is almost periodic in X, every point tx, t E T, is almost periodic in X.) Proof. Fix t E T. Consider the canonical flows y = (S, E) and Z = (S, E13.) By Example 2.1.5 and Proposition 2.2.1 we have: x is almost periodic in X5 <=> x is almost periodic in y (i) tx is almost periodic in Z (:1 tx is almost periodic in X 5. E] 51 Proposition 2.2.6. Let X = (T, X,71), )7 = (T, Y, p) be two flows with the same acting group T, r, s e T. Consider a continuous group homomorphism h : T —> TXT, given by h(t) = (rtr-1,sts-1) = (Int.(t),Int,(t)). Suppose that the subgroup h(T) of T x T has the topology induced from T x T and consider the flow 2 = (h(T), X x Y), defined by (t17t2)($7y)=(W(tlr$)P(tziy))=(1519371231), where (11,12) 6 h(T) and (x,y) e X x Y. Then a point (x,y) is almost periodic m X X y if and only if (rx, sy) is almost periodic in Z. Proof. (h,7rt X p,) : X X y ——> Z is a skew-isomorphism (by 0.2.2 and a routine checking) and (71, X p,)(x,y) = (rx,sy). [:1 Corollary 2.2.7. Let X = (T,X), )1 = (T, Y) be two abelian flows with the same acting group T. (i) If a point (x, y) is almost periodic in X X y, then every point (rx, sy), r, s E T, is almost periodic in X X y. (ii) If a point x is almost periodic in X, then every point (rx,sx), r,s E T, is almost periodic in X X X. Proof. (i) The diagonal of T X T can be identified with T. (ii) x is almost periodic in X if and only if (x, x) is almost periodic in X X X, so (ii) follows from (i). [:1 Remark 2.2.8. The statement (i) is used in [1]. The statement (ii) is Lemma 8 from [8]. 52 Proposition 2.2.9. Let X = (T,X) be a flow, f : X —> X an endomorphism of X. Then ifx is almost periodic in X, (x, f(x)) is almost periodic in X X X. Proof. Consider the subflow y = (T, Gr(f)) ofX X X. Let g : X —> Gr(f) be given by g(x) = (x, f(x)). Then (idT,g) : X —> y is a skew-isomorphism (using 0.2.2 and the assumption that f is a morphism). Hence, since x is almost periodic in X, (x, f (x)) is almost periodic in y. By Proposition 2.2.1, (x, f(x)) is almost periodic in X X X. C] Remark 2.2.10. Note that, using Proposition 2.2.9, we can again deduce (ii) from Corollary 2.2.7, if we observe that in the case of an abelian flow X all transition homeomorphisms x t—> tx are endomorphisms of X. Proposition 2.2.11 ([2I,[13I,[22I,[38I). Let X = (T,X) be a compact flow. Then: (i) a point x E X is almost periodic if and only if it is discretely almost periodic; (ii) X is pointwise almost periodic if and only if every orbit closure in X is min- imal; (iii) if X is minimal, every point x E X is almost periodic; (iv) there is at least one almost periodic point of X; (v) let S be a syndetic normal subgroup of T, X5 = (S,X) a restriction of X, x E X; then x is almost periodic in X if and only if x is almost periodic in X 5. Remark 2.2.12. All statements from Proposition 2.2.11 can be easily proved us- ing 0.2.17 and the natural connection 0.2.28 between almost periodicity of a point and minimality in the case of compact flows. Proposition 2.2.13 ([12], [38, II(7.10)] (for morphisms)). Let X : (T,X), y = (T’, Y) be two compact flows, (h, f) : X —-) y a skew-morphism with h surjective. 53 Let y E Y be an almost periodic point of )7. Then the set f "1(y) contains an almost periodic point of X. Proof. Let N 2 TE. This is a minimal subset of Y by Proposition 2.2.11. The set f‘1(N) is a nonempty closed invariant subset of X. Also f‘1(N) is compact. Hence (by 0.2.17) f‘1(N) contains a minimal subset M. Then f(M) is a closed nonempty invariant subset of N. Hence f(M) = N. In particular, there is a point x E M such that f(x) = y. Since we must have T; = M, x is almost periodic by Proposition 2.2.11. [:1 Remark 2.2.14. The above proof is the same as the proof in the case of morphisms. 2.3 The notion of a skew-morphism good over a point with respect to orbit-closures Definition 2.3.1. Let X and Y be topological spaces, y E Y. A continuous map f : X —> Y, is said to be good over y if the fiber f"1(y) = {:L'; I i E I} is nonempty finite and if given neighborhoods U.- of x,-, i E I, there exist neighborhoods W,- of x,, i E I, and V of y, such that: (G1) W,- C U,, i E I; (G2) ifiyéj then VIC-OW,- =0, i,j E I; (G3) RU... W.) = v; (G4) f"(V) = UrerWr. Example 2.3.2. Any homeomorphism f : X —> Y is good over any y E Y. More 54 generally, if X is a topological space and F a finite (discrete) space, then prl : X X F —> X is good over any x E X. Remark 2.3.3. Let f"1(y) = {x1,-~ ,xn} and suppose that there exist neighbor- hoods U, ofx;, i = 1,--- ,n, and V ofy, so that each f: U,- —> V, i = 1,--- ,n, is a homeomorphism. Still f is not necessarily good over y. (Consider the subsets of R”: X = {(a,b)I — 1 S a _<_ 1, b E {0,1}} \ {(0,1)}, Y = {(a,0)I — 1 S a S 1}, the map f 2 pr,, and the point y = (0,0).) Proposition 2.3.4. Let X and Y be compact spaces, f : X —+ Y a surjective con- tinuous map, y E Y. Then if the fiber f‘1(y) is finite, f is good over y. Proof. Let f“(y) = {x1,-~ ,x,,} and let U,- be an open neighborhood of x,, i = 1, - - - , n. We may asssume that the U,- are pairwise disjoint. The set X’ = X\UL, U, is compact. For any point z E X’ choose disjoint open neighborhoods 0; of y and 0 of f(z). Then Az = f‘1(0) and B2 = f"l(0,) are disjoint open neighborhoods of f‘l(f(z)) and f“(y) respectively. The set X’ is covered by UzEX’ Az, so there are finitely many points 21,. ~ ,zk e X’ such that X’ c UL. A2,. Consider (73;, 3,, That’s a saturated (with respect to f) open neighborhood of f ‘1 (y) (as an intersection of saturated neighborhoods). Also (flL, sz)n(UL, A2,.) = 0. Since UL, Az, D X \ U?=r 11,-, we have (7;, 3., c UL. 11,-. Put W,- = (0;, B,,)nU,-, i = 1,... ,n. Now UL, W, 2 (IL, Bz, = 0L, f‘1(0z,) = f‘l(flL, 02,). Since f is surjective, f(UL, W.) = f(flL, 82,.) = 0;, 0.,. Put v = nfz, o,,. The neighborhoods W,- i = 1, - -- , n, and V satisfy the conditions (G1)—(G4). E] Remark 2.3.5. If the condition (G3) from the previous definition is replaced by 55 (G3’) f(W,-) = V for all i E I, (and if the fiber f‘1(y) is not necessarily finite), f is said to be locally surjective over y. This notion was considered in [35]. Other than this definition, the line of investigation we pursue has no connections with this paper. The map f from the previous proposition is not necessarily locally surjective over y. (Consider the subsets of R”: X = {(a,0)I — 1 S a S l} U {(0,1)}, Y = {(a,0)I — 1 S a S 1}, the map f : pr,, and the point y = (030).) Definition 2.3.6. Let X = (T,X), y = (T’, Y) be two flows. A skew-morphism (h, f) : X -—> y is said to be good over y with respect to orbit closures if the following two conditions hold: (GR) for any x E f‘1(y), the restriction f1 : T; —+ TE of f is good over y; (CC) for any x,x’ E f"1(y), x’ E T? implies x E fl. A morphism f : X —> y of flows X = (T,X) and y = (T, Y) is said to be good over a point y E Y with respect to orbit closures if the skew-morphism (idT, f) : X —) y is good over y with respect to orbit closures. Example 2.3.7. If (h,f) : X —) y is a skew-isomorphism of flows X = (T,X) and y = (T’, Y), then for any y E Y, (h, f) is good over y with respect to orbit closures. Example 2.3.8. Let X = (T,X) be a flow and let X, = (Td,X). Let (idT, f) : X; —> X be a skew-morphism with f a homeomorphism. Then for any y E Y, (h, f) is good over y with respect to orbit closures. Example 2.3.9. More generally than in the previous example, let X = (T,X) and y = (T’, Y) be two flows, (h,f) : X —> y a skew—morphism with h surjective and 56 f a homeomorphism. Then for every y E Y, (h, f) is good over y with respect to orbit—closures. (Indeed, let f‘1(y) = {x}. Since f is a homeomorphism, f(T—x) is a closed subset of Y, hence of TE. Since it contains a dense subset T’ y of TE, we have f (TE) 2 T77. So f1 : T; —> T73] is a homeomorphism. Hence (GR) holds. Also (OC) holds since each fiber has exactly one element.) 2.4 Examples of skew-morphisms good over a point with respect to orbit-closures Proposition 2.4.1. Let X = (T, X), y = (T’,Y) be two flows, (h,f) : X —> y a skew-morphism with h surjective. Suppose that (X, f) is a covering of Y whose all fibers are finite. Let y E Y. Suppose that each deck-transformation of (X, f) is an automorphism of the flow X and that the group of deck-transformations of (X, f) is transitive on f‘1(y). Then (h, f) is good over y with respect to orbit closures. Proof. Fix any x E f‘1(y). Consider the restriction f1 : E ——> T—’y of f. Let’s check that fl is surjective. Indeed, let y’ E m. Suppose to the contrary, i.e. f‘1(y’) fl T—x~ = 0. Let f'1(y’) 2 {xi- I i E I}. Take an elementary neighborhood U,’ of each of the elements xI-(i E I). We may assume that all of them are disjoint from W. Let V’ be the corresponding elementary neighborhood of y’. There is an element t’ y E V’ (since y’ E T—’y). Let t E T be such that h(t) = t’ and consider tx. We have f(tx) = t’y. Hence tx E U,’ for some i, a contradiction. Thus fl is surjective. 57 Let f,’1(y) 2 {x = x1,x2, - -- ,xn} and let U, be a neighborhood of x,- in fl (i = 1, - -- , n). There are elementary neighborhoods W,’ of these points which all corre- spond to the same elementary neighborhood V’ of y and are such that W, = WIDT—x- C U,. [Here we use finiteness of the fiber f—1(g).] Let V = V’ 11 T6. We want to show that these W]- and V satisfy (G1)-(G4) (in that way the condition (GR) for (h,f) will be checked). Let x’ E W,. Then f(x’) E f(W,) C f(VVg’) = V’. Also there is a net tax —> x’. Hence f(tax) —> f(x’), i.e. h(ta)y ——> f(x’). Hence f(x’) e T15. Thus f(x’) E V’ (WT—’y = V. So f(Wg) C V for i = 1,-~ ,n. Let now b E V. Since fl is surjective, there is an a e T? such that f1(a) = b. This a must belong to f‘1(V’), hence to the one of WY, hence to the one of 147,. Let’s check (OC). Observe that for every g E A and x’, x” E X, x” E W im- plies gx” e W since g is an automorphism of X. Consider r,- e f-'(y) 1) Ti. Let g E A be such that gx = x,-. From gx E T—x we have (using the observation) g2x 6 Tot? c E. Then g3x e Tim—r c T7,}, etc. Since all elements r,gr,g2r,.-. are in the finite set f‘1(y), there is a smallest n 2 1 such that g"x = x. We have fl=T—g"x—CTgTa—:C CTEECfi. Hencefi=TE=Tx_,-. Hence (OC) holds. [:1 Lemma 2.4.2. Let X = (T,X), y = (T’,Y) be two flows, let x E X, y E Y and suppose that X = T—x, Y 2 TE. Let (h,f) : X —-) y be a skew—morphism with h surjective and f (x) = y. Suppose that y has a neighborhood V such that K :2 f‘1(V) is compact. Then the restriction f’ : K —> V of f is surjective. In particular, V is compact. 58 a Proof. Let z E V. Since T’y fl Int(V) is dense in V, there is a net t’ y —> z in Int(V). For each t; let to, be an element of T such that h(ta) = t;. Since f(tax) = tgy E V, tax E K. Since K is compact, there is a convergent subnet tgx. Let th —> w E K. Then f(tflx) ——+ f(w), i.e. tby ——) f(w). Hence f(w) = 2. Cl Proposition 2.4.3. Let X = (T,X), y = (T’,Y) be two flows, (h,f) : X —-> )7 a skew-morphism with h surjective. Suppose that whenever x1, x2 E X are in the same fiber, their orbit-closures are either equal to each other or disjoint. Let y be a point of Y which has a neighborhood V such that f"1(V) is compact and let x E f’1(y) be such that fin f'1(y) is finite. Let f’ : E —> TTy be the restriction off and let X’ = (T, TE) and y’ = (T’,T’_y) be the canonical flows. Then (h,f’) : X’ ——> 37’ is good over y with respect to orbit closures. Proof. Let f‘1(y) (1 Ta? = {x = x1, x2, . -- ,xn}. Since, by assumption, T; = W for i = 1, 2, - - - , n, (h, f’) is good over y with respect to orbit closures iff f’ is good over y. The set V’ = V D m is a neighborhood of y in TE. Note that for every subset of m its closures with respect to T—’y and with respect to Y are the same. Since V7 C Vfl TE, f’—1(V) C f'1(V). Since f’_1(W) is closed in T}, it is closed in X. Hence it is compact. Let U.- be a neighborhood of x.- in E (i = 1,2,--- ,n). Denote K = f’-1(V). We may assume that U.- C K for all i since K is a neighborhood of f”‘1(y) in T—x-. The restriction f” : K —> V7 of f’ is surjective by Lemma 2.4.2. Hence, by Proposition 2.3.4, f” is good over y. So there are open neighborhoods W; C U.- of x, in K and V” of y in V7, which satisfy (G1)—(G4). They are at the same time 59 neighborhoods in T; and in T73]. So f’ is good over y and consequently (h, f’) is good over y with respect to orbit closures. [:1 Proposition 2.4.4. Let X = (T, X) be afiow all of whose orbit closures are compact (for example a compact flow) and let y = (T’, Y) be a compact flow. Let (h, f) : X —> y be a skew-morphism with h surjective and f locally injective. Let y E Y be a point with a nonempty fiber. Then if y is almost periodic in y, (h,f) is good over y with respect to orbit-closures. Proof. Suppose that y is almost—periodic in y. Let x E f‘l(y). Since y is a compact flow, T—'y is minimal. Hence the restriction f1 : E —> T73; of f is surjective. For each point z E T; we can choose an open neighborhood Oz of z in 27.1.: such that fl is injective on Oz. Since Ta: is compact there are finitely many points z], 22, - - - . 2,, such that 0;, U - - - U 02,, covers T—x-. Each of these sets can contain at most one element from ffl(y). Hence ff1(y) is finite. By Proposition 2.3.4, fl is good over y. So the condition (GR) is satisfied. Let x’ be another point from f ‘1(y) and suppose x’ E Tit. Suppose that x E T337. Let ffl(y) = {x 2 x1,x2,--- ,x’ :- xm,xm+1, - - - ,xn}.Without loss of generality we may assume that T170 ffl(y) = (23",, xm+1, - -- ,xn}. Using compactness of E and the fact that fl is good over y, we can find open pairwise disjoint neighborhoods W,- of 13,-, i 2 1, 2, . - - , n, and V of y, so that at the same time the conditions (G1)—(G4) are satisfied, fl is injective on each of W,, i = 1,2, - -- , n, and T—x’ is disjoint from every W, i = 1,2,~~- ,m — 1. Let S’ = D(y,V). Then by Proposition 2.2.11 T’ = F’S’, where F’ is a finite subset of T’. Hence by Lemma 0.2.7, T = Fh"1(S’), where F is a 60 finite and S' = h‘1(S’) a syndetic subset of T. There is a net tasaxl —> xm with to E F and so, E S. The net (to) in F has a convergent subnet t3 -—> t. Since thgfIII —+ xm, we have ts/gxl ——) xm. Hence 352:1 —-> t‘lxm. Since f(sBxl) = h(sg)y E V, 33131 E UL, IV.- = f”1(V). At the same time t'lxm E Tx_’. Since T737 is disjoint from each W,- for i = 1,2,--- ,m — 1, we have that for E 2 ,80 (for some fig) all 33x1 are in UL", W5. Fix some 33:13] E Wj,j E {m,m+1,--- ,n}. For each i = m,m+1,-~- ,n, sax,- E U:___m Wp (must be in T? and in UL, W.- at the same time). So there are two of the points 33x1, sme, sme+1,- - - , sgxn in one of the sets Wm, - -- , W". The image under f1 of each of them is h(sg)y. Since fl is injective on each of Wm, - - - , Wn, these two points should be equal to each other, a contradiction. Hence x E W, i.e. the condition (CC) is satisfied. [:1 2.5 A criterion for lifting of almost periodicity of a point Theorem 2.5.1 (criterion for lifting of almost periodicity of a point). Let X = (T,X), y = (T’, Y) be two flows, (h,f) : X —-> y a skew-morphism with h surjective. Let y E Y be a point such that (h, f) is good over y with respect to orbit- closures and let x E f‘1 (y). Then y is almost periodic in y if and only ifx is almost periodic in X. Proof. (é) : Suppose y is almost periodic in y. The restriction f1 : T; —> T—’y of f is good over y. In particular the fiber f,_l(y) is finite. Let ff1(y) = {x = 61 x1,x2,-~ ,xn}. Fix any neighborhood U of x1 in W. Put U1 2 U and t1 = e. For each i E {2,3, - -- ,n} we have .731 E T; (since (h,f) is good over y with respect to orbit-closures). Hence for each i E {2,3, - -- ,n} there is an open neighborhood U.- of x; in Ti: and t,- E T such that thg C U. Choose open neighborhoods W; C Ug, i = 1, 2, - ~ . , n, of the points x,- in fl and an open neighborhood V of y in T—’y so that the conditions (G1)-(G4) are satisfied. By Lemma 0.2.19, there is a neighborhood V’ of y in T737 and a neighborhood 0 of the unit element er in T’, such that 0D(y, V’) C D(y,V). Also there is a compact K’ C T’ such that T’ = K’D(y,V’). We have A” C F’O for some finite subset F’ of T’. Thus T’ C F’0D(y, V’) C F’D(y, V) C T’, so T’ = F’D(y,V). By Lemma 0.2.7, there is a finite subset F of T such that T = Fh'1(D(y,V)) = FS, so S = h'1(D(y,V)) is syndetic in T. We have 5171 C UL, W,- (since for every 3 E S, f(sx1)= h(s)y E V). Let S.- = {s E S|sx1 E 147,-}, i : 1,2,--- ,n. Iffor s E S, sxl E W.- for some i = 1,2,--- ,n, then tgsxl E til/V.- C t,-U.' C U, hence for every 3 E S, s E S.- implies t,s E D(x1,U). Cosequently D(x1,U) 3 UL, t,S,-. Since S = UL, 5,, the set UL, th, is syndetic in T by Lemma 0.2.9. Hence D(xl, U) is syndetic and so x = x1 is almost periodic. (=>) : Follows from Proposition 2.2.3. D 62 2.6 Applications of the criterion for lifting of al- most periodicity of a point Corollary 2.6.1. Let X = (T,X) be afiow whose all orbit-closures are compact and let y = (T’,Y) be a compact flow. Let (h,f) : X —) y be a skew-morphism with h surjective and with f locally injective. Let y E Y be an almost periodic point in y with a nonempty fiber. Then every x E f"1(y) is an almost periodic point of X. Proof. By Proposition 2.4.4, (h, f) is good over y with respect to orbit—closures. Hence by Theorem 2.5.1, every x E f ‘1 (y) is an almost periodic point of X. [:1 Corollary 2.6.2 ([12, Proposition 3]). Let X = (T,X), y = (T, Y) be two com- pact flows and f : X —> y a surjective locally injective morphism. Let y be an almost periodic point of 3). Then every x E f_1(y) is an almost periodic point of X. Proof. Follows from Corollary 2.6.1. [:1 Corollary 2.6.3. Let X = (T, X), y = (T’, Y) be two flows, (h,f) : X ——> y a skew- morphism with h surjective and f a homeomorphism. Let y E Y and let x E f’1(y). Then y is almost periodic in y if and only if x is almost periodic in X. Proof. By Example 2.3.9, (h, f) is good over y with respect to orbit closures. So the statement follows from Theorem 2.5.1. [3 Corollary 2.6.4 ([30, Theorem]). Let X = (T,X) be a flow and x a point of X. Then x is almost periodic if and only if it is discretely almost periodic. 63 Proof. Consider a skew—morphism (idT,idx) : Xd -—) X, where Xd = (Td,X) and apply Corollary 2.6.3. Cl Corollary 2.6.5 ([26, Proposition 4.3] (with T = T’ and h = idT)). Let X = (T, X), y = (T’, Y) be two flows, (h,f) : X —> y a skew-morphism with h surjective. Suppose that whenever x1, :1:; E X are in the same fiber, their orbit-closures are either equal to each other or disjoint. Let y be a point of Y which has a neighborhood V such that f'1(V) is compact and let x E f‘1(y) be such that Tin f‘l(y) is finite. Then y is almost periodic in y if and only if x is almost periodic in X. Proof. Let f’ : T; —> m be the restriction of f, X’ = (T, fi), 37’ = (T’,—T_’y) the canonical flows. Then, by Proposition 2.4.3, (h,f’) : X’ —> y’ is good over y with respect to orbit-closures. Hence, by Theorem 2.5.1, y is almost periodic in 37’ iff x is almost periodic in X’. Also, by Lemma 2.4.2, y is almost periodic in )2 iff y is almost periodic in y’ and x is almost periodic in X iff x is almost periodic in X’. Thus y is almost periodic in y iff x is almost periodic in X. [:1 Corollary 2.6.6 ([27, Theorem 2.1] (with T = T’ and h = idT)). Let X = (T, X), y = (T’, Y) be two flows, (h, f) : X —+ y a skew-morphism with h surjective. Suppose that (X, f) is a covering of Y all of whose fibers are finite. Let y E Y and let x E f’1(y). Suppose that each deck-transformation of (X, f) is an automorphism of the flow X and that the group of deck-transformations of (X, f) is transitive on f ’1(y). Then y is almost periodic in y if and only if x is almost periodic in X. Proof. By Proposition 2.4.1, (h, f) is good over y with respect to orbit-closures. So the statement follows from Theorem 2.5.1. Cl 64 Bibliography [1] J. Auslander, On disjointness in topological dynamics and ergodic theory, Lecture Notes in Mathematics, Vol 729, Springer Verlag, New York, N.Y., 1979, p.28—33. [2] J. Auslander, Minimal flows and their extensions, North - Holland, Amsterdam, 1988. [3] J. Auslander, Weak mixing and pure weak mixing flows, Contemp. Math. 215(1998), 13-17. [4] G. Bredon, Topology and geometry, Springer - Verlag, New York, N.Y., 1993. 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