Vries, J. (Jan) de

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Centrum voor Wiskunde en lnformatica Centre for Mathematics and Computer Science J. de Vries A proof of the relativized, non-metric form of Furstenberg's structure theorem Department of Analysis, Algebra and Geometry Report AM-R9022 October The Centre for Mathematics and Computer Science is a research institute of the Stichting Mathematisch Centrum, which was founded on February 11 , 1946, as a nonprofit institution aiming at the promotion of mathematics, com-puter science, and their applications. It is sponsored by the Dutch Govern-ment through the Netherlands Organization for the Advancement of Research (N.W.O.). Copyright © Stichting Mathematisch Centrum, Amsterdam A Proof of the Relativized, Non-Metric Form of Furstenberg's Structure Theorem Jan de Vries Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands In this note we present a rather direct proof of the result mentioned in the title. It combines methods of McMahon, McMahon and Wu, Glasner and Furstenberg. No use is made of auxiliary topologies (-r-topologies, or Furstenberg topologies), but instead the notion ot a relatively invariant measure plays a cen-tral role. 1980 Mathematics Classification: 54H20 Keywords & phrases: extension of minimal flows, equicontinuous, distal, proximal, weakly mixing, relatively invariant measure. 1. INTRODUCTION In this paper we assume that the reader is familiar with the basic facts from abstract Topological Dynamics, as. e.g. exposed in the first chapter of [14]. Save a few exceptions, notation and terminol-ogy will be as in [14]. By a.flow we shall mean a topological transformation group (ttg) with a com-pact Hausdorff phase space, and instead of the term "homomorphism of compact ttg' s" we shall use "morphism of flows". For the convenience of the reader we recapitulate a few definitions. All flows are assumed to have phase group T, except in Section 3, where we also consider subgroups of T (in that case, we add phrases like" ... under H' or "H-.... "). A set Bin a flow .5r(re~ the flow .5ritscli) is said to be minimal wp.enever for every point x EA (resp. x EX) the equality Tx =A (resp. Tx = X) holds. An extension is a surjective morphism of flows. If cp:.51:~.fl'is an extension then Rq, (or R(cp)): = {(xi,x 2)EXXX: cp(x 1)=cp(x2)} is a non-empty closed invariant set in the flow .51:X.51: (i.e., the space XXX on which T acts coordinate-wise); the subflow of .51:X.51: on Rq, is denoted by Blq,. Note that an extension ip:~.fl'is an isomorphism (i.e., injective, hence a homeomorphism) iff Rq,~6.x: = {(x,x): x EX}. An extension cp:~.fl'is said to be equicontinuous (in [14] the term "almost periodic" is used) when-ever the action of Ton X is "equicontinuous relative Z", i.e., uniformly equicontinuous on fibers of cp and uniformly so with respect to the fibers; more precisely: V'a:E621 3/3 e621x : T/3 n R <1> <;;,a. Here 621x denotes the (unique) uniform structure generating the topology of X, and T/3 has to be understood in the flow ;rx,qr (each f3 E621x is a subset of X X X). We say that an extension cp:~.2" is weakly mixing whenever the flow Blq, is ergodic, i.e., TU is dense in Rq, for every non-empty open sub-set U of Rq, (in its relative topology in XXX, i.e., U= VnRq, with V open in XXX). It follows in a straigthforward way from these definitions that an extension cp:~.2" that is both weakly mixing and equicontinuous has Rq,=6.x, i.e., is an isomorphism. A pair of points (xi.x 2)EXXX, where .5ris a flow, is said to be proximal (to each other) whenever T(xi,x 2)n6.x=#=0 (here T(xi,x 2 ) has to be understood in ~El); it is called a distal pair whenever x 1 =x2 or (xi,x 2) is not proximal. An extension cp:~.fl'is called proximal (resp., distal) whenever each pair of points in a fiber of cp is proximal (resp. distal), i.e., each (xi.x 2)ERq, is a proximal (resp., distal) pair. Report AM-R9022 Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands 2 By a strictly-I-extension we mean a "transfinite composition" of equicontinuous extensions. For the technical definition we refer to [13], p. 800/801 (where this notion is called "I-extension") or to [14], definition 4.1 with in condition (b) the words "either proximal or" omitted. The general form of Furstenberg's Structure Theorem (FST) now reads: THEOREM (FST). Every distal extension of minimal flows is a strictly-I-extension. In this general form the result was proved in [11]. For earlier versions, see [6], [3], [7] and [5]. By using an extension of the techniques of [5], McMahon and Wu in [11] reduce the general FST to one of the key results of [4]. The proof of this result in [4] depends heavily on Ellis' algebraic approach, includ-ing various T-topologies. We shall use the construction of [11] in a different way and prove that every extension of minimal flows that is both distal and weakly mixing is an isomorphism: the "relativiza-tion" of the key result of [5]. Then an application of a construction from [8] reduces the proof of FST to the fact that an open RIM extension() that has no non-trivial equicontinuous factor is weakly mix-ing. We outline a proof of this fact, which is a simplification and clarification of the proof in [ 10] ( cf. also [14], VII 3.11). 2. RELATIVELY INVARIANT MEASURES An extension O:f!f..-7W is called a RIM-extension whenever there exists a continuous function A.: Y-rM 1 (X) (called a section for 0) such that (writing A.y for A.(y)): (a) V(t,y)ETX Y: A1y=tA.y, (b) VyEY: SuppA.v~O,__[Y]. Here M 1(X) is the.space of all probability measures on X and for p.EM1(X) and teT, tp. is defined by (tp.)(j): =p.(f), where f (x): = f(tx) for fEC(X) and x EX. We quote the following result from [8]. (2.1) THEOREM [8]. Let O:!!C-rW be an extension of minimal flows. Then there is a commuting diagram of the form O'J, J,O C[!f' -17 tf!I where a and 'T are proximal extensions and 0' is a RIM-extension of minimal flows. The flow !!C' is obtainted as subflow of !!CX W', namely on a minimal subset X' of the set RuT: = {(x,y)EXX Y': O(x) = T(y')} , and a and {)' are the restrictions to X' of the canonical projections of X X Y' onto X and Y', respectively. (2.2) REMARKS. 1. It is easy to verify that if in the above 0 is a distal extension then 0' is distal as weIL hence()' is an open mapping ([1], p. 142), i.e. 0' is an open RIM-extension. (By [14], VII. 1.10 one may always assume that()' is open, but that proof is much less elementary than the above). 2. If 0 is distal then (as (J is proximal) RuT is minimal ("relativize" the proof of [9], II. 1.3), hence X'=RoT· (2.3) We shall use sections of RIM-extensions to construct continuous invariant fibre-wise pseudo-metrics (CIFP's). Let O:!!C-rW be an extension of flows. A CIFP for 0 is a continuous mapping p:R 11--r!R + such that (a) \fy E Y: rlrtvJXO'"""[yJ is a pseudo-metric on O<-[Y ], (b) V(x1>x2)ERe 'rltET:p(txi.tx2)= p(xi,x 2). If p is a CIFP for 0 then D(p):={(x1,x2)ERe:P(xl>x2)=0} is a non-empty closed invariant (in the 3 flow flfX!?l) equivalence relation on X. (2.4) LEMMA. Let 0:!!£41[j/ be a RIM-extension and let A. be a section for 0. For every closed invariant subset N of Ro, put N[x ]: = {x' E!!t::(x,x') EN} (x EX) and PN(x1ix2): = AIJ(x,) ((N[x i] \ N[x2]) U (N[x2] \ N[x i])) for (X1iX2)ER1J. Then PN is a CIFP for 0. PROOF. It is elementary to prove that PN fulfills (a) and (b) of (2.3). That PN is continuous is impli-citly proved on p. 228 of [10] (alternative proof: "relativize" p. 128/120 of [l]). D We say that an extension 0:!!£41[j/ has no proper equicontinuous factors whenever in every factorization 0=82°01 with 81 an extension and 82 an equicontinuous extension we have that 82 is an isomorphism. (2.5) LEMMA. Let 0:!!£41[j/ be an extension without proper equicontinuous factors. Then every CIFP for () is zero on R 0. PROOF. D (p) is a closed invariant equivalence relation on X so one can form the flow !!£! D (p) on the quotient space X I D(p) such that the quotient map 'TT:!!f:.-7!!£/ D (p) is an extension of flows. It is not hard to see that p('TTXJ,'TTx 2): = p(x1ix2) ((xJ.x2)ERo) defines a CIFP p on R(j= (7TX'T1)[Ro] such that D(p)=!::.x1n(p)· By a straightforward compactness argument one shows that for every open aEG/.Lxm(p) there exists t:>O such that U,: = {(i1i:X2)ER(j:p(i1,i2).;;;;E} <;;;;a. As U, is a nbd of !::.xtD(p) in Re there is /3EGl1xiD(p) such that j3nR9<;,;;;U,. But TU=U, hence Tf3nR0<;,;;;U<:;;,a. So 0 is equicontinuous, hence 8 is an isomorphism, i.e., Ro=flx1ncp)· Hence R 0 =('7TX'7l').,_[R0]= ('TTX'TT).,_ [6.xm(p)]= D(p). D (2.6) LEMMA. Let 0:!1£,41[J/ be an open RIM-extension of minimal flows. If every CIFP for 0 is zero on R 0 then 0 is weakly mixing. PRooF. let U 1 and U2 be open subsets of X such that (U1 XU2)nR 0=f=0. We have show that N: = T( U 1 X U 2) n R 0 equals R 0, i.e., that (V 1 X V 2) n N =f= 0 for every pair of open sets V 1 and V 2 in X such that (V 1 X V2)nR 0=f=0. We may and shall assume that O[Ui]= O[U2] and O[Vi]= O[V2] [for i = 1,2, replace U; by U;nO<-[U], where U:=8[Ui]ITO[U2]=f=0 is open (because 8 is open); similar for V;]. As !!£is minimal there exists t 0 ET with W: = t 0 U2 n V2=f=0; select w EW such that w E Supp A.o(w)· [ Note that the set of points x EX with x E Supp ~x) is dense: pick any x 0 EX, x' E Supp Aocx,) c;,;;;8<-[8xo]; then 8(x')= 8(xo), so x' E Supp AIJ(x')' hence tx' E Supp t"Aocx') = Supp AIJ(tx')> and Tx' is dense.] By the choice of U; and V; (i=l,2) there are x 1 EV1 and x 2 Et0 U1 with 8(x 1)= O(w)= 8(x2). Then 0=¥=({x2} x W)nR 8 <:;;, t 0(U 1 x U2)nRo <;,;;; N, hence W n tJ<--[ Ox 2] ~ N [ x 2 ], and therefore (Wn8 ...... [8x 2 ]) \N[xi] <;,;;; N[x 2 ] \N[xi]. (1) (2) By (2.4) and the hypothesis of the lemma it follows that the right-hand side of (2) has measure zero under AIJ(x,)· As the left-hand side of (2) is open in rr-[8x 2 ] and the support of Ao(x,> is included in 8,_[8x 2] it follows that the left-hand side of (2) is disjoint from Supp Ao(x,)• so Wn Supp "Aecx,)<:;;,N[xi]. As wEWn Supp A.o(x,) [for8(x 2)=8(w)] it follows that (xi,w)EN. But also (xi,w)EV1 X V2, so (V1 X V2)nN:¥=0. D 4 (2.7) COROLLARY [10]. Let (}:~<!JI be an open RIM-extension of minimal flows. If() has no proper equicontinuous factors then (} is weakly mixing. PROOF. Clear from (2.5) and (2.6). D REMARK. It is one of the central problems in Topological Dynamics for which type of extensions of minimal flows the conclusion of (2.7) holds. See [14], VII. 3.17. The following theorem is a special case of this more general result. Its proof was motivated by Lemma 5.1 of [7]. (2.8) LEMMA. Let 8:!!l-7W be a distal extension of minimal flows. If(} has no proper equicontinuous fac-tors then (} is weakly mixing. PRooF. Consider the diagram in (2.1). Then (}' is an open RIM-extension (cf. (2.2)1 ) and (aXa)[R({}')]= Ro (an easy consequence of (2.2) 2). It is sufficient to show that every CIFP p' for (f factorises as p' =pa( a X a) with p:R 11 -7~ + : then p is a CIFP for (}, hence zero on R 9 by (2.5), so p' is zero on R 8., and (since this holds for every CIFP for (}') by (2.6), 8' is weakly mixing. As aX a:~(}')-7~0) is an extension of flows it follows that ~O) is ergodic, so (}is weakly mixing. It remains to show that if (x'1>x'2)ER((J') and (z'i,z'2)ER((J') with (aXa) (x'i,x'2)= (aXa)(z'1>z'2) then p'(x'"x'2)= p'(z'i.z'2) (this value then unambiguously defines p(ax'1>ax'2)). For i = 1,2, let xi:= a(x';)= a(z';), y':= (}'(x' 1)= (}'(x'2) and w':= (}'(z' 1)= (}'(z'2); then x';= (x;,y'), z';= (x;,w') and (y', w') ER n so (y', w') is a proximal pair. Hence there are a net {ta} a in T and a point y" E Y' such that tay'"""y" and taw'"""Y"· It follows that for i = 1,2, assuming that taX;"""X; in X, tax'; """(X;,y") and t aZ1;N't(X;,y''). So p'(x'1,X12)= p'(taX'1,laX12) """p'((X1>y"), lx2,y"))= :d, hence p'(x'1>x'2)=d. Similarly, p'(z'"z'2)=d and, indeed, p'(x'i.x'2)= p'(z'i,z'2)- D REMARK. The remarks in (2.2) hold also for so-called RIC-extensions. So the proof and the conclu-sion of (2.8) are valid in that case as well. Details will be published elsewhere. 3. THE CONSTRUCTION OF ELLIS/MCMAHON & Wu Aim of this section is to outline a proof of (3.1) THEOREM. Let 8:!!l-7W be a distal extension of minim,gl flows. If(} is weakly mixing then (/ is an isomorphism. Clearly, if we combine this theorem with (2.8) then we get: a distal extension of minimal flows that has no proper equicontinuous factors is an isomorphism. As by a transfinite construction one easily shows that every distal extension cp:!!l-7.2' of minimal flows factorizes as cp=tf;00 °800 with i/;00 a strictly-I-extension and 800 an extension without proper equicontinuous factors which is, in addition, distal (for R (()00 ) c;;;;R4'), the general FST easily follows. In order to prove (3.1) we quote from [11] the following result. (3.2) LEMMA. let (/:!!l-?<!J/ be an open extension of minimal flows and let p be a continuous pseudo-metric on !7£. Then for every countable subgroup H of T there exists a commutative diagram x -4 y 5 of continuous surjections with the following properties: 1. ~ and Y n are compact metric spaces, 2. 'Vy EY: an[tr-"y]= 0.li[TnY] 3. (anXan)[Ro]= R(On), 4. There exist continuous actions of Hon ~ and Y n such that an, On and Tn are extensions of H-fiows. D REMARK. Condition 3 is not in [ 11 ], but follows easily from equality 2. We do not need that an X an maps Q(O) onto Q(On) for suitable H. Note that (3.2) can be applied when 0 is distal (for distal implies open). (3.3) LEMMA. Let 0:!!£4lf!I be an open weakly mixing extension of minimal flows. Then for every count-able subgroup H of T there exists a countable subgroup K of T with K";d H such that OK:~4 Y K is a weakly mixing extension of K-minimal K-flows. PRooF. Apply the method of [5] in the context of the--eonstruction of (3.2). (To justify the method one has to observe that for an ascending sequence { Hn }n eN of countable subgroups of T and K: = U nHn one can obtain not only ~ as inverse limit of the spaces Xn •• but also R(OK) as the inverse limit of the spaces R (On).) D (3.4) LEMMA. Let B:!!£4lf!I be a distal extension of minimal flows. Then for every countable subgroup K of T such that Xi:- and YK are K-minmal OK:~4YK is a distal extension of minimal K-flows. PROOF. Let K be a countable subgroup of T, K;;JH, such that~ (hence YK) is minimal under K. In [11] only point-distality (and openness) of fh could be obtained. We obtain distality, as follows. In Y there exists a K-minimal closed subset Y'; let X:=B~[Y'] and B':=Olx<X74Y'. Then 0' is a distal extension of K-fiows, and as Y' is minimal under K' the set R (8') is a union of K-minimal subsets [ apply [9], H. 1.2 to (x'1>x'2)~0'(x' 1)= 0'(x'2):R(0')4Y']. As YK is minimal under K,TK maps Y' onto YK, so by condition 2 of (3.2), aKXak maps R(O') onto R(OK) and, consequently, R(OK) is a union of K-minimal sets. It follows easily that the (minimal!) orbit closure of any point in R (BK) is either included in or disjoint from Lix~. hence such a point is a K-distal pair. D (3.5) COROLLARY. Let 0:!!£4rJ.!/ be a distal and weakly mixing extension of minimal flows. Then for every countable subgroup Hof T there exists a countable subgroup K of T with K"d_H and BK an isomorphism. PROOF. By (3.2), (3.3) and (3.4) there is a countable subgroup K"d_H such that BK:~4YK is both a weakly mixing and a distal extension of metric K-minimal flows. But in the metrizable case it is easy and well-known that this implies that OK is an isomorphism [ briefly: R (BK) has a point with dense orbit (because R (BK) is second countable), and it is a union of minimal sets (use [9], II.1.2) ]. D (3.6) LEMMA. Let B:!!l--7>lf!I be an extension of minimal flows. If B is not an isomorphism than in (3.2) the pseudo-metric p can be chosen such that On is not an isomorphism for any coimtable subgroup H of T. PRooF. Cf. [11], the proof of remark (i) on p. 286. D (3.7) PROOF OF THEOREM (3.1): clear from (3.5) and (3.6). D AKNOWLEDGEMENTS. The author is indepted to Jaap van der W oude for some of the ideas in this paper. 6 REFERENCES [l] AUSLANDER, J.: Minima/flows and their extensions, Amsterdam, 1988. [2] BRONSTEIN, I.U.: Extensions of minimal transformation groups, Alphen aan den Rijn, 1979. [3] ELLIS, R.: Lectures on topological dynamics, New York, 1969. [4] ELLIS, R.: The Veech structure theorem, Trans. Amer. Math. Soc. 186 (1973), 203-218. [5] ELLIS, R.: The Furstenberg structure theorem, Pacific J. Math. 76 (1978), 345-349. [6] FURSTENBERG, H.: The structure of distal flows, Amer J. Math. 85 (1963), 477-515. [7] FURSTENBERG, H. & S. GLASNER: On the existence of isometric extensions, Amer. J. Math. 100 (1978), 1185-1212. [8] GLASNER, S.: Relatively invariant measures, Pacific J. Math. 58 (1975), 393-410. [9] GLASNER, S.: Proximaljlows, Berlin, 1976. [10] McMAHON, D.C.: Relativized weak disjointness and relatively invariant measures, Trans. Amer. Math. Soc. 236 (1978), 22-237. [11] McMAHON, D.C. & T.S. Wu: Distal homomorphisms of non-metric minimal flows, Proc. Amer. Math. Soc. 82 (1981), 283-287. [12] VEECH, W.A.: Point-distal flows, Amer. J. Math. 92 (1970), 205-242. [13] VEECH, W.A.: Topological dynamics, Bull. Amer~ath. Soc. 83 (1977), 775-830. [14] WouDE, J.C.S.P. VAN DER: Topological dynamix, Amsterdam, 1986. 

A proof of the relativized, non-metric form of Furstenberg's structure theorem

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