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Restricted Orbit Equivalence for Actions of Discrete Amenable Groups
Restricted Orbit Equivalence for Actions of Discrete Amenable Groups
Janet Whalen Kammeyer, Daniel J. Rudolph
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This monograph offers a broad investigative tool in ergodic theory and measurable dynamics. The motivation for this work is that one may measure how similar two dynamical systems are by asking how much the time structure of orbits of one system must be distorted for it to become the other. Different restrictions on the allowed distortion will lead to different restricted orbit equivalence theories. These include Ornstein's Isomorphism theory, Kakutani Equivalence theory and a list of others. By putting such restrictions in an axiomatic framework, a general approach is developed that encompasses all of these examples simultaneously and gives insight into how to seek further applications.
หมวดหมู่:
ปี:
2002
ฉบับพิมพ์ครั้งที่:
First Edition
สำนักพิมพ์:
Cambridge University Press
ภาษา:
english
จำนวนหน้า:
208
ISBN 10:
0521807956
ISBN 13:
9780521807951
ซีรีส์:
Cambridge Tracts in Mathematics volume 146
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equivalence^{238}
theorem^{232}
entropy^{221}
ergodic^{200}
lemma^{180}
actions^{169}
orbit^{158}
hence^{152}
proof^{148}
map^{137}
measure^{136}
invariant^{135}
finite^{131}
metric^{126}
partition^{126}
sequence^{114}
polish^{109}
definition^{108}
suppose^{107}
measures^{106}
continuous^{105}
joinings^{95}
subset^{93}
rearrangements^{83}
arrangements^{81}
define^{78}
sets^{78}
topology^{77}
elements^{75}
thus^{73}
axiom^{73}
examples^{73}
theory^{71}
tower^{70}
equivalent^{69}
arrangement^{68}
bounded^{68}
corollary^{67}
exists^{66}
joining^{65}
preserving^{64}
sizes^{62}
finitely determined^{61}
polish space^{60}
free and ergodic^{60}
weak^{56}
obtain^{54}
orbit equivalence^{54}
ornstein^{54}
partitions^{53}
copying^{52}
follows^{51}
weiss^{49}
bernoulli^{49}
spaces^{48}
rearrangement^{48}
defined^{47}
functions^{46}
identity^{46}
notion^{46}
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J. Kammeyer would like to thank the U.S. Naval Academy, in particular for their sabbatical support during the course of this research. D. Rudolph gratefully acknowledges the support of NSF grants DMS9401538 and DMS9706829. The purpose of this work is to lift the notion of restricted orbit equivalence to the category of free and ergodic actions of discrete amenable groups. The axiomatics of a size m and the nature of the associated equivalence relation, mequivalence, are established. An extensive list of examples of sizes and the corresponding equivalence relations are described. An entropy, called mentropy, associated with each size, is defined as the infimum of the classical entropy over the mequivalence class. It is proven that a restricted orbit equivalence is either entropy preserving, in that the mentropy is simply the classical entropy, or entropy free, in that on a residual subset of the equivalence class, the entropy is zero and hence the mentropy of all actions is zero. The notion of mfinitely determined is introduced, and some of its basic properties are developed, in particular that it is an mequivalence invariant. Finally, the equivalence theorem is proven, that any two mfinitely determined actions of equal mentropy are mequivalent. This is carried out using category methods, following the BurtonRothstein approach, within a natural Polish space of mjoinings. In the Appendix, it is shown that previous size axiomatizations give rise to essentially the same notion of mequivalence. Restricted Orbit Equivalence for Actions of Discrete Amenable Groups Janet Whalen Kammeyer Daniel J. Rudolph CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 100114211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Wate; rfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 2002 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002 Printed in the United Kingdom at the University Press, Cambridge Typeface Times System I£Tr£X2£ [UPH] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Kammeyer, Janet Whalen, 1963— Restricted orbit equivalence for actions of discrete amenable groups / Janet Whalen Kammeyer, Daniel J. Rudolph. p. cm Includes bibliographical references and index. ISBN 0 521 80795 6 1. Measurepreserving transformations. 2. Entropy (Information theory) I. Rudolph, Daniel J. II. Title. QA613.7.K36 2001 515'.42dc21 2001025506 ISBN 0 521 80795 6 hardback Contents 1 Introduction 1.1 Overview • 1.2 A roadmap to the text 1.3 History and references 1.4 Directions for further study page 1 1 3 5 11 2 Definitions and Examples 2.1 Orbits, arrangements and rearrangements 2.2 Definition of a size and mequivalence 2.3 Seven examples 13 13 20 32 3 The OrnsteinWeiss Machinery 49 4 Copying Lemmas 65 5 mentropy 91 6 mjoinings 6.1 Polish topologies 6.1.1 Overview of the topology on mjoinings 6.2 Modeling pairs of arrangements 6.3 Modeling rearrangements 6.4 Adding sizes to the picture 6.5 More orbit joinings and mjoinings 100 100 107 108 113 129 132 7 The Equivalence Theorem 7.1 Perturbing an mequivalence 7.2 The mdistance and mfinitely determined processes 7.3 The equivalence theorem 139 139 150 161 vi Appendix 1sizes A.I A.2 psizes Bibliography Index Contents 167 168 189 196 199 1 Introduction 1.1 Overview The purpose of this work is to lift the notion of restricted orbit equivalence to the category of free and ergodic actions of discrete amenable groups. We mean lift in two senses. First of course we will generalize the results in [43], where Rudolph developed a theory of restricted orbit equivalence for Zactions, and in [25] where both authors later established a similar theory for actions of Z'', d > 1 to actions of these more general groups. However, we will also lift in the sense that we will develop the axiomatics and argument structures in what we feel is a far more natural and robust fashion. Both [43] and [25] were based on axiomatizations of a notion called a "size" measuring the degree of distortion of a box in Z'' caused by a permutation. It is not evident that on their common ground, Zactions, these two theories agree. Hence we refer to the first as a 1size and the second as a psize, p for "permutation". Here we will establish the axiomatics of what we will simply call a size. We ask that the reader accept this new definition. In the Appendix we show that any equivalence relation that arose from a psize will arise from a size as we define it here. The same is not done for 1sizes, but for a slight strengthening of this axiomatics that includes all the examples in [43]. We will work on the level of countable and discrete amenable groups, as the work of Ornstein and Weiss [37] has shown this to be a natural level on which all the basic dynamics and ergodic theory we need holds sway. We will not push beyond this to locallycompact amenable groups as the formalism of both orbit equivalence and entropy theory require basic work before our approach appears feasible. Outlining our approach, in Section 2.1 we will set up the basic 1 2 Introduction vocabulary we will use for our work, the vocabulary of arrangements and rearrangements of orbits, and describe the natural topologies on these spaces. In Section 2.2 we will establish the axiomatics of a size m and the nature of the associated equivalence relation, mequivalence. One sees immediately here the change in perspective from [43] and [25] in that a size m is now a family of pseudometrics on the fullgroup of a free and ergodic action, one for each arrangement of the orbit as an action of the group G. The mequivalence class of an arrangement will appear here as a certain G$ subset of the completion of the fullgroup relative to this pseudometric. We end Chapter 2 with a list of seven equivalence relations, some well known others not so well known, which can be described as mequivalences for an appropriate m. We also present one "nonexample" that uses the methods developed here but does not fall directly under our development and indicates one of several directions in which to further broaden this approach. In Chapter 3 we present the fundamental results that we will need from the Ornstein and Weiss work on the ergodic theory of actions of amenable groups. In Chapter 4 we present a variety of copying lemmas that will be essential to our progress both in developing an entropy theory for restricted orbit equivalences and for our proof of the equivalence theorem. Chapter 5 contains our development of an entropy theory for restricted orbit equivalences. We define an entropy, called mentropy, associated with each size as the infimum of the classical entropy on the mequivalence class. The principle result we obtain, (as was done in [43] and [25] for the cases they considered) is that a restricted orbit equivalence is either entropypreserving, in that mentropy is simply the classical entropy, or entropyfree in that on a residual subset of the equivalence class the entropy is zero and hence the mentropy of all actions is zero. From this point our goal is to prove the natural generalization of Ornstein's isomorphism theorem for Bernoulli shifts for our restricted orbit equivalences. That is to say, we wish to show that there are certain distinguished free and ergodic actions, intrinsically recognizable, for which mentropy is a complete invariant of mequivalence. As we indicated earlier our goal in lifting results is both to demonstrate that they hold in the more general context and also to raise the general level of argument to a more robust form. In particular the approach to the equivalence theorem we take is to bring to bear the categorical approach that Burton and Rothstein [42] brought to the isomorphism theorem. To accomplish this it is necessary to build up a certain topological 1.2 A roadmap to the text 3 perspective. We will be considering Polish spaces and Polish actions. Recall that these are topological spaces that can be imbedded as a Gs subset of a compact metric space, and homeomorphisms of them. Chapter 6 presents this development from its foundations through to the proof that the space of mjoinings of free and ergodic Gactions form a Polish space of measures. Although we cannot expect the reader to have any idea at this point what precisely an mjoining is, the point of such a result should be clear to the reader familiar with the BurtonRothstein approach. Any mequivalence between two actions will sit as a subset of this space of mjoinings. Just as Burton and Rothstein show that for Bernoulli actions of equal entropy, the conjugacies are a residual subset of the space of joinings, our aim is to show that for any two mBernoulli actions of equal mentropy, the mequivalences sit as a residual subset of the space of mjoinings. To obtain this in Section 7.2 we introduce the notion of an mfinitely determined action and develop some of its basic properties, in particular that it is an mequivalence invariant and is inherited by factor actions. What remains to complete the equivalence theorem is to define a list of open sets in the space of mjoinings whose intersection is precisely the mequivalences, and to show that in the case of mfinitely determined actions they are dense. The first part of this is easy. It is the second part that takes some work. Central to this proof of density is of course the copying lemmas which we developed in Chapter 4. With them we show how to perturb an arbitrary mjoining to lie in one of our open sets. Section 7.1 presents the basic structure theory for the notion of perturbation we will use. Section 7.3 finally completes the equivalence theorem. To connect this work with earlier work, in the Appendix we will demonstrate that psizes give sizes in our sense with the same equivalence classes, and that 1sizes essentially do in that all known examples do, and in general the mfinitely determined classes are the same. This means the large classes of examples discussed there are restricted orbit equivalences according to the definition used here. 1.2 A roadmap to the text We offer the readers an indication of how they might best benefit from this text depending on the level and nature of their interest. This text is not intended as an introduction to the isomorphism theory of Ornstein. A reasonable preparation is necessary before the material presented 4 Introduction here will be comfortably accessible. To the reader interested in a broad overview of dynamics we recommend the texts of Walters [62], Petersen [39], Cornfeld, Fomin and Sinai [7], Hasselblatt and Katok [17], and Brin and Stuck [4]. We also recommend the reader consult Ornstein [35], Shields [50], Rudolph [44], and Ornstein, Rudolph and Weiss [38]. Of these [44] is perhaps the most important reference as it is intended as a basic technical introduction to fundamentals at work here. For the reader seeking a deep understanding of our presentation these older works will provide valuable perspective and background. We envision two audiences for this text, those seeking a basic technical overview of the tools and methods of this theory for use in their own work but without an intention to work in this area and students and researchers seeking a deep understanding of this area with the intention of working in it. We give here an abbreviated path through the text for the former audience and a recommended first reading for the latter. All readers should spend time on Chapter 2. For a reader only interested in what the text is about, Chapter 2 offers a sufficient treatment. The material of Chapter 3, through Theorem 3.0.6, is the now classical treatment of the ergodic theory of discrete amenable groups due to Ornstein and Weiss [37]. Understanding this is a must for anyone interested in modern ergodic theory. The next few pages introduce the vocabulary of names for the entropy theory of these actions and it is important to understand them. Beyond this, the important conclusions of Chapter 3 are Theorem 3.0.9 and Corollary 3.0.10 and the reader should be familiar with their meaning. Chapter 4 is quite technical. What is essential are the two Theorems 4.0.5 and 4.0.13 and one should understand their meaning. The technicalities of proof can be absorbed later if needed. What is essential from Chapter 5 concerning mentropy is contained in its first paragraph. Chapters 6 and 7 are really only appropriate for the reader wanting a detailed understanding of the equivalence theory. Chapter 6 develops the space of mjoinings of two actions as a Polish space of measures on a symbolic representation. This provides a framework on which the equivalence theorem of Chapter 7 can be reached without much ado. Chapter 6 though is quite heavy going. The reader should read Section 6.1 as an introduction to Polish spaces ending with a brief and vague description of the succession of spaces constructed to reach the notion of an mjoining. From here one can proceed to Chapter 7, skipping over Section 7.1 and instead focusing on the definitions and results of Section 7.2. Continue with Section 7.3 through the statement of Theorem 7.3.3. 1.3 History and references 5 The path just described gives a complete overview of the text. To continue the reader should now return for a careful reading of Chapter 6 and then Section 7.1. 1.3 History and references The notion of restricted orbit equivalence can be traced back some decades, and the techniques used here can be traced back somewhat further. We take this opportunity to outline our understanding of the history of these ideas and to acknowledge the sources of our work. There are many significant parts of this broad area that we will not mention here. We focus on those particular ideas central to the evolution of this particular part of the theory, the construction of general orbit equivalence theorems, and the tools of those constructions. Certainly the first pieces, historically, of this story do not relate directly to notions of orbit equivalence, but rather are the basic technical pointset tools of ergodic theory, in particular the Rokhlin lemma [31], [41], the mean ergodic theorem [61], and the entropy theory of Kolmogorow and Sinai [28], [29], [51] (we cite the earliest references we know to the basic results and methods). Of these three the Rokhlin lemma may seem the most mundane but in the long run it is in fact the deepest of the three. Looking forward to the seminal work of Ornstein and Weiss [37] on actions of amenable groups, it is their ability to realize a Rokhlin lemma that in fact carries forward their entire program. Dye's proof, in 1959 [10], that any two ergodic measure preserving actions on nonatomic standard probability spaces are orbitequivalent is the first instance of an orbit equivalence theorem, and of course a very startling one. The core technical pieces of this proof are of course the Rokhlin lemma and the ergodic theorem. Dye's original techniques are still visible in all the copying lemmas that follow. The only issue that he did not have to address was entropy. Dye's result was simply too profound and final in the case of measurepreserving actions and so has had much more impact in nonsingular dynamics and Von Neuman algebras than in ergodic theory per se. Krieger [30], [54] was able to characterize the orbit equivalence classes of all nonsingular ergodic actions by careful study of the obstacles to Dye's argument. The obstacle to moving our work in the direction of nonsingular dynamics is the issue of entropy. Certainly, though, one might consider developing the notion of entropyfree restricted orbit equivalences in this setting. On the other hand, perhaps the attempt to lift these methods to the non 6 Introduction singular category will give some insight into how to approach entropy in this category as entropy in the measurepreserving category can be defined by a restricted orbit equivalence (see Section 2.3). The next step, and truly the pivotal one, in this development was Ornstein's proof that any two Bernoulli shifts of equal entropy were isomorphic [32], [33], [34]. One must remember Sinai [52] had already shown they were weakly isomorphic, that is to say each sat as a factor action of the other. Ornstein's real contribution, from our perspective, is his general characterization of the Bernoulli shifts via the notion of finitely determined actions and more generally the wide range of powerful constructive tools he laid out. At this time the notion of finitely determined was given in a finitistic form, not in terms of joinings as is more common now; but the lift to the joinings perspective is a modest contribution in comparison to the significance of the original concept. This theorem is a phenomenal piece of technical work. Others have had profound insights since then but this result showed what the path to an equivalence theorem would look like. All one had to do was see how to take the steps. One also should note that the Ornstein machinery provides tools for showing not only that Bernoulli shifts of equal entropy are conjugate, but also that various collections of actions are not conjugate. Parallel to the positive side of equivalence/isomorphism results, one could use these methods to develop nonisomorphism results. For example, Ornstein and Shields [36] constructed uncountable families of nonconjugate ergodic Ksystems, all of the same entropy. Relating back to Sinai's theorem, Polit [40] produced a pair of weaklyisomorphic but nonisomorphic actions (in this case zeroentropy mixing actions). This side of ergodic theory is extensive. During the same period Vershik [60] began an investigation building on Dye's work, but taking a different focus. In orbit equivalence terms he was considering actions of groups other than Z, in particular actions of infinite sums of finite cyclic groups. Such a group is the union of a sequence of finite groups 34?„, where each J^n/^fn+i is a cyclic group of order rn. The simplest nontrivial case is where the r,,'s are all two, the dyadic case. What he considered was in fact a notion of restricted orbit equivalence, asking that the orbit equivalence between two such actions should be an orbit equivalence of each of the M',,subgroups. Heicklen has completely translated this work into the vocabulary of restricted orbit equivalence as we develop it here (see [19], [18] and Example 6 in Section 2.3). In truth this is more naturally described in terms of 1.3 History and references 1 the ^  i n v a r i a n t ualgebras J v Notice that these form a decreasing sequence (or reverse filtration) of algebras !Fn+\ s J%. Ergodicity of the action is equivalent to the algebras intersecting to the trivial algebra. Vershik's notion of equivalence is simply that there should be a map between the measure spaces respecting the two reverse filtrations. In this sense Vershik's study has a similar feel to that of Dye, the natural generalizations move away from measurepreserving actions. One is led to consider general reverse filtrations, without regard to their arising from a measurepreserving action. In terms of orbit equivalence theory though, this particular development has a particularly prescient nature. Vershik showed that there was an entropy associated with this relation and, that if the r,,'s did not grow too quickly, it was the standard entropy of the action, and if they grew quickly enough it would be zero. Using this observation both he and Stepin [53] were able to construct nonequivalent reverse filtrations as they had distinct entropies. Central to Vershik's study is the notion of standardness. This also points to a central aspect of later work, and certainly ours, that there will be certain distinguished classes, the "Bernoulli" class of the given equivalence relation. (Vershik's "standard" class is the zeroentropy mfinitely determined class for the associated size m.) This work gives the first clear indication of how entropy might enter in a general picture. The next major contribution in the direction of our work also had two sides, one in the west and one in the east. The notion of Kakutani equivalence had arisen some decades earlier in the study of measurable crosssections of measurepreserving actions of K [2], [3], [22]. It was known that although entropy was not an invariant of Kakutani equivalence, entropy class was (zero, finite, or infinite entropy) and that entropy changed in a simple way when moving among equivalent actions (Abramov's formula [1]). What Feldman did in the west and what Katok did in the former Soviet Union was to introduce the /metric (Feldman's notation) on names. Feldman used this to show that there were many distinct Kakutani equivalence classes of the same entropy class [11]. He also saw the possibility that / might plug into Ornstein's isomorphism machinery and lead to an equivalence theory parallel to Ornstein's conjugacy theory. Katok [26], in part jointly with Sataev [49], and Ornstein, Weiss and Rudolph fulfilled that expectation [38]. Katok and Sataev, of course, were working completely independently of Feldman, Ornstein and Weiss. Building on Feldman's original examples, and Ornstein's observation 8 Introduction that the Cartesian square of a rank1 and mixing map would not be finitely fixed, many exotic examples were constructed. This constructive side was also pursued by Katok, leading to his construction, via this theory, of the first smooth K and not Bernoulli action [26]. Building on the existence of three theorems, Dye's, Ornstein's, and the Feldman, Katok, Ornstein, WeissKakutani equivalence theorem, Feldman proposed in 1975 the potential for a general theory of equivalence relations based on the common structures in these three results. There were two essential gaps in the picture. Kakutani equivalence is not a restricted orbit equivalence in that it is not an orbit equivalence. This seemed a minor issue in that inducing on a subset is not so far from an orbit equivalence. More critical though was to understand what role entropy would play. The isomorphism theorem and Kakutani equivalence theorem both use entropy in much the same way, following Ornstein's basic plan. But entropy does not enter Dye's theorem at all. Of course Vershik's work had already indicated what the answer might be, but this was not well known in the west at the time. The one real gap in the picture was exactly how to phrase a general theorem, although the thought certainly was to create the needed material to apply Ornstein's method, that is to say, define some analogue of d or / . On a more technical level, in the late 1970s Burton and Rothstein gave an approach to Ornstein's proof that recast the focus to joinings [42], [43]. In these terms what one sees is not the detailed construction of a single conjugacy, but rather the description of all conjugacies as a residual subset of a space of joinings. Although in a pure sense there was nothing really new in what Rothstein did, it made the whole path to the result much clearer. The critical technical piece of the isomorphism theorem was the copying lemma which forced the denseness of certain open sets in a space of joinings. The intersection of these sets were precisely the factor maps, or projections, of some fixed but arbitrary system of sufficient entropy onto a finitely determined system. In this light one could say that Sinai had it right, that although weak isomorphism did not imply isomorphism, if the class of projections of a general system on a finitely determined system of equal entropy could be shown large enough (that is, shown residual in the compact space of all joinings) then there would be lots of isomorphisms to choose from between two finitely determined systems of equal entropy and residual set of them. It was not clear until our work here that this perspective could be lifted to the more general restricted orbit equivalence level. In particular [25], [37] and [43] all follow the original direct constructive approach of Ornstein. 1.3 History and references 9 When del Junco and Rudolph showed that Kakutani equivalence could be given a natural characterization in terms of orbit equivalence [9] the general picture became much clearer. As we pointed out earlier, from Abramov's formula we know that the entropy of an induced map varies inversely proportionally to the measure of the set on which one induces. If one defines a notion of "even equivalence" of two actions to mean that one induces conjugate actions in each on subsets of the same measure, then this equivalence relation preserves entropy. Moreover, they showed that the conjugacy could be extended from the subsets to the rest of the two ambient spaces as an orbit equivalence precisely because the two sets have the same measure. An orbit equivalence arising in this fashion could be characterized in a variety of ways; in particular in ways that extended to higher dimensional actions. A variety of authors have pursued this area (Katok [27] for example). The last piece in this particular development is HasfuraBuenaga's proof of an equivalence theorem in Zrf [16]. One could now see all three, orbit equivalence, even Kakutani equivalence, and conjugacy, as restricted orbit equivalences and look for the common thread in the corresponding equivalence theorems. This is what was attempted in [43]. The basic structure laid out there was that one should axiomatize a notion that measures how badly one is distorting an orbit. In both [43] and [25] this is based on an axiomatization of how one would measure the wildness of a permutation of a large block of the acting group Z or Zd. These axiomatizations were rather elaborate and technical, especially in [43]. Very quickly it became evident that the formulation in [43] was quite flawed. The author admits there that the basic structures will not lift reasonably to larger group actions, and natural examples arose of equivalence relations, that were restrictions on orbit equivalence, that "ought" to be but could not be brought into the framework of [43]. For example, aequivalences described here in Examples 4 and 5 cannot arise from a 1size. At a more subtle level, Fieldsteel indicated quite rightly that the definition given for an mjoining was not sufficiently robust. This is because an mjoining was required to be a joining perturbed by a "bounded coboundary", not by an mequivalence. In particular an mequivalence itself was not necessarily an mjoining. The reason for this concern over basics is that as we indicated earlier there is a good deal more to the isomorphism theory than just Ornstein's theorem itself. There are all the constructive examples built to show it attains the best one could hope for. There is the "relativized theory" of 10 Introduction Thouvenot [55] leading to his deep study of the weakPinsker property [56]. In particular one has Thouvenot's result that the property of being Kakutani equivalent to a map with the weakPinsker property is inherited by factor actions. Fieldsteel and one of the authors generalized this to the [43] theory of entropypreserving sizes [14]. One also has the theory of isometric and affine extensions of Bernoulli actions [23], [24], [46]. One could hope to generalize all of this work from conjugacy, and Bernoulli actions, to restricted orbit equivalences and mfinitely determined actions. This has been done for some examples [43]. Fieldsteel [12] showed that if one took compact group extensions of two ergodic actions, by the same group, so that the extensions were ergodic, then one could construct an orbit equivalence between the two extensions that preserved the group extension structures. In a result which really deserves deeper study Fieldsteel and Friedman [13] showed that Belinskaya's amazing theorem (if the generating functions of an orbit equivalence were integrable, then the equivalence was essentially trivial) was false in higher dimensions even if integrable was replaced with bounded. This indicates that there is perhaps a nontrivial L1 and even U° orbitequivalence theory in higher dimensions which is vacuous in 1. That is to say, there may be families of sizes in 7LA for all d, which for some d, say d = 1, give the same equivalence relations, but for larger d do not. At this point the authors put forward the development in [25] as a restricted orbit equivalence theory for actions of Z(/. This work is quite parallel to [43]. In [43] one takes an injection from a block of integers (/',_/) and "pushes together" the range set to obtain a permutation of (i,j). In [25] one constructs permutations of boxes in Z'' from injections by "filling in" that is by taking those points that the map throws out of the box and placing them on the points in the box which have no preimage. These are similar, but far from the same notion. Certainly this new picture was much more generalizable, but still was tied to basic structures in the group Z'', in particular boxes. In the meantime Ornstein and Weiss [37] had finished their seminal study of the ergodic theory of amenable group actions. This profound work makes it evident that the natural level at which the three basic tools of ergodic theory (Rokhlin lemma, ergodic theorem and a ShannonMcMillan theorem) apply is that of amenable groups. To be more precise, it is clear that they all apply at the level of discrete amenable groups. It still remains unclear to what degree the entropy theory for general amenable groups is wellfounded. As our work on orbit equivalence will 1.4 Directions for further study 11 certainly also have technical problems at the level of uncountable groups, we gladly limited our work here to discrete amenable groups. 1.4 Directions for further study We have indicated a number of areas that deserve further study in the light of our work here. We sketch a few of these here. 1. A relativized theory parallel to that of Thouvenot is probably directly accessible. We propose a natural notion for this. If m is a size, then in any given action (X, cF,\i, T), with H some fixed finite partition of X, consider the subgroup FH of the fullgroup of elements with the property that for ^ta.e. x, both x and (j){x) lie in the same element of H. This is an L'closed subgroup. We say two arrangements are H relatively mequivalent if they are mequivalent via a sequence of elements 0, in T H . 2. The "nonexample" described in the next section, based on work of Hoffman and Rudolph [20], proving an isomorphism theorem for certain endomorphisms is one direction for further work. Although these endomorphisms are measurepreserving, the natural orbit relation here is not, it is nonsingular. Furthermore the natural group acting on these orbits is nonamenable, although the action is amenable. That is to say, to bring this example under a general umbrella such as we do here for the measure preserving case, one would have to consider a restricted orbit equivalence theory for nonsingular amenable actions of perhaps nonamenable groups. This example and its generalizations (for example Kakutani equivalence for endomorphisms) makes this a significant direction for further study. 3. Following this direction further, so far we have always assumed that the orbits under consideration were organized as a copy of a group. One wants the orbits organized this way as our restrictions will all take the form of a metric on how badly this structure is distorted. One may consider other structures on a nonsingular orbit relation. One idea is to replace the group with a lattice or graph structure. Some kind of stationarity of this structure would be needed. It could of course be a random structure, i.e. vary from point to point. 4. We have pointedly avoided nondiscrete actions in this work but obviously one must address the technicalities in this for group actions, semigroup actions and perhaps some generalization to continuous graphs. A major issue in any such generalization is the exact nature of entropy. This can be avoided initially by considering entropyzero classes. For 12 Introduction example, although the endomorphisms studied in [20] are of positive entropy all the entropy is carried by the trees of inverse images and relative to these trees the dynamics is zero entropy. 5. Certain notions exist that seem to have the character of either a restricted orbit equivalence, or of the finitely determined class in one. For example the weakPinsker property seems to have much the character of the finitely determined class of some entropypreserving size. Is this the case? Feldman's notion of preBernoulli also can perhaps be brought under this heading. Certainly one can look quite broadly for useful examples and applications of our general machinery. 2 Definitions and Examples 2.1 Orbits, arrangements and rearrangements Let (X, SF,\.i) be a fixed nonatomic Lebesgue probability space, that is to say it can be represented up to measure zero by Lebesgue measure on the unit interval. Let G be an infinite discrete amenable group. Suppose G acts ergodically on X as a group of measure preserving transformations. More precisely, we have measurepreserving maps Tg whose conjugation law is the group law of G (Tgl o Tg2 = Tglg2) and for which all invariant sets have measure zero or one. Assume the action is free, that is to say for all g ^= id, Tg(x) = x on at most a set of measure zero. (These are the standing hypotheses for the OrnsteinWeiss Rokhlin lemma.) Let (9 = {x, Tg(x)}gec £ X x X be the orbit relation generated by this action. We say two such actions are orbitequivalent if there is a measurepreserving map between their domains that carries one orbit relation to the other. We refer to such a map as an orbit equivalence between them. One case of the general theorem of Connes, Feldman and Weiss [6] is that all such orbit relations are orbitequivalent. For our purposes then there is only one space and orbit relation. We study how these orbits can be arranged and rearranged to form orbits for an action. Definition 2.1.1. Let G be an infinite countable discrete amenable group. A Garrangement a is any map from & to G that satisfies: (i) a is 11 and onto, in that for a.e. x £ X, for all g S G, there is a unique x' € X with a(x,x') = g. We write x' = T*{x); (ii) a is measurable and measure preserving, i.e. for all A € ^,g both T«(A) € & and n{T«(A)) = n(A); and (iii) a satisfies the cocycle equation a(x2,x^)a(x\,X2) = a(.\'i,X3). 13 € G, 14 Definitions and Examples As G will not vary for our considerations we will abbreviate this as an arrangement. Let si denote the set of all such arrangements. Lemma 2.1.2. A map a is a Garrangement if and only if there is a measure preserving ergodic free action of G, T, whose orbit relation is 0 such that gforall(x,Tg(x))e&. a(x,Tg(x)) = Thus the vocabulary of Garrangements on (9 is precisely equivalent to the vocabulary of Gactions whose orbits are (9. For a Garrangement a, we write Tx for the corresponding action. For a Gaction T, we write <xj for the corresponding Garrangement. This concept is very similar to that of a Gcocycle on (9. The cocycle would map the pair (xi,g) to xi where the arrangment maps the pair (x\,xi) to g. Although the vocabulary of actions and cocycles is much more well known and all our work could be translated into either of these, the vocabulary of arrangements will make many formulations much simpler and more transparent. Definition 2.1.3. The fullgroup of (9 is the group (under composition) F of all measurepreserving invertible maps 4> '• X —* X such that for /.ia.e. x G X, (x, 0(x)) G (9. Note that it would be sufficient in this definition to assume that (j> is measurable and 11, since the fact that (9 is a measurepreserving orbit relation forces 0 to be measurepreserving. Also note that the orbits of (j) are a subrelation of (9 and need not be all of (9. Definition 2.1.4. A Grearrangement of (9 is a pair (a, (j>), where a is a Garrangement of & and 4> € F. As G is fixed for our purposes we will abbreviate this to a rearrangement. Let 2. denote the set of all such rearrangements. Intuitively, a rearrangement is simply a change of an orbit from the arrangement a to the arrangement <x$, where a<t>(x, x') = a((f)(x),(j>(x')). One can formalize such a rearrangement in three different ways. Set 38 to be the set of bijections of G and (S the subgroup of 38 fixing the identity. Both are topologized via the product topology on G c . Notice there is a homomorphism H : 08 —• (S given by H(q)(g) = q(g)q(id)~\ Observe that information is lost in mapping via H. The kernel of H consists of the left translation maps. 2.1 Orbits, arrangements and rearrangements To a rearrangment we can associate a family of functions q^ where 15 G 3$ Now suppose a and /? are two arrangements of the orbits (9. Regard the first as an initial and the second as a terminal arrangement. We can associate to this pair and any point x a bijection from G, fixing the identity that describes how the arrangement of the orbit has changed: hf(g) = P(x, 7£(x)). Notice here that H{q^) = h***. Write ha''! :X^>y. The third way to view a rearrangement pair has a symbolic dynamic flavor. For each orbit (9(x) = {x';(x, x1) S &}, a rearrangement (a, 0) also gives rise to a natural map G —> G (not a bijection though), given by Visually, regarding (9(x) laid out by a as a copy of G, (j> translates the point at position g to position /"'*(g)g. Notice in particular that the definition of f^ is stationary in that Thus if we map a point x to the infinite word w(x) = {/?*(g)} g 6 G G GG then w(Tg(x)) = ag(w(x)) where a% is the shift action on G G , ag(h)(k) = h(kg). There is a natural link between the three functions / i 0 ^ , q*<<t> and as follows. For any map / : G —» G we define Q(/)(g) = / ( g ) g It is an easy calculation to see that and 16 Definitions and Examples Let {F,} be a fixed Folner sequence for G. We will describe a number of concepts in terms of the F, but will note at appropriate points where they are, in fact, independent of the Falner sequence. Lemma 2.1.5. Let {F,} be a Folner sequence in G, and let (a, 0) be a rearrangement. Define b,{x) = #{g G Fr,ff*(g)g t F,} = #{g $ F,;ff*(g)g G F,}. Then lim bi(x) = 0. I—>00 Proof For a finite set K £ G set EK = ( x ; / ^ ( i d ) £ K}. For e > 0 choose K so that /j(£ K ) < e/4. Let /j,(x) = #{g G F,;Tga(x) G £ K } . By the L2ergodic theorem, there exists / , such that for / > / , <e/2. By the F0lner property, we may further select / such that for i > I, #U, e ^AF,.) < £ / 1 Now for i > / , select g € F, such that /"'*(g)g £ F,. For such g, either (2) Thus bj(x) < /J,(X)+# completes the proof. \JkeK(kFiAFj), so that lim sup II ^ < e, which • We now consider three metrics on the set of rearrangements. These all arise from natural topologies on functions G —> G, that is to say on GG. As G is countable the only reasonable topology is the discrete one, using the discrete 0,1 valued metric. This topologizes GG as a metrizable space with the product topology. This is the weakest topology for which the evaluations g : / —• /(g) are continuous functions. Notice that H is a continuous map from G c to itself and the map h —•> h~] on <8 is continuous. List the elements of G as {gi = id,g2,...} and let do be the 0,1 valued metric on G. Define a metric d on <§ as follows. Set d(huh2) = 2.1 Orbits, arrangements and rearrangements 17 Notice that d(h\,li2) < 2~' if h\, h2, h\\ and hj* agree on gi,...,g,. On the other hand if d(h\,h2) < 2~' then h\, h2 and their inverses agree on this list of / terms. Lemma 2.1.6. The metric d on<$ gives the restricted product topology and makes <S a complete metric topological group. Proof It is evident that the group operations are continuous and that d gives the product topology. We show that d makes (S complete. Suppose the h\ are a dCauchy sequence. It follows that for all g G G, for all i sufficiently large, both ft,(g) and /if'(g) remain constant. Hence h,• —» h € GG and h~{ —> k e G°. But for any i sufficiently large k(k(g)) = g and then clearly h o k = id and h^(S. D A simple corollary of this is that 2? is a G<5 subset of GG as we have just seen it to be topologically complete in the product topology. We can use this to define an L1 metric on arrangements: As d{h{h2,id) < d(huid) + d{h2,id) and {hf)~l a metric. = hP'a we see that this is Lemma 2.1.7. The metric space {$#, , h) is complete. Proof Suppose the a, form a Cauchy sequence of arrangements in , i. There will then exist a subsequence i(j) converging pointwise, i.e. so that for /ia.e. x € X, h*fA>) are Cauchy in the metric d. As (^,d) is complete, we conclude that for fia.e. x e X, hTiU) > hx. j Setting P(x, Tg«(x)) = hx(g) it is straightforward to see that /? is an arrangement. As ay(,),j8i —> 0 the proof is complete. • 18 Definitions and Examples We can also define a metric similar to d on GG itself making it a complete metric space by just taking half of the terms in d: This also leads to an L1 metric on Gcvalued functions on a measure space: Wfuf2h=Jdl(f],f2)dn. These two L 1 distances now give us two families of L' distances on the fullgroup, one a metric the other a pseudometric, associated with an arrangement a: 2 =J Hi d(H•(/«•*' )'ff (/«•*') ,id) dfi, and = II fx,<l>\ fa'^2,. The weak L1 distance, , ^, is only a pseudometric but the strong L1 distance, vlls> ^s a metric. Notice that Tinvariance of /i tells us that Wt < 2Jdo(f**<(id)J"'+i{id))dii Thus, in fact, the topology generated by the strong L' distance on the fullgroup is independent of the arrangement a. Before moving on to the weak* pseudometric we point out that it would perhaps be more correct to call the above weak and strong "distribution" metrics as they measure how closely the two rearrangements distribute the mass of X on the space of functions GG. We of course call them L1 pseudometrics as they arise from integrals of "distances". To describe the weak*distance between two arrangements we let G* = G U {*} be the one point compactification of G. Now (G*)G is a compact metric space and hence the Borel probability measures on (G*)G, which we write as Jf\{G"), are a compact and convex space in the 2.1 Orbits, arrangements and rearrangements 19 weak*topology (that is to say the topology induced on Borel measures as the dual of the continuous functions). We can put an explicit metric on this space as follows. For any finite subset F £ G and / e GG let fF be the restriction of / to F. As fF can be one of at most a countable collection of values, / —> fF partitions GG into a countable collection of clopen sets. If two measures on (G*)G agree on these sets, that is to say on all cylinder sets that do not have a "*" in their name, then they agree. Moreover the characteristic functions of these sets are continuous. Hence if / ( , ( / F ) —» M / F ) then /*, —> /.i weak*. To turn this into a metric, let F, be an increasing sequence of finite sets that exhaust G, for example a Felner sequence. For each F, let P(F,) be the partition of G c according to the values fFl. These partitions refine and for any fixed F, once F <= F,, fF will be P(F, )measurable. Set Notice that £ and so the ith term in this sum is bounded by 2~'. Moreover as the partitions P(Fj) refine, the values increase. It follows that for all / Before continuing we make two remarks. First it is clear from the discussion that this metric gives the weak*topology on (G*)G. Second, it is not too difficult to argue that those measures which put no support on * are a G$ subset of the measures in (G*)G. We will take a broader approach to this particular issue later in Section 7, showing that not only are the measures supported on GG a G$ but the weak*topology here is independent of the way we choose to compactify GG as long as the compactification is metric. At this point these issues are not important. We now define the distribution pseudometric between rearrangements by (a, <£),(/?, ip)\\. = 20 Definitions and Examples We can combine the two L1metrics on arrangements and the fullgroup to define a product metric on rearrangements in the form We end this Section by relating this complete L1metric on rearrangements to the distribution pseudometric. Lemma 2.1.8. The map (a, 0) —> (/a'*)*(ju) is uniformly continuous as a map from (.2, llvlli) t0 (2,  •,•!!•)• That is to say, given any e > 0 there Grearrangements exists a d > 0 such that if (a.\,<f>\) and (x2,4>2) are two that satisfy (ai,0i),(a2,</>2)lli < $ then )U <£• Proof Let F E {gi,g2,• • •,gi<} be any finite set. Suppose d > 0 and llocazlli < 5/(^2*). Then H({x : f#*( gl ) ± g, for some i < K}) < 5. Thus, + /I({JC : /iav"a2(g/) ^ g/ for some i < K}) #F+K2 K )(a l ; </,,),( a <5{#F + K2K). Thus for all i, where Ff £ {gi...g^}. Let £ > 0. Select i so that 2~' < e/2 and let <5 = e/(2(#F, + 2K'/C,)). The result follows. D 2.2 Definition of a size and mequivalence In this section we define the notion of a size m on rearrangements (a, 0) as a family of pseudometrics ma on the fullgroup satisfying some simple relations to the metrics and pseudometrics defined in the previous Section. We then define the mequivalence class of an arrangement 2.2 Definition of a size and mequivalence 21 a to be those arrangements ft for which the corresponding ma and /^completions of the fullgroup are isometric in a canonical fashion. We let F denote the fullgroup of &, as before. Recall that si denotes the set of arrangements and 2L denotes the set of rearrangements. A size is a function m : J2 > R + such that, if we write ma(4>\,4>i) = m(a0i,07'</>2), defn then m satisfies the following three axioms. Axiom 1. For each a 6 rf, nij is a pseudometric on T. Axiom 2. For each a £ r f , the identity map is uniformly continuous. In particular if ma(0i,<^2) = 0 then the two arrangements a<p\ and a<j>2 are identical. Axiom 3. The function m is upper semicontinuous with respect to the distribution metric. That is to say, for every e > 0, there exists a 5 = (5(s, a, </>), such that if \\(a, <j>), (/?,i/>). < S then m{fi,\p) < m(a, </>) + e. This last condition tells us that if the two measures (/a'^)*(/«) and (/ ' T( V ) a r e t n e same, then m{a,4>) = m(fi,ip). Hence the value m is well defined on those measures on GG which arise as such an image, and we can write /; V Later on we will find ourselves in the situation where the rearrangement (a, 4>) is welldefined pointwise and the measure ).i is allowed to vary. In this case we will be more specific and write m/((a,<£) or mai/,($i,$2)Lemma 2.2.1. Let m be a size. The identity map is uniformly continuous. 22 Definitions and Examples Proof Let 0i,02 € F. As Lemma 2.1.8 tells us that for any <5i > 0, there exists 5 > 0, such that if 0i,02? < 5 then (a0,,0j02),a0i,id), < Sx. Fix an arrangement a0 and 0o = id. Let e > 0. Now select (5] = <5(e,ao,id) from Axiom 3. It follows that if 0i,02lls < <5 then (a0i,0[0 2 ),a0i,id). = (a0i,0]0 2 ),ao,id). < 5u and hence • Definition 2.2.2. Let a £ r f , {0,} £ F and m be a size. Define F a to be the equivalence classes of elements ofY at an ma distance zero. Thus (F a ,m a ) A A is a metric space and we let (Ta,ma) be its completion. That is to say, F a consists of sequences {0,} which are mxCauchy, modulo the equivalence relation {0,} ~ {y>,} if ma(0,, xpi) > 0. We write the elements of Fa as (0,)a. Lemma 2.2.3. Fixing an arrangement a and size m the map 0 —> <x0 from V to si is welldefined as a map Fa —> si and extends to a uniA formly continuous map from (Fa, ma) —> (si, , •li) We refer to this map as P m,a((0i)a ) = lim,a0/. Proof As the map 0 —> a0 is uniformly continuous F a —• si it extends to the completion of F a (remember, si in the ,imetric is complete). a Thus if (0,) a € F a then a0, —» j3 for some /J in j / . Hence we will at times abbreviate this element (0,) a G F a as j6, indicating that it is a lift to F a of the arrangement p. The arrangements in the range of P,m are a first cut toward the mequivalence class of a. The subset which is the actual equivalence class may be somewhat smaller. 2.2 Definition of a size and mequivalence 23 Next, we see that since m satisfies Axiom 3, right multiplication in the fullgroup F is an isometry. Specifically, we have the following Lemma. Lemma 2.2.4. Let a be any arrangement. For all 0o € F, the map 0 —» 00o is an maisometry, that is to say: mM\ 0o, 020o) = m«(#i, 4>2)Proof For all 0o, 0i S F, we have that (oc0o,0o'0i0o),(a,0i). = ° Thus, by Axiom 3, for any 0o,0i,02 G F, That is to say, m a (0i0 o ,020o) = «a(0i,02) • It is not necessarily the case that left multiplication is continuous. In fact, left multiplication need not preserve the equivalence classes of the pseudometric ma. For example, consider the size For G = Z and 0, = Tf, we have that m°(id,0,) = 0, but m°(0,00,) = m°(id, 00i 0~'). Here one can prove that m^(id, ip) = 0 if and only if y; = r t a . So if m°(0,00i) = 0 then 0 0 1 0 " ' = 0 7 Y 0  ' = 7£. Thus, (using ergodicity and freeness), fc = 1 and 0 commutes with T". Hence 0 is itself T" for some n, yet the full group is much larger than this. We do have something akin to semicontinuity of right multiplication. Suppose a0,,/?£, * 0. Then, for a.e. x, we have that a(0,0,(x),0,02(x))  /?(0,(x),02(x)), and so (a0,0i,(0f0i)~'(0,02)) = (a0,0i,0]~'02) converges in distribution to (/?0i,0f'02). Axiom 3 now tells us that limma(0i0,,0(02) < m/i(01,02). (#) I—>O0 Definition 2.2.5. H'e saj' t/iat m is a 3 + size if whenever {0,} is maCauchy and hence a0,, /?i —> 0, t/ie inequality (#) is o/i equality. i In particular, this will be true if Axiom 3 is replaced by the following: 24 Definitions and Examples Axiom 3 + . Let a be any ordering, <fi e F and e > 0. There exists § so that if \\(a.,(f>),(P,[p)\\, < 5, then \m(a,4>) — m(fi,\p)\ < e; i.e. m is continuous in the distribution metric. The isometry of right multiplication on F (Lemma 2.2.4), implies that the action of F by right multiplication extends as an isometric action of F on (f a , ma). This action is clearly topologically transitive, since F is an orbit closure. In particular, we get the following: Lemma 2.2.6. Let a be an arrangement and m a size. For any (<j)j)a in Proof Let {$,} be an maCauchy sequence, (representing the class (0,)a). Fix j and using Lemma 2.2.4 compute that Since {</),} is maCauchy, for any e > 0, there exists / such that for all i,j > I, ina((f)i,(l)j) < e. Thus, for all j > I, The result follows. D In particular, this tells us that the action of the fullgroup F on {f^mx) is minimal, i.e. every orbit is dense. For a size m, we would like to define a notion of mequivalence between two arrangements a and /?. Within the context of rearrangements, a natural candidate would be to say that two arrangements a and fi are mequivalent if P is in the range of PmA. That is, there is a sequence of rearrangements (a, 4>>) with a$, —» fi in L1, such that {<£,} is maCauchy. The problem with this definition is that on the face of it the "equivalence relation" is not symmetric. More precisely, it is not clear that, in general, the waCauchiness of {<£,} will imply m^Cauchiness of {0~'}The following theorem describes this situation more precisely. 2.2 Definition of a size and mequivalence 25 Theorem 2.2.7. Suppose a is an arrangement, m is a size, {</>,} is maCauchy and a</>, —• /? in L1. The map i P : (T,mp) > (f a ,m a ), given by is a Fequivariant contraction, so that Hence P extends to a Tequivariant contraction P :(fp,mp)»(f a) ma ). The following are equivalent: (1) (2) (3) (4) id € Range(P); P is onto; P is an isometry; {4>~1} is mpCauchy. Lastly, if m is a 3 + size, then for all fi in the range of PmA, P is an isometry. Proof That P is a Tequivariant contraction we verified earlier (in (#)) as a consequence of Axiom 3. This certainly implies that P extends to a Fequivariant contraction P : (tii,mp) »• (f a ,m a ). Moving on to the four equivalent statements: if id e Range(P) then there exists y E (tp,mp) such that P(y) = id. Suppose y = (xpt)p. To show that P is onto, we need only show that P maps onto F. Let <t> e r . We will show that P(y0) = 4>. In fact, P{y(t>) = (P(Vi0),)«. We need only prove that Compute that l\mma{P(\pi<l>),<t>) = \\ i—co i i—> coy—+oo = limlimma((i,y>i, id) i>ooy>oo 26 Definitions and Examples since F acts isometrically, = limma(P(y;,), id) 1—>00 = 0, since P(y) = id. Hence if id s Range(P) then F £ Range(P) and P must be onto. Thus (1) and (2) are equivalent. Next, we argue that if id e Range(P) then the sequence {0, r '} must be m/?Cauchy. Suppose P{y) = id, where y = (tp,)/;. The fact that P is an equivariant contraction implies that To see this, simply compute that Y\mma{(j)j,\p~[) = l = \imma(P(ipi),P({p)) l»0O = 0. Define g :(r,ma) by Exactly as for P, argue that Q is a Fequivariant contraction, so that mp(Q(a),Q(b))<ma(a,b). Hence Q extends as a Fequivariant contraction Q :{r,ma)^(r,mp). 2.2 Definition of a size and mequivalence 27 Since y G (t,mp), by Lemma 2.2.4, we see that Q({vr1)*) = idThe above argument (applied to Q) now shows that In particular this implies that {0j~'} is m/jCauchy. Thus, since Q is a contraction, for (f> G F, mp(Q(P(cl>)),cl>) = 0. Hence for all <> / G {tp,mp), we see that Thus, for a,b £ (Yp,mp), mp(a,b) = mll{Q(P(a)),Q(P(b))) <mx(P(a),P(b)) <mp(a,b), so that P is an isometry. If P is an isometry, then of course, {</>"'} is m^Cauchy, by Lemma 2.2.4. Hence, id e Range(P). This completes the proof that statements (l)(4) are equivalent. If m is a 3 + size, then P is directly seen to be an isometry for all fi in the range of PmA. • Definition 2.2.8. There are three natural levels now on which to define mequivalence classes. The first is the most functorial, as a subset of ta we can set G fa : afa » 0 and (^p G Second, we can consider the relation on arrangements given by Em(a) = {P\a ~P} = P,,,,a(£,,,(a)). Third, we can consider the category of free and ergodic Gactions T and S and say S is inequivalent to T if there is an arrangement ft G Em(ar) with T^ conjugate to S. We indicate all three of these relations by the symbol ~. Thus we will write (0,) a ~ (i/j,)a, a ~ fl and T ~ S. 28 Definitions and Examples We investigate the first two of these. We will show that Em(a) is a dense G$ subset of Ta directly by exhibiting it as a countable intersection of open sets. We will also show that each equivalence class Em(a), as a subset of the arrangements, can be endowed with a natural wvmetric making it a universal Gs We obtain this latter result by showing the map A A Em(oC) —> £,,,(a) is obtained by considering £m(<x) modulo a natural group of isometries of £,,,(a). Our first task is to see that on the level of arrangements ~ is an equivalence relation. It then follows automatically for free and ergodic actions. We begin by putting a natural metric on Em(a). As PmA is continuous from fa —y #/, for any P in the range of PmA, its pullback P^liP) will be a closed set. We can use the Hausdorf metric on these closed sets to put a metric on the equivalence class Em(a). More precisely, for P\,Pi € Em(a), define At this point it is not quite evident that this is a metric and not just a pseudometric. We have the following trivial consequence to Axiom 2. Lemma 2.2.9. For every e > 0, there exists a 8 > 0, such that ifm{a., /?) < 5 then \\a,P\\[ < e. That is to say, the identity is uniformly continuous from m to Ki. Hence m(a.,f5) is a metric. In particular, a ~ P if the map P, defined in Theorem 2.2.7, is an isometry. We can now put this together in a simple form: Lemma 2.2.10. There exists a Tequivariant isometry P : (F, ma) —> (F, nip), with PmypP = PmA, if and only if a ~ p. Proof The existence of such a P follows from Theorem 2.2.7 if a ~ p. Conversely, suppose such a P exists. As PmA(id) = a, we see that Pm,p(P(id)) = a. Now P(id) = <#')/, e (f/j.mp). Thus, P^d^)^ = a. 2.2 Definition of a size and mequivalence 29 For any j , we compute that ] ' (id)), —• 0 in j , by Lemma 2.2.4. Thus p'(id) = (4>j)a e (ta,ma), and id e Range(P). • The following theorem is now evident. Theorem 2.2.11. The relation ~ on srf is an equivalence relation. From this we see that ~ breaks si into disjoint equivalence classes on each of which we have defined a metric. Our final step is to see that relative to this metric each of these classes is a Polish space. We begin with the classes Em(a). Theorem 2.2.12. The set Em(a) is a G$subset ofrx. maseparable the matopology on Em(a) is Polish. Proof As the fullgroup is For any cf> and \p in the fullgroup and e > 0 let &(<j),\p,s) = {(#,)« e fa : a<t>i > /} and j), \p) < ma({^i)a4>, ((j)i)xy) + e}. Notice that: (1) Em(a) £ &{(p,ip,e) for all 4>,\p and e; and (2) if (</>,)a 6 &(^>,y),e) for all cj>,\p and e then mp((t>,ip)<ma(P(<t>),P(ip)). As we always have mp((j),ip) > ma{P((f>),P{xp)), we conclude P is an isometry and hence (0,)a e Ea. The fullgroup is separable in the L 1topology and so, by Axiom 2, is separable in the matopology and we can find a countable collection dense in all the mx or m^topologies. Hence It remains to see that the sets &(4>,\p,£) are open in fx. Suppose (4>i)a e &(4>,ip,s) and hence there is an e > 0 with , \p) < ma(((/),)o,0, ((t>i)a\p) + e + e. 30 Definitions and Examples By Axiom 2, Lemma 2.1.8 and Axiom 3 there is a do > 0 such that if then jS,jS'i will be sufficiently small to imply that mp>((t>,ip) < As making sure that <50 < e/3 we will have mir((t),\p) <mli((t>,\p)e/3 ) + e + 2e/3 and /?' e G(<j>,\p,e). D Suppose that ^i = ($•)« and ^2 = (^)« are in £,,,(a) with a0 —> y5 and a0? —> j3. That is to say, PmA(P\) = PmAfc) — ft This means that for all 4> and ip that Begin the definition of an m^isometry /g g by setting The above calculation implies that !„ % is an maisometry where it is defined, and, as the fi\(j) are dense in fa, 1% g will extend to an isometry of fa. Notice that this makes /g g commute with the action of e full group on fa. Lemma 2.2.13. For all /?i, fa in Em(a) with Pm,x{P\) P Proof = PmAfc), w e have I  — P As PmA is equivariant with the action of the fullgroup, PmAiKh{h<l>) = Pm^Upj^m = H = PmAMY This now extends to all of £,,,(a) as the fi\(f) are dense. • 2.2 Definition of a size and mequivalence 31 A Definition 2.2.14. Let J consist of those maisometries of Fx commuting with the action of the fullgroup and satisfying P I— P This is a complete and separable metrizable space under pointwise convergence. Notice then that for any ft\ G Em(a) and / e / , setting ft = l(ji\), we will have ft € Em(a) and Furthermore, as right multiplication by elements of the fullgroup is an maisometry of F a , is a constant on fa. We now argue that the orbits of J are closed sets. Suppose /,(/?o) A A converges to some ft. Then in particular the /,(/?o) will be m^Cauchy. But the above remark implies that /,(/?) is maCauchy for all fi and in particular converges to some /(/?) G Fa. That is to say, /, —> / uniformly and we conclude that / e / and all orbits are closed. Lemma 2.2.15. Using the infimum metric, the space t^/J separable metric space. Proof is a complete To see that the infimum is a metric suppose then of course ley and there will be a sequence /,(ft) —> ft. But this says /,(ft) is i A A and hence /,(/?) is m^Cauchy for all ft That is to say, the /, converge uniformly to another I e J implying ft = /(ft). Thus the infimum is a metric on Ta/J. Separability follows from the fact that f;, is separable. • We now show that Em(a) is also a Gssubset of j / by showing that it embeds as Em(u)/J which we show to be a residual subset of ta/J. We achieve the embedding by lifting /? e Em(a) to P^la(P) n £,,,(a) which we note maps to a singleton in Va/J. It follows directly from the definitions 32 Definitions and Examples that this is a continuous embedding. This implies Em{a) is a universal G$, that is to say a G$ subset of any metric space in which it is embedded. Lemma 2.2.16. The sets 0(0, tp,e) of Theorem 2.2.12 are Hence ^invariant. is a Gssubset. Proof Remember that &(4>, v ,e) = {/? : P^CP) = P and For / e / w e have both Fm?a7 = Pm>a and / is an maisometry. These combine to say that if ft = l(fi) then PmA{P') = P and It is now clear that It is obvious that if (9 is an ./invariant open set then (91J is open, and that the collection of ./invariant sets form a Boolean algebra, with moding out by ./equivariant with the Boolean operations. This completes the result. D Perhaps the best heuristic to take away from this Section is the image of the mequivalence classes as foliating the set of arrangements. Each leaf of this foliation is metrized by its ma as a Polish space and T acts on each leaf minimally and isometrically. 2.3 Seven examples Having developed the axiomatics of mequivalence we now give a list of examples to indicate the range of equivalence relations that can be brought under this perspective. In [43] a number of examples and classes of examples are discussed. Some of those are quite speculative. The Appendix of this work demonstrates how to bring all those examples under the umbrella we open here. The examples we discuss in this Section are those which are most obviously significant and directly related to classical issues in ergodic theory. As part of this discussion we give 2.3 Seven examples 33 some general principles that underlie many of these examples as the beginning of we expect a fruitful study of what a size might look like in general. Many examples of sizes have the common feature of being integrals of some pointwise calculation of the distortion of a single orbit. To make this precise we first review some material about bijections of G. Remember that 38 is the space of all bijections of the group G with the product topology, ^ is the space of bijections fixing id and we metrized both with a complete metric d. The group G can be regarded as a subgroup of 38 acting by left multiplication, (g(g') = gg') The map H : 38 —> 10 given by H(q) = qq(\d)~[ is a contraction in d. Also G acting by right multiplication conjugates 38 to itself giving an action of G on 38. (Tg(q){g') = q(g'g)g~l.) We view this action by representing an element q e 38 by a map f : G —> G, /(g) = <?(g)g~'. Those maps / e GG that arise from bijections are a G$ and hence a Polish space we call F. The map q —> / is obviously a homeomorphism from 38 to F. For / e F let Q(f) be the associated bijection and for q e 38 let F(q) be the associated name in G c . The action of G on 38 in its representation as F is the shift action <xg(/)(g') = /(g'g) Any rearrangement pair (a, </>) then gives rise to an ergodic shift invariant measure on this Polish subset of GG and any ergodic shift invariant measure is an ergodic action of G with a canonical rearrangement pair. The probability measures on a Polish space are weak* Polish [57] and hence the invariant and ergodic measures on this Polish space are weak* Polish. We will now define a general class of sizes that arise as integrals of valuations made on the bijections q'd. Definition 2.3.1. A Borel D : 38 —> K + is called a size kernel if it satisfies: D(q)>0; D(id) = 0; D(q(id)lqlq(id)) = D(q); D(ql(id)q2qYl(id)ql) < Dfo,) + D(q2); for every e > 0 there is a 5 > 0 so that if D(q) < 5 then d(id,H(q))<s; (6) the function fi —> f D(q(f))d[i is weak* continuous on space of shift invariant measures \i on the Polish space F. (1) (2) (3) (4) (5) Note: an element of G regarded as an element of 38 acts by left multiplication. 34 Definitions and Examples The complex form of conditions (3) and (4) arise from the following considerations. When an orbit is viewed as a copy of G the base point x sits on the identity element of the group. When acted on by some rearrangement the identity moves, i.e. the point x now is based at a different point in G. Hence it is necessary to view both q~l and q as based at this new origin when they act. Writing it out explicitly on an orbit we have the identities and and now conditions (3) and (4) become and For size kernels D, defined solely in terms of H(q), these two become even simpler as and For a size kernel D we define mD{a,(j>)= ID(q«' We call such an mD an integral size. In condition (6) on D we could have asked for only upper semicontinuity and still obtained that mD was a size. All examples though are continuous here so we ask for the stronger condition and obtain a stronger conclusion: Theorem 2.3.2. For D a size kernel, mD is a 3 + size. 2.3 Seven examples 35 Proof First note that i and so and from the above identity and condition (3), Condition (3) gives symmetry as and m® is a pseudometric on F. Axiom 2 of a size follows directly from condition (5). Condition (6) is precisely that Axiom 3 should hold. • Examples 1 & 2 (Conjugacy and Orbit Equivalence). These first two examples are the extremes of what is possible. For one the equivalence class will simply be the fullgroup orbit and for the other it will be the entire set of arrangements. Both of the pseudometrics d(q, id) and d(H(q), id) are easily seen to be size kernels and so both m\a,(j>) =  (a, 0), (a, id) C are 3 + sizes. As d makes 38 complete, relative to m1 a class of sequences ((/>,)„ e F a iff (j>j —> (f> in probability. Thus a ~ /} iff /? = oup i.e. they differ by an element of the fullgroup and the equivalence class of a is exactly its fullgroup orbit. Note in particular that T a and T^ will be conjugate actions. To tie this relation into our work here notice that in Chapter 7, where we define the notion of an mfinitely determined action, this 36 Definitions and Examples definition reduces to Ornstein's classical characterization of the Bernoulli actions as the finitely determined actions for m1. As for m°, for any a and fi one can use the OrnsteinWeiss Rokhlin Lemma (Lemma 2.1.5) to construct a sequence of (/>, with <x$; —• fi in L' with the sequence </>, an maCauchy sequence. Thus all arrangements are m°equivalent. Dye's Theorem [10] and the Theorem of Connes, Feldman and Weiss [6] now tell us that any two ergodic actions of G are m°equivalent. We tie this example into our work. First a reminder of the distribution topology. For G a countable and amenable group and £ a finite labeling set, the space of probability measures on £ G forms a compact metrizable space. For any measurepreserving action of G and £ valued partition the map from points to £, Gnames will project the invariant measure to a measure on £ G and will make a pseudometric space of such processes (pairs of actions and partitions). For two such pairs (T,P) and (S,Q) let \\(T,P),(S,Q)\\, be some metric giving this weak* or distribution pseudotopology. We state a lemma concerning ergodic actions of G. Lemma 2.3.3. Let G be a countable and amenable group, Ta a free and ergodic action of G on the standard space {X,2F,\i) and P a finite Hvalued partition of X. For each e > 0 there is a 5 > 0 so that for any other free and ergodic action S^ of G on (Y,$, v) and partition Q : Y —> £ satisfying: (1) \\(T«,P),(SP,Q)\\t<5 and for every 5\ < 0 there is a 0 in the fullgroup of fi and a partition Q' of Y with (a) m%S, <£)<£(b) v(QAQ') < s and (V) \\(T',P),(Sl>*,Q:)\\.<5i. We leave the proof as an exercise for the reader. A version of this fact for Kakutani Equivalence is found in lemma 4.3 of [38]. The reader can use this as an outline of to how to proceed. One concludes from this that all ergodic actions of G are weakly m°finitely determined (see Definition 7.2.5) and, as m° is a 3 + size, Theorem 7.2.6 now implies all ergodic actions are m°f.d. giving an alternate albeit elaborate proof of Dye's theorem using our machinery. Before we continue to other examples we make a few general observations concerning size kernels. First we can w.l.o.g. assume that all size kernels are bounded by 1, as replacing D by the supremum of D and 1 will maintain the axioms and will not change the associated equivalence 2.3 Seven examples 37 relation. Notice next that one evaluates the size mD of a rearrangement by calculating / D(Q(f))d/i(f) where fi is some shift invariant measure on F. Suppose Fo £ F is a shift invariant set with H(FQ) = 1 for all shift invariant probability measures \i. Assume as well that FQ contains the identity and if it contains F(q) then it also contains F(q~l). Notice that changing D outside of Fo will have no effect on the evaluation of mD. In particular if D is initially only defined on Fo, is bounded by 1 there and satisfies the axioms of a size kernel (where applicable), then if we set D(Q(f)) = 1 for / ^ Fo we would extend D so as to be a size kernel. We now give an explicit example of such an Fo. Suppose G is Z" and BN = [—N,N]" is the standard Felner sequence of boxes. For shift invariant measures on GG the pointwise ergodic theorem holds along this sequence BN For each / £ F set A M ( / ) to be the upper density of the set {v\f(v) £ BM} calculated along the sequence of sets BN as N s oo. Let Fo consist of those / for which lim A M (/) = 0. The pointwise M>co ergodic theorem tells us that this set has measure 1 for all shift invariant measures. Hence when working in Z" one need only define a size kernel D on such / . Notice that for such / one will have im N>C» #{i3 € BN\Q(f)(v) £ BN} = Q #BN We describe a class of examples that take advantage of these observations and this choice for an FQ. Example 3 (Kakutani equivalence). The development of Kakutani equivalence in Z" can be found in [9] and a complete development of the equivalence theorem for it in [16]. What we present here is an approach that brings this example into our context. For this example let G = Z" and BN = [—N,N]n be the standard Folner sequence of boxes centered at 0. We begin with a metric on Z" given by x{u,v) = min(K«/M)  (S/t51) +  (assuming 0/0 = 0). What is important about T are the following two properties: (1) T is a metric on Z" bounded by 1; and (2) w and v are x close iff the norm of their difference is small in proportion to both of their norms. 38 Definitions and Examples For h e % set BN(h) = {v e mapped into BN by /i). Now set BN/I(2) e BN} (those elements of = sup N & N ieBN(h) and /<(<?) = k(H(q). Lemma 2.3.4. The function K is a size kernel. Proof That k(h) = k(h~]) is a calculation as in fact this equality holds already for each TV. That k{h2 o h\) < k(h2) + k(h\) is also true as it is true for each TV before taking the sup N . That K satisfies the first five conditions of a size kernel is now direct. We get (3) and (4) from the observation that K(q) only depends on H{q). As x is a metric, for K(q) to be small H(q) must fix a large (finite of course) number of vectors v. To obtain (6) suppose [i is some shift invariant measure on GG, hence supported on FQ. For q € FQ as TV —> oo we have \BN(H(q))\/\BN\ —> 1. Moreover for each TV this value is continuous and so its expected value relative to fi is weak* continuous. For any fixed D, {q\H(q)(id) = v} is a clopen set and hence its measure is weak* continuous in p.. As i; varies these sets form a countable partition of GG and so for any e > 0 there is an No and a neighborhood U of j.i so that for v e U also invariant and TV > No, letting h = h(q(f)), I ueBN(h) For each N < No the calculation ieBN(h) is continuous and hence the supremum of these values for N < No is, continuous and so its integral is weak* continuous in fi. It follows that in some subneighborhood U' £ U we will have a variation of at most e in the value jK(q{f))dv. • The use of T here is not the usual calculation taken to construct Kakutani equivalence, but noting that for T to be small simply means the distance between two vectors is small relative to their lengths makes it clear that it is equivalent to earlier presentations. One finds without 2.3 Seven examples 39 mK much effort that a ~ /? iff for a.e. x Although the vocabulary of [9] is somewhat different it is shown there that this is equivalent to saying: Proposition 2.3.5. For every e > 0 there is a (/> e F and a subset A with fi(A) > I — 6 so that for all x,y £ A, x{a4)(x,y),tx(x,y))<£. For G = Z this implies that T^ and T& induce the same map on A and hence Ta and T^ are evenly Kakutani equivalent in the classical sense. In [9] the converse of this is proven, i.e. this is precisely even Kakutani equivalence in Z and a broad exploration of this equivalence relation in Z'' is made connecting it to Katok crosssections of Rd actions. As mK is entropypreserving we know the Bernoulli actions are mKfinitely determined and hence there exist mK finitely determined actions. By Theorem 7.2.6 they are characterized by the condition of being weakly mK finitely determined. Notice that for actions of Z this precise fact is proven in lemma 4.3 of [38]. Examples 4 & 5 (a equivalences). Once more take G = Z" and choose a vector a = {<xi,a2,...,a,,} of nonzero real numbers. Set A : K" —• T" to be A(vi,...,vn) = (vi/ai,V2/<*2,...,vn/<xn) mod Z" and on T" use the natural metric p ( M 2 ) = e 2 *' v 'e 2 *' T ; 2 '. Notice that for p o A to be small means the two vectors differ approximately by a vector {«iai,...,n,a,} where the ;i, are integers. For q e 88 set k(q) = p(A(q(0)),A(0)). A is not a size kernel as it fails to satisfy (1) although it does satisfy both (2) and (3). To obtain a size kernel all we need do is add to A some other size kernel. We currently have two choices giving the two size kernels Dsfa) = d(H(q),id) + A(q) K&(q) = K(q) + A(q). and 40 Definitions and Examples (Notice it makes no sense to use d(q, id) as adding on A(q) would add no further restriction to the already minimal equivalence class.) Both of these examples give interesting equivalence relations. The second has been well studied under the name of aequivalence (see [8] and [48]). Because of the standard use of a to represent the parameter of this relation we will use /? to represent an arrangement throughout the discussion of these two examples. The first example, D&, has not been discussed in the literature so we present a brief discussion here. What we obtain is a refinement of simple orbit equivalence that splits the ergodic actions into a countable list of equivalence classes characterized spectrally. Remember a function / : X —> C is an eigenfunction of the ergodic action T with eigenvalue 1 if / is of norm one and Fixing i 0 = ( l / a i , l / a 2 , . . . , l / a , , ) , those values (kuk2,...,kn) € Z" for is an eigenvalue for T form an additive which (k\/a\,k2/<X2,...,kn/an) subgroup we will call As(T). Two ergodic actions U and V of Z" are mDi equivalent iff A 5 ([/) = Aj(K). In particular we see that there are at most countably many mDiequivalence classes. We will indicate the proof of parts of this characterization, leaving much to the reader. Proposition 2.3.6. If fix "~ /?2 then Aa(T^) = A 5 ( 7 % Proof Suppose X = {k\/a\,ki/a.2,...,kn/a.n) is an eigenvalue for the eigenfunction / of T^'. We now compute that for all v = I f ° Tp' HJ ,, „ B+/?(v^(x))/j(r'1(x)0(rflw)) < \\p{A{q*'*(d)),A(6))h Thus if /?i m~ fo, for all v we will have  f 2.3 Seven examples 41 By the mean ergodic theorem there must be an / with veBN and by the above, We know / must have constant norm and as long as it is not identically 0> / / I / I will be an eigenfunction for T"1 with eigenvalue I As all TM are conjugate to T^1 we can assume mDi(fS\,fi2) < 1/2, forcing / ^ 0. • Lemma 2.3.7. Suppose T^ is a free and ergodic action of Z" on (X, J*, /i) with As(T^) = {6}, and P : X » Z is a finite partition. For each e > 0 there is a 5 > 0 so that for any other free and ergodic action Sy on (Y,(S, v) and partition Q : Y —> Z satisfying: an>> value 5\ < 0 t/iere is a 0 in the fullgroup of Sy and a partition Q' : Y » Z wit/i (a) m D »(y,0)<£ (b) v(QAQ') < £ and (l') ^ We once more leave a complete proof of this copying lemma to the reader. We do point out the ingredient used to obtain (a) beyond the construction of Lemma 2.2.3. For a fixed consider the group rotation on T" given by (xi,...,x n ) —> (xi + a[,...,xn + a,,) mod 1. This is not necessarily ergodic but all its ergodic components are conjugate to some group rotation we call (Rz,Z) where Z is a compact subgroup of T". To say A 5(T^) = {6} is equivalent to saying R& x T& is ergodic. Partition Z into sets of diameter less than e/2 by a partition H. Consider now H V P,Bjvnames arising from the action of R& x T^. The pointwise ergodic theorem tells us that if we fix a choice of h £ H and cylinder set C in the process (T$,P) then for N large the relative density of C just at indices of an H\/ P,BNname whose H term is h will be very close to n(C). Fixing the value N, if {Sy,Q) satisfies (1) for a small enough 5 then this same fact will be true for H V gnames relative to v (even if R$ x Sy is 42 Definitions and Examples not ergodic). The fullgroup element 4> will now be constructed on some Rokhlin tower of size BM, (M » N) in the action Sy by overlaying the Sy, Q names with a template R&, //name and constructing <f> to move this name close to some T^,Pname in d while simultaneously preserving the template name. The remark in the previous paragraph guarantees that this can be done. This preservation of the template name is the new ingredient needed to ensure mDi(y,(j)) < e. One now changes Q slightly according to the necessary d error to be an exact copy of the target T^Pname. This Lemma and the previous Proposition now imply that the ra°5finitely determined actions are those with Aa(T^) = {0} (see Definition 7.2.5 and Theorem 7.2.6) and any two such actions will be mDiequivalent. One approach to showing that the group A^T^) is a complete invariant even outside the finitely determined class is to fix a choice A for As and generalize Lemma 2.3.7 and the theory of mDijoinings of Chapter 6 to a relativized version relative to the nontrivial isometric factor algebra generated by the eigenfunctions whose eigenvalues lie in A. As indicated earlier the size kernel K% leads to a rather well studied area. Certainly mKiequivalence refines even Kakutani equivalence. The argument in Proposition 2.3.6 applies here as well to say the group A5 is an mKi invariant. Hence each even Kakutani equivalence class is cut into at least countably many mKi classes. In [8] it is shown that in one dimension, and for the "loosely Bernoulli" class (the mK finitely determined class), this is the full refinement. That argument will push through to all dimensions. As a consequence of the construction in [15] there are other mK classes which contain uncountably infinitely many mKi classes. There exists a connection to sections for actions of R" as well, although it is understood only in 1 and 2 dimensions. The following is what is known in one dimension. For a irrational and > 0 any measurepreserving flow can be represented as a flow under a function taking on only the two values 1 and 1 + a [45]. Just as for Kakutani equivalence itself one can define here a relationship between Z actions by saying they are arelated if they arise as the return maps from a common flow where the return times take on only these two values 1 and 1 + a. We say they are evenly arelated iff the integrals of these return time functions agree. In [8] it is shown that two Z actions are arelated iff they are mKiequivalent. Moreover it is shown that if U and V are mK5equivalent, then any flow for which U arises as such a section, V does as well. 2.3 Seven examples 43 Here is what is known in two and higher dimensions. In [47] it is shown that any R" action can be represented as a special sort of "Markov" tiling suspension of an action of Z" where the tiles are rectangles whose length in dimension k is either only 1 or 1 + a* (we assume all a/< are irrational and > 0). We say two actions U and V of Z" are a related if they arise as such sections of a common W action. We say they are evenly 5equivalent if the proportion of the space occupied by each of the tile shapes is the same for both representations. The argument in [8] extends to higher dimensions to show that if U and V are evenly arelated then they are mKiequivalent. In two dimensions Sahin [48] shows that the converse is true, i.e. if two actions U and V are mKiequivalent then they arise as Markov tiling sections of a common K" action. It remains open however whether any R" action for which U is such a section must also have V as such a section. Our last two examples exhibit another general context in which a restricted orbit equivalence relation can arise. Suppose that to each arrangment a we can assign a subgroup Fg of the fullgroup F with the equivariance property that FQ* = (^"'Fg^. In particular the choice of subgroup does not change when we perturb a by an element of its subgroup. What interests us are those /? reachable as limits of sequences of rearrangements a</>, where <f>j £ Fg. We write a size for such a relation as a sum of two pieces, one measuring the ml distance from <f> to Fg and the other measuring some chosen size of (a, cf>). As a simple example of this consider aequivalence in Z for a = 2. Here the groups Fg = {cj)\ix(x, (/)(x)) = 0 mod 2}. Using the two sizes m° and mK within the subgroups will yield the two examples described of aequivalence. This idea becomes cumbersome for a, irrational. We do not attempt to axiomatize precisely what is needed of the family of subgroups, leaving this to a more general study of sizes. Our final two examples will offer an indication of the range this idea covers. Example 6 (Vershik's f Equivalence). The work described here can be found in [18] and [19] of Heicklen. Suppose (X, 3F,\t) is a standard nonatomic probability space and J*, is a sequence of sub <ralgebras with 3F = J^o and !Fi+\ £ !F[. We refer to such a sequence as a reverse filtration. Two such sequences are conjugate if there is a measurepreserving bijection between the measure spaces carrying one reverse filtration, term by term, to the other. To remove some trivial issues and make this subject addressable by our methods we make two assumptions. First, we assume that for each /' the conditional fiber measures of J% over 44 Definitions and Examples #",_! are atomic with a fixed number of atoms k, and that each atom has a constant mass l//c,. We call such a filtration uniform. Next we assume that the 3Fi decrease to the trivial algebra. A filtration with this property is called exact. One natural way for such a filtration to arise is from an action of a group of the form G = Y^=i Z/r n Z. What matters here is that G is the increasing union of the finite groups Hi = Yl'n=\ Z/r n Z. If we have a measurepreserving and free action of this group and we set 2Fi to be the algebra of H\ invariant sets then we obtain a uniform reverse filtration. It is exact iff the action is ergodic. Conversely, given any uniform and exact reverse filtration, using the Rokhlin decomposition of each successive #", over J ^  i , we can place on the space an action of G for which the filtration is obtained as this list of invariant subalgebras. The action of G here is not unique and this leads to a natural relation: we say two actions of G are Vershik related if conjugate versions of both of them can be placed on the same space, giving rise to the same reverse filtration. Notice in particular that the two actions will be orbitequivalent and what characterizes the particular orbit equivalence is that it preserves the orbits of all the subgroups //,. For the purpose of our discussion it will be useful to assume only that G is the increasing union of finite abelian groups //, without assuming that each is cyclic over its predecessor. Such a G is countable and amenable. Notice that for any increasing subsequence {_/,} we could define //, = Hjt and get another representation of G as an increasing union of finite subgroups. These distinct representations will give distinct values for the vector f = {\H{\, \H2/H\\,...} and so we can represent a choice for such an increasing subsequence of subgroups by its vector f. This is consistent with the usage when //,///,_! is cyclic of order r,. We say two actions of G are Vershik rrelated if they are Vershik related for the choice of subgroups H; determined by the values of f. We describe Vershik relatedness indexed by the choice of subsequence r a s a family of restricted orbit equivalences on G. Although one can use a size kernel here we follow [19] and give the size directly. Notice that for a fixed arrangement a and choice for f the fullgroup F contains closed subgroups F0' a consisting of those cj> which preserve the Tx orbits of all //,. (This is equivalent to saying that either q*'^ or equivalently h^ permutes cosets of //, for all i and a.e. x.) If we have a sequence of rearrangements a</>, converging to some fi where all the (pj £ rf^ then T a and T1* will have identical H, orbits for all i and hence be Vershik f related in this very strong sense. To define a size giving this relation, for a and r fixed, we first calculate the distance some 2.3 Seven examples 45 is from the subgroup FrA as its m\ distance: c?(a,4>)= inf n{x\4>{x) (This is not the definition of c> given in [19] but is equivalent by the Flattening Lemma proven there.) One can now define the family of sizes Heicklen proves this to be a size but does not show it to be 3 + . We will not present the details showing that it is in fact 3 + . This can be done either by suitably expanding Heicklen's argument or by showing that the sizes mr arise from size kernels. Heicklen's conclusion is that two actions a ~ /J iff there is a ip G F so that Tai' and T'1 have identical Ht orbits for all i. As Ta and T0"" are conjugate (by \p of course) T01 and T^ are f related and if two actions are f related they can be realized as two such actions. The family of sizes exhibits two very interesting properties. The first is due to Vershik who proved a lacunary isomorphism theorem for such groups [58]: for any two actions U and V, if the /, are chosen to grow rapidly enough, then the two actions are f related. Notice that the allowed choices for the sequence //, are partially ordered under containment. As one goes further out in this partial order more and more actions become equivalent and Vershik's result says any two will become equivalent once one is far enough out in this net. The second very interesting property follows from this. Vershik has shown that for very slowly growing sequences, like r, — 2 for all /, the entropy of a Gaction is an invariant of requivalence. On the other hand, Vershik's lacunary isomorphism theorem tells us that for some choices of r, entropy definitely is not an invariant. As we learn in Chapter 5 a restricted orbit equivalence either preserves entropy or generically in the ma topology an action has zero entropy. Vershik [59] conjectured and proved the sufficiency and Heicklen [19] proved the necessity of the following characterization of the boundary between these two regimes: The size mr is entropypreserving iff Example 7 (Entropy as a Size). We discuss this example only for actions of Z although the ideas extend to general countable amenable groups. The 46 Definitions and Examples results described here are found in [46]. Before examining this example in detail consider the following observations. Two major goals of this current work are to demonstrate: (1) A size m is either entropypreserving in that two equivalent actions have the same entropy, or entropyfree in that residually in each class actions have zero entropy. In the first case we say an action's mentropy is its entropy and in the latter that its mentropy is always zero. (2) Each size possesses a family of distinguished classes, characterized by their mentropy, called the mfinitely determined classes. Any two mfinitely determined actions of the same mentropy are mequivalent. Notice that this implies the possibility of two sizes m for which all actions are finitely determined, one that is entropyfree and one that is entropypreserving. Dye's theorem, here done via the size m°, shows that there is an entropyfree size for which all actions are mfinitely determined. What the example we now discuss shows is that the other size also exists, relative to which two actions are equivalent iff they have the same entropy. The size at its base will simply be the entropy of the rearrangment itself. We make this precise as follows. Note g(«,0)(x) = a(x,(j)(x)) takes on countably many values and hence can be regarded as a countable partition g ^ j of X. Set Fg to be those <> / for which g ^ j is finite. It is not difficult to see that Fjj is a subgroup and moreover FQV) = I/;~'FQ1/; as g(av,,v,^VI)(v~'(A)) = g(«,^)(x). It is shown in Theorem 4.0.2 that the Fg are all ml dense in F. For <> / e Fg one can use the entropy of the process h(Tx,g{a,<t>)) t 0 start the definition of a size defining e(a, 4>) = m^ h{T\ g M )) + n{x\4>{x) + <j>' (x)}. Now set the size to be To see that this is a size, Axiom 1 follows from basic conditional entropy considerations and Axiom 2 is directly due to the second term. Axiom 3 here follows from upper semicontinuity of entropy and for this reason this is not on the face of it a 3 + size. To see that in fact it is not, note that those a for which T a is zero entropy are m° residual. Hence a rearrangment (a, <j>) can be perturbed by as little as we like in distribution to an (a', </)') with m"(a', 0') = 0. It is this example which motivated the weakening of Axiom 3 + to Axiom 3. 2.3 Seven examples 47 The form of the size makes it reasonable to believe and easy to prove that m''equivalence will be entropypreserving. A more subtle combinatorial argument leads to the reverse conclusion as well, that any two ergodic actions of equal entropy are in fact /n''equivalent. A NonExample. Hoffman and Rudolph [20] have presented an isomorphism theory for measurepreserving endomorphisms that can be viewed as an extension of the methods here but cannot be viewed as an application of restricted orbit equivalence as we develop it. This work grows naturally from the Vershik equivalence theory described in Example 6. Consider the standard example discussed there, {0,1,...,p— 1 }N with uniform Bernoulli product measure (1/p,..., l/p). In Example 6 this gives the standard filtration where all k, = p. Here we consider in addition the shift map, giving a p  1 Bernoulli endomorphism. We call it "uniform" as all p inverse images of a point are equally likely. An endomorphism conjugate to this standard one would also have to be uniformly p— 1 in that almost every point must have p inverse images and all must be equally likely. This standard example has entropy logp so this also would be true of any endomorphism conjugage to it. What is shown in [20] is that for the class of uniformly p — 1 endomorphisms of entropy logp there is an isomorphism theory completely analogous to that of Ornstein for Bernoulli automorphisms and following the same outline as our work here. We will describe enough of [20] to indicate why the theory is parallel and why the results here simply do not apply. Let T acting on (X, 3F, \i) be an ergodic and uniformly p— 1 endomorphism of entropy log p. For a.e. x € X we can consider the set of all inverse images T~i(x), x > 0 organized as a pary tree rooted at x. The points in T~>(x) are at "level / ' of the tree and each x\ € T~'(x) is connected by one edge to T(x\) e T~j+l(x). We refer to a map from the nodes of such a tree to itself that preserves the edges as a "tree automorphism" and between two such trees as a "tree isomorphism". A conjugacy between two uniformly p — 1 endomorphisms will give tree isomorphisms between the trees of inverse images attached to matched points. For P a finite partition of X label each node of a tree of inverse images by the set in P containing it. Given two such labeled trees of inverse images one can ask how closely they can be matched by a tree isomorphism. To be more precise, each node at level j is given a mass of 2~J so that the set of nodes at each level has total mass 1. Relative to this weighting one seeks to match the nodes of two labeled trees by 48 Definitions and Examples a tree isomorphism minimizing the proportion of nodes in the trees with mismatched labels. This minimum is called the t distance between the two labeled trees. This extends in a standard fashion to provide a f distance between measures on labeled trees and hence between uniformly p — 1 Fvalued stationary stochastic processes. Notions of t finitelydetermined and t very weakly Bernoulli follow directly and can be shown to be equivalent. Both these are true of the standard example and are conjugacy invariants. Finally, one can give a natural weak* Polish space of joining measures for this theory, called the "onesided joinings", and show that for the t finitely determined endomorphisms those onesided joinings which arise from conjugacies are a residual subset. This development follows the outline of our work here and is in fact much simpler both because it is just a single example of an equivalence relation and because it is a "zero entropy" theory. Example 6 of Vershik equivalence was brought under the restricted orbit equivalence umbrella by the choice of an action whose orbit structure mirrored the fibers of the filtration. Here one can also find such a group action. Construct the natural orbit relation setting x\ ~ xj if there are j \ > 0 and ji > 0 with T'1 (xj) = T' 2 ^) The equivalence classes are organized naturally as a complete pary tree (each node has p + 1 edges attached to it, p going back in time and 1 going forward). These naturally organize as the orbits of a free group on p involutions where two points are connected by an edge in the tree iff they are interchanged by one of these p involutions. This is not an amenable group so our work here does not apply. This action also is not measurepreserving. It is in fact an amenable action of type IIIi/ p . As such it could be given as an orbit of a Z action but this would lose the tree structure essential to the conjugacy theory of the original endomorphisms. What is needed to make this nonexample an example is to lift the work here to nonsingular amenable actions of groups that are not necessarily amenable. This nonexample gives evidence for, and an approach to, such a generalization. 3 The OrnsteinWeiss Machinery In this section, we describe the constructive tools we will need in order to continue with our work. From the beginnings of the Ornstein approach to constructive ergodic theory and in particular the Isomorphism Theorem it has been understood that there are three basic tools necessary to work constructively with dynamical systems: a version of the Rokhlin lemma; a version of the Ergodic theorem; and a version of the ShannonMcMillan theorem. It has also been understood for some time that a natural context in which all these results hold is that of locally compact and amenable groups. The results described here are lifted almost verbatim from the seminal work of Ornstein and Weiss on this subject [37]. We include them here to provide the reader with ready access to them and as we vary their statements slightly in places. Since we will consider only countable discrete amenable group actions, we do not need the most general form of their results. Thus, for clarity, we have stated these results in the context of discrete group actions. Furthermore, we have opted for a classical description of entropy, using finite partitions and namecounting techniques. Two notions of essential invariance of finite subsets F £ G are central to [37]. Definition 3.0.1. Let d > 0. Let K £ G be a finite set. A subset F £ G is called (6, K)invariant if f(KK~lFAF) To say that a set F is sufficiently invariant means that there exists a 5 > 0 and a finite set K s G such that F is (S,K)invariant. 49 50 The OrnsteinWeiss Machinery is sufficiently invariant is to say To say that a list of sets F\,F2,,Fii that there exists a 8 > 0 and a finite set K £ G, such that, setting FQ = K, for each j G {1,2,...,k}, the set Fj is (d,Fj\)invariant. We now describe our version of the OrnsteinWeiss quasitiling theorem. Definition 3.0.2. A finite list of sets H\,H2,..,Hk £ G, with id € H,, for all i, is said to \,2,...,k, £quasitile a finite set F £ G if there exist "centers" Cjj, i = j = 1,2,...,/(/), and subsets Hjj £ H, such that: (1) (2) the HJJCJJ £ F are disjoint; (3) and ( y y ) Theorem 3.0.3 ([37]). Given e > 0, there exists N = N(e) such that is any suffiin any countable discrete amenable group G, if H\,...,H^ ciently invariant list of sets, then for any £> £ G that is sufficiently invariant (depending on the choice of H\,...,H^), D can be equasitiled by H\,...,Hs This theorem is the essential content of Theorem 6, 1.2 [37]. Our definition of equasitiling is slightly different; weaker in that we do not ask that H,c,j n Hkcuj = 0, /' ^ k, and stronger in that we require HjjCjj £ F. Obtaining the latter from Theorem 6, 1.2 [37] is easy if D is sufficiently invariant and N is fixed. We have described this result, as the picture it gives makes much of [37] more accessible. We will not go further into the development of families of tilings, which are the essential tools of their proofs. Rather, we will move on to state their principle results. In particular, we will discuss their version of the Rokhlin and strong Rokhlin lemmas, which can be regarded as dynamical versions of the tiling theorem. Suppose (X,£8,n) is a standard probability space. Suppose I is a measurepreserving free action of G on X. For a finite set F £ G and measurable subset / l e ^ with /i(A) > 0, consider F x A £ G x X. As a measure on F x A, put the direct product c x \i of counting measure c and fi. Consider the map T : F x A —> X given by T(g, x) = Tg(x). On each level set g x A, T is 11 and measurepreserving. On any fiber set F x x, T is again 11. We definitely do not expect T to be 11 on F x A. It is clear, though, that T is nonsingular and, at most, # F to 1. The OrnsteinlVeiss Machinery 51 In particular, if T is y'tol at x e X, then Rokhlin lemmas concern the degree to which maps T, as above, can be made 11. In particular, within a set F x A, one can look for large subsets S on which T is 11. We will ask that S be large in a rather strong sense. For S £ F x A, we get a counting function, defined on X, given by Of course (cx/i)(S)= / cs{x)dii(x). JA Set c(S) = min cs(x). Definition 3.0.4. We say that F x A maps an £quasitower if there exists a measurable subset S S F x A such that: (1) T\s is 11; and (2) c ( S ) > ( l  e ) # F . The equasitower itself is T(F x A) £ X. Notice that we may always assume T(S) = T(F x A). Note that if there exists an S s F x A, such that T is 11 on S and (c x JI)(S) > (1 e 2 )(c x /i)(F x /I), then there must exist an A' £ /I, with /((/I') > (1 — e)fi(A), such that cs(x) > (1 — e)#F, for all x e A'. Hence F x A' maps to an equasitower. We now state our version of the OrnsteinWeiss Rokhlin lemma. It is only a minor modification of Theorem 5, II.2 of [37]. Theorem 3.0.5 ([37]). Suppose G is a discrete amenable group. For any e > 0, there exist 5 > 0, K <= Q and N = N(s) such that for any sequence H\,..., HN of(5,K)invariant subsets ofG, and any free measurepreserving Gaction T = {Tg}geC> acting on (X,3S,^t), there exist sets A\,...,AN e & such that: (1) each Hi x A, maps to an squasitower £%, in X; (2) for i ^= j , 3#i n 3kj = 0; and (3) 52 The OrnsteinWeiss Machinery A collection of sets of the form {H, x Aj}f=l satisfying (1), (2) and (3), we call an eRokhlin tower. As we indicated earlier, Ornstein and Weiss prove this for a slightly different notion of equasitower. We commented above that using e2 in their result gives the e in ours. Their statement differs slightly in another respect. They partition each Ai further into sets Ajj with T actually 11 on each H, x A,j. In fact, for any set A £ 3S, and finite set H £ G, A can be partitioned into a countable list of sets Aj with T 11 on each H x Aj simply because T acts freely. Hence this added structure is automatic. Theorem 3.0.6 ([37]). Given any finite partition P of X, one can select the sets Aj in Theorem 3.0.5 with Proof This result is Theorem 6, II.2 of [37]. Again, we have stated a slightly strengthened version. The important observation from Theorem 6, II.2, is that setting "e"= e/10, one obtains N and {S,K) for Theorem 3.0.5. one is now at liberty to choose Then for ((5,X)invariant sets H\,...,HN, a finite partition Q, which we set to be V T.i(P). geytf, g Theorem 6, II.2 contains an extra parameter 5 with Aj ±5 Q. But as Ornstein and Weiss point out following the proof, by slightly shrinking the Ai, one obtains strict independence. • We now describe the entropy of an ergodic, measurepreserving Gaction. We use a namecounting approach, as described in [44]. Because we are considering only discrete amenable group actions, the entropy fun