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Alexander S. Kechris Benjamin D. Miller

Topics in Orbit Equivalence

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Author Alexander S. Kechris Department of Mathematics California Institute of Technology 253-37 Pasadena, CA 91125 U.S.A. e-mail: [email protected] Benjamin D. Miller Department of Mathematics University of California at Los Angeles 6363 Math. Sci. Bldg. Los Angeles, CA 90095-1555 U.S.A. e-mail: [email protected]

Library of Congress Control Number: 200410415

Mathematics Subject Classification (2000): 03E15, 28D15, 37A15, 37A20, 37A35, 43A07, 54H05 ISSN 0075-8434 ISBN 3-540-22603-6 Springer Berlin Heidelberg New York DOI: 10.1007/b99421 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to proscution under the German Copyright Law. Springer is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004
Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors 41/3142/du - 543210 - Printed on acid-free paper

Preface

(A) These notes provide an introduction to some topics in orbit equivalence theory, a branch of ergodic theory. One of the main concerns of ergodic theory is the structure and classification of measure preserving (or more generally measure-class preserving) actions of groups. By contrast, in orbit equivalence theory one focuses on the equivalence relation induced by such an action, i.e., the equivalence relation whose classes are the orbits of the action. This point of view originated in the pioneering work of Dye in the late 1950’s, in connection with the theory of operator algebras. Since that time orbit equivalence theory has been a very active area of research in which a number of remarkable results have been obtained. Roughly speaking, two main and opposing phenomena have been discovered, which we will refer to as elasticity (not a standard terminology) and rigidity. To explain them, we will need to introduce first the basic concepts of orbit equivalence theory. In these notes we will only consider countable, discrete groups Γ . If such a group Γ acts in a Borel way on a standard Borel space X, we denote by EΓX the corresponding equivalence relation on X: xEΓX y ⇔ ∃γ ∈ Γ (γ · x = y). If µ is a probability (Borel) measure on X, the action preserves µ if µ(γ · A) = µ(A), for any Borel set A ⊆ X and γ ∈ Γ