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Ergodic theory via joinings. (English) Zbl 1038.37002

Mathematical Surveys and Monographs 101. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3372-3/hbk). xi, 384 p. (2003).
The idea of joining in ergodic theory evolved from the work of Furstenberg, Ornstein and Rudolph, and was put into the mainstream of ergodic theory by Rudolph. This is one of the first books to look at ergodic theory from the point of view of joinings, although an earlier book of D. J. Rudolph [Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces, Oxford Science Publication. Oxford: Clarendon Press (1990; Zbl 0718.28008)] also treated joinings, and some sections of this book are inspired by the earlier book, particularly those on the isomorphism theory of Bernoulli shifts.
Given two dynamical systems \({\mathcal X}= (X,{\mathcal F},\mu,T)\) and \({\mathcal Y}= (Y,{\mathcal B},\nu, S)\) (so that \((X, {\mathcal F},\mu)\) is a Lebesgue probability space, and \(T: X\to X\) is a measure-preserving transformation and similarly for \((Y,{\mathcal B},\nu)\)), a joining of \({\mathcal X}\) and \({\mathcal Y}\) is a measure \(\lambda\) defined on \({\mathcal F}\otimes{\mathcal B}\), which is \(T\times S\) invariant and for which the marginals are \(\mu\) and \(\nu\) (i.e., \(\lambda(A\times Y)= \mu(A)\) and \(\lambda(X\times B)= \nu(B)\), for all \(A\in{\mathcal F}\), \(B\in{\mathcal B}\)).
The dynamical system \({\mathcal Y}\) is said to be a factor of \({\mathcal X}\) if there is a measurable map \(\pi: X\to Y\) with the properties \(\pi(\mu)= \nu\) and \(T\circ\pi= \pi\circ S\). A common factor of \({\mathcal X}\) and \({\mathcal Y}\) is a third system \({\mathcal Z}\) which is a factor of both.
Furstenberg defined the notion of disjointness for \({\mathcal X}\) and \({\mathcal Y}\). This amounts to the product system \({\mathcal X}\otimes{\mathcal Y}\) (i.e., \(\lambda= \mu\times\nu\)) being the only joining. He asked whether \({\mathcal X}\) and \({\mathcal Y}\) having no common factors and being disjoint are equivalent.
Rudolph, using a class of examples which he showed to have “minimal self-joinings”, (i.e., the only ergodic joinings of \(T\) with itself are \(\mu\times\mu\) and those arising from the powers of \(T\)), gave examples of ergodic transformations with no common factors, without being disjoint. This led to the realization that joinings were a useful tool for answering open questions and for illuminating the theory. Burton and Rothstein used joinings to prove the isomorphism theorem of Ornstein. Lemańcyk gave a new proof of the Halmos-von Neumann discrete spectrum theorem. Host was able to use joinings to show that a mixing transformation having singular spectrum is mixing of all orders. Joinings have even found application to questions on spectral aspects of ergodic theory (see the survey article of the reviewer [J. Dyn. Control Syst. 5, 173–226 (1999; Zbl 0987.37004)] in this regard).
The book under review covers many aspects of the use of joinings in ergodic theory. The results in the first part of the book are given in considerable generality, for general group actions. Basic definitions are presented in terms of the Koopman unitary representation associated with a dynamical system. This makes the text difficult for the beginning student in ergodic theory, who would probably feel more at home with books such as those by P. Walters [An introduction to ergodic theory, Graduate Texts in Mathematics, Vol. 79. New York etc.: Springer-Verlag (1982; Zbl 0475.28009)] or K. Petersen [Ergodic theory,Cambridge Studies in Advanced Mathematics, 2. Cambridge etc.: Cambridge University Press (1983; Zbl 0507.28010)]. Many of the standard results of the subject (in the early chapters) are relegated to the exercises. This enables the author to cover a lot of material with a high degree of generality. The book highlights the analogies between results from topological dynamics and ergodic theory. This is an excellent book for someone who has completed a first course in ergodic theory and also for the specialist in ergodic theory wishing to learn more about joinings.
Contents: Chapter 1: Topological dynamics, including Furstenberg’s distal structure theorem and van der Waerden’s theorem.
Chapter 2: General aspects of dynamical systems on a Lebesgue probability space, including the Poincaré recurrence theorem.
Chapter 3: Ergodic and mixing properties of dynamical systems. Although joinings proofs of some of these results are available, joinings are not used in this chapter. Results are proved generally using unitary representations. The Koopman representation (the idea that a measure preserving action gives rise to a unitary representation) is defined, the ergodic decomposition theorem and Rohlin’s skew product theorem are given.
Chapter 4: Ergodicity and mixing/weak mixing are defined using unitary representations, and the usual definitions are deduced as theorems. The pointwise ergodic theorem is proved and \(K\)-automorphisms and Gaussian automorphisms are studied.
Chapter 4: Invariant measures and the unique ergodicity of the geodesic and horocycle flows are studied, and \(E\)-systems are defined.
Chapter 5: A brief look at the spectral theory of measure preserving transformations. This chapter is mostly restricted to \(\mathbb{Z}\)-actions, although there is a discussion of irreducible representations.
Chapter 6: In this chapter, we meet joinings for the first time. In addition to the usual definition, an equivalent operator theoretic definition is given: corresponding to a joining \(\lambda\) of \({\mathcal X}\) and \({\mathcal Y}\) is a Markov operator \(P_\lambda: L^2(X,\mu)\to L^2(Y,\nu)\) which intertwines the actions on \({\mathcal X}\) and \({\mathcal Y}\). This different approach is used when convenient. Graph joinings, factor joinings and relatively independent joinings are defined and Veech’s characterization of group extension in terms of joinings is given and then extended to isometric extensions. Disjointness is introduced and studied in this chapter.
Chapter 7: Applications of joinings include Lemańczyk’s proof of the Halmos-von Neumann discrete spectrum theorem using joinings, the mixing of all orders of the horocycle flow, and a study of \(\alpha\)-weak mixing.
Chapter 8: Factors and quasi-factors are studied, together with a proof of the Finetti-Hewitt-Savage theorem.
Chapter 9: This chapter gives a detailed study of the notions of isometric and weakly mixing extensions.
Chapter 10: The Furstenberg-Zimmer structure theorem is proved, and a discussion of the multiple recurrence theorem of Furstenberg and its application to generalize Szemeredi’s theorem are given.
Chapter 11: A proof of Host’s theorem concerning pairwise independent joinings and its application to showing that a mixing system with singular spectrum is mixing of all orders is given.
Chapter 12: Simple systems are those which have only the obvious selfjoinings (the centralizer need not be trivial). Here, it is shown that, for \(\mathbb{Z}\)-systems which are weakly mixing, simplicity of order three implies simplicity of all orders, and rigidity, together with simplicity of order two implies simplicity of all orders.
Chapter 13: Kazhdan’s property \(T\), a theorem of Bekka and Valetta, and the geometry of certain spaces of measures are studied.
Chapter 14: Topological and measure theoretic entropy are defined and their properties studied. Both the Adler, Konheim and McAndrew, and Bowen definitions of topological entropy are given. The Kolmogorov-Sinai and Shannon-McMilllan-Breiman theorems are proved. The chapters on entropy theory are mainly restricted to \(\mathbb{Z}\)-actions.
Chapter 15: Symbolic representations, Kakutani and Rohlin towers and a generalization of the Jewett-Krieger theorem due to Weiss are given.
Chapter 16: Rank one systems are defined, shown to be ergodic with simple spectrum, and they are used to construct the Chacon transformation, and \(\alpha\)-weakly mixing transformations.
Chapter 17: The Goodwyn-Goodman-Dinaburg theorem giving the relation between measure theoretic and topological entropy is proved (that the topological entropy is the supremum over the measure theoretic entropies – the variational principle). This is applied to expansive dynamical systems.
Chapter 18: The dichotomy between zero entropy systems and those having positive entropy is studied. The Pinsker algebra is defined and the Rohlin-Sinai theorem is proved.
Chapter 19: The notions of topological and measure entropy pairs are studied. The question of the dichotomy between results in topological dynamics and ergodic theory is considered.
Chapter 20: Krieger’s and Ornstein’s theorems are proved using the ideas of Burton and Rothstein, which were made coherent by Rudolph in his text on Measurable Dynamics.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A15 General groups of measure-preserving transformations and dynamical systems
37A25 Ergodicity, mixing, rates of mixing
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B99 Topological dynamics
28Dxx Measure-theoretic ergodic theory
54H20 Topological dynamics (MSC2010)
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