Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory.

Topics include:

• an intuitive introduction to ergodic theory

• an introduction to the basic notions, constructions, and standard examples of topological dynamical systems

• Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem

• measure-preserving dynamical systems

• von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem

• strongly and weakly mixing systems

• an examination of notions of isomorphism for measure-preserving systems

• Markov operators, and the related concept of a factor of a measure preserving system

• compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition

• an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy)

Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory

(20.1) into a property of this dynamical system. By (20.1), we obtain for and the following equivalences: Since each τ −j (M) is open and is dense in K, we have that (20.1) is equivalent to (20.2) The strategy to prove (20.2) is now to turn the topological system into a measure-preserving system by choosing an invariant measure μ in such a way that for some the set has positive measure. Since , there is a subsequence in such that We define a sequence of probability measures on K by

3.12). Now, by taking a suitable δ > 0 depending on the coefficients c 1, … c j , we obtain for all n ∈ B. Moreover, for n ∈ B one has Altogether we obtain for n ∈ B and (20.17) is proved. Since for some and all , it follows that Thus the limit in (20.16), which exists by the considerations in the previous subsection, is positive. □ Combining the above results leads to the classical theorem of Roth (1953), a precursor of Szemerédi’s Theorem 20.1. Theorem 20.20 (Roth). If has positive

compactifications , Acta Math. 105 (1961), 63–97. J. de Vries [1993] Elements of Topological Dynamics , Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. A. Deitmar and S. Echterhoff [2009] Principles of Harmonic Analysis , Universitext, Springer, New York, 2009. C. Demeter [2010] On some maximal multipliers in L p , Rev. Mat. Iberoam. 26 (2010), no. 3, 947–964. C. Demeter, M. T. Lacey, T. Tao, and C. Thiele [2008] Breaking the duality in the

2008/2009, we profited enormously from the participants, not only by their mathematical comments but also by their enthusiastic encouragement. This support continued during the subsequent years in which a part of the material was taken as a basis for lecture courses or seminars. We thank all who were involved in this process, particularly Omar Aboura, Abramo Bertucco, Miriam Bombieri, Rebecca Braken, Nikolai Edeko, Tobias Finkbeiner, Retha Heymann, Jakub Konieczny, Joanna Kułaga, Henrik Kreidler,

, f n ≥ f n+1, one has This is a direct consequence of the monotone convergence theorem (see Appendix B.5) by considering the sequence . Actually, the monotone convergence theorem accounts also for the following statement. Theorem 7.6. Let X be a measure space and let . Let be a ∨-stable set such that Then exists in the Banach lattice and there exists an increasing sequence in such that and . In particular, if is closed. Proof. Take a sequence in with . By passing to the sequence we may