Stunning recent results by HostKra, GreenTao, 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 examplesof topological dynamical systems
Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the GelfandNaimark theorem
measure-preserving dynamical systems
von Neumanns Mean Ergodic Theorem and Birkhoffs 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 Jacobsde LeeuwGlicksberg 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, Furstenbergs Correspondence Principle, theorems of Roth and FurstenbergSarkozy)
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