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
