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Operator Theoretic Aspects of Ergodic Theory (Hardcover, 1st ed. 2015)
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Operator Theoretic Aspects of Ergodic Theory (Hardcover, 1st ed. 2015)
Series: Graduate Texts in Mathematics, 272
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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-Sarkoezy) 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
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