Operator Theoretic Aspects of Ergodic Theory
Authors: Eisner, T., Farkas, B., Haase, M., Nagel, R.
Free Preview Treats both classical and recent results in ergodic theory from a modern analytic perspective
 ?Assumes no background in ergodic theory, while providing a review of basic results in functional analysis
 Provides a foundation for understanding recent applications of ergodic theory to combinatorics and number theory
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 About this Textbook

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 specialtopic 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
• measurepreserving 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 measurepreserving 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  About the authors

Tanja Eisner is a Professor of Mathematics at the University of Leipzig. Bálint Farkas is a Professor of Mathematics at the University of Wuppertal. Markus Haase is a Professor of Mathematics at the Delft Institute of Applied Mathematics. Rainer Nagel is a Professor of Mathematics at the University of Tübingen.
 Reviews

“This book can serve as a good introduction to an active research area. Each chapter ends with a nice list of exercises. At the end of the book complementary material can be found on measure theory, functional analysis, operator theory, the Riesz representation theorem, and more. This makes the book selfcontained. The book has the potential to become a basic reference in this field.” (Idris Assani, Mathematical Reviews, January, 2017)
 Table of contents (21 chapters)


What Is Ergodic Theory?
Pages 17

Topological Dynamical Systems
Pages 932

Minimality and Recurrence
Pages 3344

The C ∗Algebra C(K) and the Koopman Operator
Pages 4570

MeasurePreserving Systems
Pages 7194

Table of contents (21 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Operator Theoretic Aspects of Ergodic Theory
 Authors

 Tanja Eisner
 Bálint Farkas
 Markus Haase
 Rainer Nagel
 Series Title
 Graduate Texts in Mathematics
 Series Volume
 272
 Copyright
 2015
 Publisher
 Springer International Publishing
 Copyright Holder
 Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel
 eBook ISBN
 9783319168982
 DOI
 10.1007/9783319168982
 Hardcover ISBN
 9783319168975
 Softcover ISBN
 9783319371054
 Series ISSN
 00725285
 Edition Number
 1
 Number of Pages
 XVIII, 628
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