Operator Theoretic Aspects of Ergodic Theory

✍ Scribed by Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 630
• Series: Graduate texts in mathematics 272
• Edition: 1st ed.
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Front Matter....Pages i-xviii
What Is Ergodic Theory?....Pages 1-7
Topological Dynamical Systems....Pages 9-32
Minimality and Recurrence....Pages 33-44
Measure-Preserving Systems....Pages 45-70
Recurrence and Ergodicity....Pages 71-94
The Mean Ergodic Theorem....Pages 95-113
Mixing Dynamical Systems....Pages 115-134
Mean Ergodic Operators on C(K)....Pages 135-159
The Pointwise Ergodic Theorem....Pages 161-189
Isomorphisms and Topological Models....Pages 191-210
Markov Operators....Pages 211-223
Compact Groups....Pages 225-247
Group Actions and Representations....Pages 249-271
The Jacobs–de Leeuw–Glicksberg Decomposition....Pages 273-289
The Kronecker Factor and Systems with Discrete Spectrum....Pages 291-315
The Spectral Theorem and Dynamical Systems....Pages 317-344
Topological Dynamics and Colorings....Pages 345-365
Arithmetic Progressions and Ergodic Theory....Pages 367-403
More Ergodic Theorems....Pages 405-432
Back Matter....Pages 433-460
....Pages 461-477

✦ Subjects

Mathematics;Dynamics;Ergodic theory;Functional analysis;Operator theory