Operator Theoretic Aspects of Ergodic Theory

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

List of contents

What is Ergodic Theory?.- Topological Dynamical Systems.- Minimality and Recurrence.- The C*-algebra C(K) and the Koopman Operator.- Measure-Preserving Systems.- Recurrence and Ergodicity.- The Banach Lattice Lp and the Koopman Operator.- The Mean Ergodic Theorem.- Mixing Dynamical Systems.- Mean Ergodic Operators on C(K).- The Pointwise Ergodic Theorem.- Isomorphisms and Topological Models.- Markov Operators.- Compact Semigroups and Groups.- Topological Dynamics Revisited.- The Jacobs-de Leeuw-Glicksberg Decomposition.- Dynamical Systems with Discrete Spectrum.- A Glimpse at Arithmetic Progressions.- Joinings.- The Host-Kra- Tao Theorem.- More Ergodic Theorems.- Appendix A: Topology.- Appendix B: Measure and Integration Theory.- Appendix C: Functional Analysis.- Appendix D: The Riesz Representation Theorem.- Appendix E: Theorems of Eberlein, Grothendieck, and Ellis.

About the author

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.

Summary

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

Additional text

“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 self-contained. The book has the potential to become a basic reference in this field.” (Idris Assani, Mathematical Reviews, January, 2017)

Report

"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 self-contained. The book has the potential to become a basic reference in this field." (Idris Assani, Mathematical Reviews, January, 2017)

Product details

• Authors: Tanj Eisner, Tanja Eisner, Bálin Farkas, Bálint Farkas, Markus Haase, Markus et al Haase, Rainer Nagel
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 20.12.2015
• EAN: 9783319168975
• ISBN: 978-3-31-916897-5
• No. of pages: 628
• Dimensions: 156 mm x 37 mm x 241 mm
• Weight: 1194 g
• Illustrations: XVIII, 628 p.
• Series: Graduate Texts in Mathematics; Graduate Texts in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; B, Funktionalanalysis und Abwandlungen, Dynamics, Mathematics and Statistics, Functional Analysis, Dynamical Systems and Ergodic Theory, Ergodic theory, Dynamical systems, Functional analysis & transforms, Operator Theory