Operator Theoretic Aspects of Ergodic Theory

  • Tanja Eisner
  • Bálint Farkas
  • Markus Haase
  • Rainer Nagel

Part of the Graduate Texts in Mathematics book series (GTM, volume 272)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 1-7
  3. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 9-32
  4. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 33-44
  5. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 45-70
  6. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 71-94
  7. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 95-113
  8. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 115-134
  9. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 135-159
  10. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 161-189
  11. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 191-210
  12. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 211-223
  13. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 225-247
  14. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 249-271
  15. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 273-289
  16. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 291-315
  17. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 317-344
  18. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 345-365
  19. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 367-403
  20. Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel
    Pages 405-432

About this book

Introduction

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


Keywords

Banach lattices Jacobs–de Leeuw–Glicksberg Decomposition Koopman operator mean ergodic operators pointwise ergodic theorem topological dynamical systems

Authors and affiliations

  • Tanja Eisner
    • 1
  • Bálint Farkas
    • 2
  • Markus Haase
    • 3
  • Rainer Nagel
    • 4
  1. 1.Institute of MathematicsUniversity of LeipzigLeipzigGermany
  2. 2.Faculty of Mathematics & Natural ScienceUniversity of WuppertalWuppertalGermany
  3. 3.Fac. EWI/DIAMTU Delft Fac. EWI/DIAMDelftThe Netherlands
  4. 4.Mathematisches InstitutUniversität Tübingen Mathematisches InstitutTübingenGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-16898-2
  • Copyright Information Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-16897-5
  • Online ISBN 978-3-319-16898-2
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • About this book