Graduate Texts in Mathematics

Operator Theoretic Aspects of Ergodic Theory

Authors: Eisner, T., Farkas, B., Haase, M., Nagel, R.

  • 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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eBook $69.99
price for USA (gross)
  • ISBN 978-3-319-16898-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $89.99
price for USA
  • ISBN 978-3-319-16897-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $89.99
price for USA
  • ISBN 978-3-319-37105-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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 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

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


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Table of contents (21 chapters)

  • What Is Ergodic Theory?

    Eisner, Tanja (et al.)

    Pages 1-7

  • Topological Dynamical Systems

    Eisner, Tanja (et al.)

    Pages 9-32

  • Minimality and Recurrence

    Eisner, Tanja (et al.)

    Pages 33-44

  • The C ∗-Algebra C(K) and the Koopman Operator

    Eisner, Tanja (et al.)

    Pages 45-70

  • Measure-Preserving Systems

    Eisner, Tanja (et al.)

    Pages 71-94

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-319-16898-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $89.99
price for USA
  • ISBN 978-3-319-16897-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $89.99
price for USA
  • ISBN 978-3-319-37105-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Operator Theoretic Aspects of Ergodic Theory
Authors
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
978-3-319-16898-2
DOI
10.1007/978-3-319-16898-2
Hardcover ISBN
978-3-319-16897-5
Softcover ISBN
978-3-319-37105-4
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XVIII, 628
Topics