Operator Theoretic Aspects of Ergodic Theory

1. Front Matter

   Pages i-xviii

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2. What Is Ergodic Theory?

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

   Pages 1-7

3. Topological Dynamical Systems

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

   Pages 9-32

4. Minimality and Recurrence

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

   Pages 33-44

5. The C ^∗-Algebra C(K) and the Koopman Operator

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

   Pages 45-70

6. Measure-Preserving Systems

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

   Pages 71-94

7. Recurrence and Ergodicity

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

   Pages 95-113

8. The Banach Lattice \(\mathrm{L}^{\!p}\) and the Koopman Operator

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

   Pages 115-134

9. The Mean Ergodic Theorem

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

   Pages 135-159

10. Mixing Dynamical Systems

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

    Pages 161-189

11. Mean Ergodic Operators on C(K)

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

    Pages 191-210

12. The Pointwise Ergodic Theorem

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

    Pages 211-223

13. Isomorphisms and Topological Models

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

    Pages 225-247

14. Markov Operators

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

    Pages 249-271

15. Compact Groups

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

    Pages 273-289

16. Group Actions and Representations

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

    Pages 291-315

17. The Jacobs–de Leeuw–Glicksberg Decomposition

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

    Pages 317-344

18. The Kronecker Factor and Systems with Discrete Spectrum

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

    Pages 345-365

19. The Spectral Theorem and Dynamical Systems

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

    Pages 367-403

20. Topological Dynamics and Colorings

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

    Pages 405-432

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

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

“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)

• Institute of Mathematics, University of Leipzig, Leipzig, Germany

  Tanja Eisner

• Faculty of Mathematics & Natural Science, University of Wuppertal, Wuppertal, Germany

  Bálint Farkas

• Fac. EWI/DIAM, TU Delft Fac. EWI/DIAM, Delft, The Netherlands

  Markus Haase

• Mathematisches Institut, Universität Tübingen Mathematisches Institut, Tübingen, Germany

  Rainer Nagel

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. 

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