The purpose of this book is to provide an invitation to the beautiful and important subject of ergodic theorems, both classical and modern, which lies at the intersection of many fundamental mathematical disciplines: dynamical systems, probability theory, topology, algebra, number theory, analysis and functional analysis. The book is suitable for undergraduate and graduate students as well as non-specialists with basic knowledge of functional analysis, topology and measure theory.
Starting from classical ergodic theorems due to von Neumann and Birkhoff, the state-of-the-art of modern ergodic theorems such as subsequential, multiple and weighted ergodic theorems are presented. In particular, two deep connections between ergodic theorems and number theory are discussed: Furstenbergs famous proof of Szemerédis theorem on existence of arithmetic progressions in large sets of integers, and the Sarnak conjecture on the random behavior of the Möbius function.
An extensive list of references to other literature for readers wishing to deepen their knowledge is provided.
Chapter
1. Preliminaries.- Part I. The Fundamentals.
Chapter
2.
Measure-preserving systems.
Chapter
3. Minimality and ergodicity.
Chapter
4. Some Fourier analysis.
Chapter
5. The spectral theorem.
Chapter
6.
Decompositions in Hilbert spaces.- Part II. Classical Ergodic Theorems.-
Chapter
7. Classical ergodic theorems and more.
Chapter
8. Factors of
measure-preserving systems.
Chapter
9. First applications of ergodic
theorems.
Chapter
10. Equidistribution.
Chapter
11. Groups, semigroups and
homogeneous spaces.- Part III. More Ergodic Theorems.
Chapter
12.
Subsequential ergodic theorems.
Chapter
13. Multiple recurrence.
Chapter
14. Nilsystems.
Chapter
15. GowersHostKra seminorms and multiple
convergence.
Chapter
16. Weighted ergodic theorems.
Chapter
17. Sarnaks
conjecture.
Tanja Eisner received her PhD at the University of Tübingen in 2007 and completed her habilitation in 2010. She moved in 2011 to the University of Amsterdam as assistant professor and since 2013 is a full professor at the University of Leipzig. In the meantime, she had research stays at the University of Missouri, Columbia, at the Paul Verlaine University, Metz and, repeatedly, at the University of California, Los Angeles. Her research focuses on operator theory, ergodic theory and connections with number theory. She is the author of the monograph "Stability of Operators and Operator Semigroups" published by Birkhäuser Verlag and a co-author, jointly with Bįlint Farkas, Markus Haase and Rainer Nagel, of the book "Operator Theoretic Aspects of Ergodic Theory" published in the Springer Graduate Texts in Mathematics series. She is editor of the journals "Analysis Mathematica" and "Zeitschrift für Analysis und ihre Anwendungen" and is a former managing editor of the latter.
Bįlint Farkas received his PhD from Eötvös Lorįnd University, Budapest, in 2004, with a stay at the Central European University and as a Marie Curie Fellow at the University of Tübingen during his PhD studies. Afterward, he worked as a postdoctoral researcher at the Alfréd Rényi Institute of Mathematics in Budapest, the University of Parma, and the University of Tübingen. He held faculty positions at the Technical University of Darmstadt and Eötvös Lorįnd University, Budapest. Since 2012, he has been a professor at the University of Wuppertal. His research focuses on functional analysis, operator theory, and their applications, particularly in dynamical systems. Together with Tanja Eisner, Markus Haase, and Rainer Nagel, he co-authored the book "Operator Theoretic Aspects of Ergodic Theory", which was published in the Springer Graduate Texts in Mathematics series. He serves as a deputy-editor-in-chief for the journal "Analysis Mathematica" and as an editor for "Zeitschrift für Analysis und ihre Anwendungen".
