A Journey Through Ergodic Theorems - Bálint Farkas

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Résumé :
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: Furstenberg's famous proof of Szemer?di's theorem on existence of arithmetic progressions in large sets of integers, and the Sarnak conjecture on the random behavior of the M?bius function. An extensive list of references to other literature for readers wishing to deepen their knowledge is provided....

Biographie:

Tanja Eisner received her PhD at the University of T?bingen 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 Birkh?user Verlag and a co-author, jointly with Bá...

Sommaire:
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: Furstenberg's famous proof of Szemer?di's theorem on existence of arithmetic progressions in large sets of integers, and the Sarnak conjecture on the random behavior of the M?bius function. An extensive list of references to other literature for readers wishing to deepen their knowledge is provided....