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Operator Theoretic Aspects of Ergodic Theory (eBook)

Artikelnummer: 978-3-319-16898-2
Einband: PDF
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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 researchat the intersection of ergodic theory and operator theory

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 researchat the intersection of ergodic theory and operator theory
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VerlagSpringer International Publishing
EinbandPDF
Erscheinungsjahr2015
Seitenangabe628 S.
AusgabekennzeichenEnglisch
AbbildungenXVIII, 628 p.
Masse6'465 KB
Auflage15001 A. 1st ed. 2015
PlattformPDF
ReiheGraduate Texts in Mathematics; Mathematics and Statistics; Mathematics and Statistics
AutorEisner, Tanja / Farkas, Bálint / Haase, Markus / Nagel, Rainer

Alle Bände der Reihe "Graduate Texts in Mathematics; Mathematics and Statistics; Mathematics and Statistics (R0)"

Über den Autor Tanja Eisner

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 Bálint 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. Bálint Farkas received his PhD from Eötvös Loránd 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 Loránd 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".

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