An Introduction to Ergodic Theory

Author: Peter Walters
Publisher: Springer Science & Business Media
ISBN: 9780387951522
Format: PDF, ePub, Mobi
Download Now
The first part of this introduction to ergodic theory addresses measure-preserving transformations of probability spaces and covers such topics as recurrence properties and the Birkhoff ergodic theorem. The second part focuses on the ergodic theory of continuous transformations of compact metrizable spaces. Several examples are detailed, and the final chapter outlines results and applications of ergodic theory to other branches of mathematics.

Ergodic Theory

Author: Manfred Einsiedler
Publisher: Springer Science & Business Media
ISBN: 9780857290212
Format: PDF, ePub
Download Now
This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Operator Theoretic Aspects of Ergodic Theory

Author: Tanja Eisner
Publisher: Springer
ISBN: 3319168983
Format: PDF
Download Now
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

Infinite Ergodic Theory of Numbers

Author: Marc Kesseböhmer
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110439425
Format: PDF, Mobi
Download Now
By connecting dynamical systems and number theory this graduate textbook on ergodic theory covers a highly active area of mathematics, where a variety of strands of research open up. After introducing number-theoretical dynamical systems, the text touches on foundations and renewal theory before covering infinite ergodic theory. Applications such as continued fraction expansion and sum-level sets are discussed as well.

Lectures on Ergodic Theory

Author: Paul R. Halmos
Publisher: Courier Dover Publications
ISBN: 0486826848
Format: PDF
Download Now
This concise classic by a well-known master of mathematical exposition covers recurrence, ergodic theorems, ergodicity and mixing properties, and the relation between conjugacy and equivalence. 1956 edition.

Ergodic Theory

Author: Karl E. Petersen
Publisher: Cambridge University Press
ISBN: 9780521389976
Format: PDF
Download Now
The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. The study of dynamical systems forms a vast and rapidly developing field even when considering only activity whose methods derive mainly from measure theory and functional analysis. Each of the basic aspects of ergodic theory--examples, convergence theorems, recurrence properties, and entropy--receives a basic and a specialized treatment. The author's accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

Ergodic Theory

Author: David Kerr
Publisher: Springer
ISBN: 3319498479
Format: PDF, Kindle
Download Now
This book provides an introduction to the ergodic theory and topological dynamics of actions of countable groups. It is organized around the theme of probabilistic and combinatorial independence, and highlights the complementary roles of the asymptotic and the perturbative in its comprehensive treatment of the core concepts of weak mixing, compactness, entropy, and amenability. The more advanced material includes Popa's cocycle superrigidity, the Furstenberg-Zimmer structure theorem, and sofic entropy. The structure of the book is designed to be flexible enough to serve a variety of readers. The discussion of dynamics is developed from scratch assuming some rudimentary functional analysis, measure theory, and topology, and parts of the text can be used as an introductory course. Researchers in ergodic theory and related areas will also find the book valuable as a reference.

Ergodic Theory of Numbers

Author: Karma Dajani
Publisher: Cambridge University Press
ISBN: 9780883850343
Format: PDF, Kindle
Download Now
Introduction to ergodic theory of numbers for graduate students and researchers.