An Outline Of Ergodic Theory

Author : Steven Kalikow,Randall McCutcheon
Publisher : Cambridge University Press
Page : 183 pages
File Size : 53,9 Mb
Release : 2010-03-25
Category : Mathematics
ISBN : 9781139484251

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An Outline of Ergodic Theory by Steven Kalikow,Randall McCutcheon Pdf

This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.

Author : Karl E. Petersen,Karl Petersen
Publisher : Cambridge University Press
Page : 348 pages
File Size : 40,5 Mb
Release : 1989-11-23
Category : Mathematics
ISBN : 0521389976

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Ergodic Theory by Karl E. Petersen,Karl Petersen Pdf

The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.

Author : I. P. Cornfeld,S. V. Fomin,Y. G. Sinai
Publisher : Springer Science & Business Media
Page : 487 pages
File Size : 45,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461569275

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Ergodic Theory by I. P. Cornfeld,S. V. Fomin,Y. G. Sinai Pdf

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.

Author : Peter Walters
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 42,9 Mb
Release : 2000-10-06
Category : Mathematics
ISBN : 0387951520

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An Introduction to Ergodic Theory by Peter Walters Pdf

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.

Author : Steven Kalikow,Randall McCutcheon
Publisher : Unknown
Page : 128 pages
File Size : 48,5 Mb
Release : 2012-11-01
Category : Ergodic theory
ISBN : 0521170311

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An Outline of Ergodic Theory by Steven Kalikow,Randall McCutcheon Pdf

An engaging introduction to ergodic theory for graduate students, and a useful reference for the professional mathematician.

Author : Peter Walters
Publisher : Springer
Page : 0 pages
File Size : 43,7 Mb
Release : 2000-10-20
Category : Mathematics
ISBN : 1461257751

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An Introduction to Ergodic Theory by Peter Walters Pdf

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.

Author : I︠A︡kov Grigorʹevich Sinaĭ
Publisher : Princeton University Press
Page : 156 pages
File Size : 46,6 Mb
Release : 1976
Category : Ergodic theory
ISBN : 0691081824

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Introduction to Ergodic Theory by I︠A︡kov Grigorʹevich Sinaĭ Pdf

Author : Ricardo Mane
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642703355

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Ergodic Theory and Differentiable Dynamics by Ricardo Mane Pdf

This version differs from the Portuguese edition only in a few additions and many minor corrections. Naturally, this edition raised the question of whether to use the opportunity to introduce major additions. In a book like this, ending in the heart of a rich research field, there are always further topics that should arguably be included. Subjects like geodesic flows or the role of Hausdorff dimension in con temporary ergodic theory are two of the most tempting gaps to fill. However, I let it stand with practically the same boundaries as the original version, still believing these adequately fulfill its goal of presenting the basic knowledge required to approach the research area of Differentiable Ergodic Theory. I wish to thank Dr. Levy for the excellent translation and several of the correc tions mentioned above. Rio de Janeiro, January 1987 Ricardo Mane Introduction This book is an introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems, which is sometimes called differentiable ergodic theory. Chapter 0, a quick review of measure theory, is included as a reference. Proofs are omitted, except for some results on derivatives with respect to sequences of partitions, which are not generally found in standard texts on measure and integration theory and tend to be lost within a much wider framework in more advanced texts.

Author : Feliks Przytycki,Mariusz Urbański
Publisher : Cambridge University Press
Page : 365 pages
File Size : 47,6 Mb
Release : 2010-05-06
Category : Mathematics
ISBN : 9780521438001

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Conformal Fractals by Feliks Przytycki,Mariusz Urbański Pdf

A one-stop introduction to the methods of ergodic theory applied to holomorphic iteration that is ideal for graduate courses.

Author : Robert A. Meyers
Publisher : Springer Science & Business Media
Page : 1885 pages
File Size : 44,7 Mb
Release : 2011-10-05
Category : Mathematics
ISBN : 9781461418054

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Mathematics of Complexity and Dynamical Systems by Robert A. Meyers Pdf

Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.

Author : William Parry
Publisher : Cambridge University Press
Page : 128 pages
File Size : 41,5 Mb
Release : 2004-06-03
Category : Mathematics
ISBN : 0521604907

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Topics in Ergodic Theory by William Parry Pdf

An introduction to topics and examples of ergodic theory, a central area of pure mathematics.

Author : Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel
Publisher : Springer
Page : 628 pages
File Size : 55,6 Mb
Release : 2015-11-18
Category : Mathematics
ISBN : 9783319168982

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Operator Theoretic Aspects of Ergodic Theory by Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel Pdf

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

Author : Karma Dajani,Charlene Kalle
Publisher : CRC Press
Page : 268 pages
File Size : 55,8 Mb
Release : 2021-07-04
Category : Mathematics
ISBN : 9781000402773

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A First Course in Ergodic Theory by Karma Dajani,Charlene Kalle Pdf

A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors’ own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from — designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented.

Author : Paul R. Halmos
Publisher : Courier Dover Publications
Page : 112 pages
File Size : 44,5 Mb
Release : 2017-11-15
Category : Mathematics
ISBN : 9780486826844

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Lectures on Ergodic Theory by Paul R. Halmos Pdf

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.

Author : Marcelo Viana,Krerley Oliveira
Publisher : Cambridge University Press
Page : 547 pages
File Size : 44,9 Mb
Release : 2016-02-15
Category : Mathematics
ISBN : 9781107126961

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Foundations of Ergodic Theory by Marcelo Viana,Krerley Oliveira Pdf

Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.