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Author : William Parry
Publisher : Unknown
Page : 144 pages
File Size : 42,5 Mb
Release : 1969
Category : Entropy
ISBN : UCAL:B4406945
Author : Meir Smorodinsky
Publisher : Springer
Page : 70 pages
File Size : 51,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540368793
Author : Yuri Kifer
Publisher : Springer Science & Business Media
Page : 221 pages
File Size : 49,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468491753
Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note [44] suggested to study ergodic theorems for the more general situation when one applies in turn different transforma tions chosen at random. Their program was fulfilled by S. Kakutani [23] in 1951. 'Both papers considered the case of transformations with a common invariant measure. Recently Ohno [38] noticed that this condition was excessive. Ergodic theorems are just the beginning of ergodic theory. Among further major developments are the notions of entropy and characteristic exponents. The purpose of this book is the study of the variety of ergodic theoretical properties of evolution processes generated by independent applications of transformations chosen at random from a certain class according to some probability distribution. The book exhibits the first systematic treatment of ergodic theory of random transformations i.e., an analysis of composed actions of independent random maps. This set up allows a unified approach to many problems of dynamical systems, products of random matrices and stochastic flows generated by stochastic differential equations.
Author : Karl E. Petersen,Karl Petersen
Publisher : Cambridge University Press
Page : 348 pages
File Size : 45,9 Mb
Release : 1989-11-23
Category : Mathematics
ISBN : 0521389976
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 : Paul Richard Halmos
Publisher : Unknown
Page : 90 pages
File Size : 46,5 Mb
Release : 1959
Category : Algebraic topology
ISBN : UOM:39015049386017
Author : Peter Walters
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 41,7 Mb
Release : 2000-10-06
Category : Mathematics
ISBN : 0387951520
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 : Jean Moulin Ollagnier
Publisher : Springer
Page : 154 pages
File Size : 47,6 Mb
Release : 2007-01-05
Category : Mathematics
ISBN : 9783540392897
Author : Ricardo Mane
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 50,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642703355
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 : J. Aczél
Publisher : Delhi : Hindustan Pub. ; Toronto : University Press of Canada
Page : 130 pages
File Size : 52,5 Mb
Release : 1975
Category : Entropy
ISBN : UOM:39015049073284
Author : Eli Glasner
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 52,7 Mb
Release : 2015-01-09
Category : Electronic
ISBN : 9781470419516
This book introduces modern ergodic theory. It emphasizes a new approach that relies on the technique of joining two (or more) dynamical systems. This approach has proved to be fruitful in many recent works, and this is the first time that the entire theory is presented from a joining perspective. Another new feature of the book is the presentation of basic definitions of ergodic theory in terms of the Koopman unitary representation associated with a dynamical system and the invariant mean on matrix coefficients, which exists for any acting groups, amenable or not. Accordingly, the first part of the book treats the ergodic theory for an action of an arbitrary countable group. The second part, which deals with entropy theory, is confined (for the sake of simplicity) to the classical case of a single measure-preserving transformation on a Lebesgue probability space.
Author : Paul C. Shields
Publisher : American Mathematical Soc.
Page : 249 pages
File Size : 55,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804773
This is an interesting and well-written account of the ergodic theory of stationary processes from the viewpoint of entropy theory. This book is a beautiful and very readable treatment of an important part of ergodic theory. ... strongly recommend this book to anyone who is interested in ergodic theory, stochastic processes or information theory. --Bulletin of the London Mathematical Society This book is about finite-alphabet stationary processes, which are important in physics, engineering, and data compression. The focus is on the combinatorial properties of typical finite sample paths drawn from a stationary, ergodic process. A primary goal, only partially realized, is to develop a theory based directly on sample path arguments with minimal appeals to the probability formalism. A secondary goal is to give a careful presentation of the many models for stationary finite-alphabet processes that have been developed in probability theory, ergodic theory, and information theory. The book has many attractive features for both students and researchers. An emphasis is placed on recent combinatorial results for sample paths. There are careful treatments of many models that have been found to be useful in engineering. Shields covers applications of entropy ideas to coding, sample path structure, distribution estimation, recurrence times, waiting times, and prefix trees.
Author : Mariusz Urbański,Mario Roy,Sara Munday
Publisher : Walter de Gruyter GmbH & Co KG
Page : 458 pages
File Size : 49,8 Mb
Release : 2021-11-22
Category : Mathematics
ISBN : 9783110702682
The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen’s formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub’s expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.
Author : Steven Kalikow,Randall McCutcheon
Publisher : Cambridge University Press
Page : 183 pages
File Size : 46,6 Mb
Release : 2010-03-25
Category : Mathematics
ISBN : 9781139484251
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 : I︠A︡kov Grigorʹevich Sinaĭ
Publisher : Princeton University Press
Page : 156 pages
File Size : 47,7 Mb
Release : 1976
Category : Ergodic theory
ISBN : 0691081824
Author : William Parry
Publisher : Unknown
Page : 202 pages
File Size : 43,6 Mb
Release : 1966
Category : Entropy
ISBN : STANFORD:36105000190020
