Foundations Of Ergodic Theory
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| Author | : Marcelo Viana |
| Publisher | : Cambridge University Press |
| Total Pages | : 547 |
| Release | : 2016-02-15 |
| Genre | : Mathematics |
| ISBN | : 1316445429 |
Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.
| Author | : Marcelo Viana |
| Publisher | : Cambridge University Press |
| Total Pages | : 547 |
| Release | : 2016-02-15 |
| Genre | : Mathematics |
| ISBN | : 1107126967 |
Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.
| Author | : Daniel J. Rudolph |
| Publisher | : Oxford University Press, USA |
| Total Pages | : 190 |
| Release | : 1990 |
| Genre | : Mathematics |
| ISBN | : |
This book is designed to provide graduate students and other researchers in dynamical systems theory with an introduction to the ergodic theory of Lebesgue spaces. The author's aim is to present a technically complete account which offers an in-depth understanding of the techniques of the field, both classical and modern. Thus, the basic structure theorems of Lebesgue spaces are given in detail as well as complete accounts of the ergodic theory of a single transformation, ergodic theorems, mixing properties and entropy. Subsequent chapters extend the earlier material to the areas of joinings and representation theorems, in particular the theorems of Ornstein and Krieger. Prerequisites are a working knowledge of Lebesgue measure and the topology of the real line as might be gained from the first year of a graduate course. Many exercises and examples are included to illustrate and to further cement the reader's understanding of the material. The result is a text which will furnish the reader with a sound technical background from the foundations of the subject to some of its most recent developments.
| Author | : R. Jancel |
| Publisher | : Elsevier |
| Total Pages | : 441 |
| Release | : 2013-10-22 |
| Genre | : Science |
| ISBN | : 1483186261 |
Foundations of Classical and Quantum Statistical Mechanics details the theoretical foundation the supports the concepts in classical and quantum statistical mechanics. The title discusses the various problems set by the theoretical justification of statistical mechanics methods. The text first covers the the ergodic theory in classical statistical mechanics, and then proceeds to tackling quantum mechanical ensembles. Next, the selection discusses the the ergodic theorem in quantum statistical mechanics and probability quantum ergodic theorems. The selection also details H-theorems and kinetic equations in classical and quantum statistical mechanics. The book will be of great interest to students, researchers, and practitioners of physics, chemistry, and engineering.
| Author | : Eli Glasner |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 402 |
| Release | : 2015-01-09 |
| Genre | : Mathematics |
| ISBN | : 1470419513 |
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 | : Olav Kallenberg |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 670 |
| Release | : 2002-01-08 |
| Genre | : Mathematics |
| ISBN | : 9780387953137 |
The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity.
| Author | : Steven Kalikow |
| Publisher | : Cambridge University Press |
| Total Pages | : 183 |
| Release | : 2010-03-25 |
| Genre | : Mathematics |
| ISBN | : 1139484257 |
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 | : Luís Barreira |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 295 |
| Release | : 2012-04-28 |
| Genre | : Mathematics |
| ISBN | : 3642280900 |
Over the last two decades, the dimension theory of dynamical systems has progressively developed into an independent and extremely active field of research. The main aim of this volume is to offer a unified, self-contained introduction to the interplay of these three main areas of research: ergodic theory, hyperbolic dynamics, and dimension theory. It starts with the basic notions of the first two topics and ends with a sufficiently high-level introduction to the third. Furthermore, it includes an introduction to the thermodynamic formalism, which is an important tool in dimension theory. The volume is primarily intended for graduate students interested in dynamical systems, as well as researchers in other areas who wish to learn about ergodic theory, thermodynamic formalism, or dimension theory of hyperbolic dynamics at an intermediate level in a sufficiently detailed manner. In particular, it can be used as a basis for graduate courses on any of these three subjects. The text can also be used for self-study: it is self-contained, and with the exception of some well-known basic facts from other areas, all statements include detailed proofs.
| Author | : Aleksandr I?Akovlevich Khinchin |
| Publisher | : Courier Corporation |
| Total Pages | : 130 |
| Release | : 1957-01-01 |
| Genre | : Mathematics |
| ISBN | : 0486604349 |
First comprehensive introduction to information theory explores the work of Shannon, McMillan, Feinstein, and Khinchin. Topics include the entropy concept in probability theory, fundamental theorems, and other subjects. 1957 edition.
| Author | : M. M. Rao |
| Publisher | : Courier Corporation |
| Total Pages | : 322 |
| Release | : 2011-01-01 |
| Genre | : Mathematics |
| ISBN | : 0486481220 |
Stochastic analysis involves the study of a process involving a randomly determined sequence of observations, each of which represents a sample of one element of probability distribution. This volume considers fundamental theories and contrasts the natural interplay between real and abstract methods. Starting with the introduction of the basic Kolmogorov-Bochner existence theorem, the text explores conditional expectations and probabilities as well as projective and direct limits. Subsequent chapters examine several aspects of discrete martingale theory, including applications to ergodic theory, likelihood ratios, and the Gaussian dichotomy theorem. Prerequisites include a standard measure theory course. No prior knowledge of probability is assumed; therefore, most of the results are proved in detail. Each chapter concludes with a problem section that features many hints and facts, including the most important results in information theory.
