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Author : Gerhard Keller
Publisher : Cambridge University Press
Page : 196 pages
File Size : 52,8 Mb
Release : 1998-01-22
Category : Mathematics
ISBN : 0521595347
Based on a one semester course, this book provides a self contained introduction to the ergodic theory of equilibrium states.
Author : Robert Edward Bowen
Publisher : Springer Science & Business Media
Page : 84 pages
File Size : 48,5 Mb
Release : 2008-04-18
Category : Mathematics
ISBN : 9783540776055
For this printing of R. Bowen's book, J.-R. Chazottes has retyped it in TeX for easier reading, thereby correcting typos and bibliographic details. From the Preface by D. Ruelle: "Rufus Bowen has left us a masterpiece of mathematical exposition... Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems."
Author : Robert Edward Bowen
Publisher : Springer
Page : 80 pages
File Size : 42,5 Mb
Release : 2009-08-29
Category : Mathematics
ISBN : 3540848878
For this printing of R. Bowen's book, J.-R. Chazottes has retyped it in TeX for easier reading, thereby correcting typos and bibliographic details. From the Preface by D. Ruelle: "Rufus Bowen has left us a masterpiece of mathematical exposition... Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems."
Author : Mariusz Urbański,Mario Roy,Sara Munday
Publisher : Walter de Gruyter GmbH & Co KG
Page : 458 pages
File Size : 51,5 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 : R. Bowen
Publisher : Unknown
Page : 120 pages
File Size : 54,6 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662192705
Author : Cesar E. Silva,Alexandre I. Danilenko
Publisher : Springer Nature
Page : 707 pages
File Size : 45,9 Mb
Release : 2023-07-31
Category : Mathematics
ISBN : 9781071623886
This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras
Author : Neil Dobbs,Mike Todd
Publisher : American Mathematical Society
Page : 116 pages
File Size : 51,9 Mb
Release : 2023-06-22
Category : Mathematics
ISBN : 9781470461263
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Author : Zhiqiang Li
Publisher : Springer
Page : 182 pages
File Size : 50,5 Mb
Release : 2017-04-06
Category : Mathematics
ISBN : 9789462391741
Thurston maps are topological generalizations of postcritically-finite rational maps. This book provides a comprehensive study of ergodic theory of expanding Thurston maps, focusing on the measure of maximal entropy, as well as a more general class of invariant measures, called equilibrium states, and certain weak expansion properties of such maps. In particular, we present equidistribution results for iterated preimages and periodic points with respect to the unique measure of maximal entropy by investigating the number and locations of fixed points. We then use the thermodynamical formalism to establish the existence, uniqueness, and various other properties of the equilibrium state for a Holder continuous potential on the sphere equipped with a visual metric. After studying some weak expansion properties of such maps, we obtain certain large deviation principles for iterated preimages and periodic points under an additional assumption on the critical orbits of the maps. This enables us to obtain general equidistribution results for such points with respect to the equilibrium states under the same assumption.
Author : Marcelo Viana,Krerley Oliveira
Publisher : Cambridge University Press
Page : 547 pages
File Size : 52,9 Mb
Release : 2016-02-15
Category : Mathematics
ISBN : 9781107126961
Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.
Author : Mariusz Urbański,Mario Roy,Sara Munday
Publisher : de Gruyter
Page : 0 pages
File Size : 55,6 Mb
Release : 2022
Category : Mathematics
ISBN : 3110702649
The book contains detailed treatment of thermodynamic formalism. Topological pressure, entropy, variational principle, and equilibrium states are presented in detail in the first volume. Abstract ergodic theory is also given a significant attention.
Author : Robert A. Meyers
Publisher : Springer Science & Business Media
Page : 1885 pages
File Size : 53,8 Mb
Release : 2011-10-05
Category : Mathematics
ISBN : 9781461418054
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 : David Ruelle
Publisher : Cambridge University Press
Page : 198 pages
File Size : 46,6 Mb
Release : 2004-11-25
Category : Science
ISBN : 1139455281
Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.
Author : Alexander M. Blokh,Leonid A. Bunimovich,Paul H. Jung,Lex G. Oversteegen,Yakov G. Sinai
Publisher : American Mathematical Soc.
Page : 316 pages
File Size : 43,6 Mb
Release : 2017-09-18
Category : Billiards
ISBN : 9781470427733
This volume contains the proceedings of the Conference on Dynamical Systems, Ergodic Theory, and Probability, which was dedicated to the memory of Nikolai Chernov, held from May 18–20, 2015, at the University of Alabama at Birmingham, Birmingham, Alabama. The book is devoted to recent advances in the theory of chaotic and weakly chaotic dynamical systems and its applications to statistical mechanics. The papers present new original results as well as comprehensive surveys.
Author : Peter Walters
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 47,8 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 : Anne Broise-Alamichel,Jouni Parkkonen,Frédéric Paulin
Publisher : Springer Nature
Page : 413 pages
File Size : 48,6 Mb
Release : 2019-12-16
Category : Mathematics
ISBN : 9783030183158
This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees—again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.
