Introduction To Smooth Ergodic Theory

Author : Luís Barreira,Yakov Pesin
Publisher : American Mathematical Society
Page : 355 pages
File Size : 47,6 Mb
Release : 2023-05-19
Category : Mathematics
ISBN : 9781470470654

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Introduction to Smooth Ergodic Theory by Luís Barreira,Yakov Pesin Pdf

This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided. In this second edition, the authors improved the exposition and added more exercises to make the book even more student-oriented. They also added new material to bring the book more in line with the current research in dynamical systems.

Author : Luis Barreira,Ya. B. Pesin
Publisher : American Mathematical Soc.
Page : 166 pages
File Size : 54,6 Mb
Release : 2002
Category : Ergodic theory
ISBN : 9780821829219

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Lyapunov Exponents and Smooth Ergodic Theory by Luis Barreira,Ya. B. Pesin Pdf

"The authors consider several nontrivial examples of dynamical systems with nonzero Lyapunov exponents to illustrate some basic methods and ideas of the theory.".

Author : Luís Barreira,Yakov Pesin
Publisher : American Mathematical Society
Page : 355 pages
File Size : 42,5 Mb
Release : 2023-04-28
Category : Mathematics
ISBN : 9781470473075

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Introduction to Smooth Ergodic Theory by Luís Barreira,Yakov Pesin Pdf

This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided. In this second edition, the authors improved the exposition and added more exercises to make the book even more student-oriented. They also added new material to bring the book more in line with the current research in dynamical systems.

Author : A. B. Katok
Publisher : American Mathematical Soc.
Page : 895 pages
File Size : 41,6 Mb
Release : 2001
Category : Ergodic theory
ISBN : 9780821826829

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Smooth Ergodic Theory and Its Applications by A. B. Katok Pdf

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student-or even an established mathematician who is not an expert in the area-to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference. Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincare and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of corre This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

Author : Pei-Dong Liu,Min Qian
Publisher : Springer
Page : 233 pages
File Size : 45,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540492917

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Smooth Ergodic Theory of Random Dynamical Systems by Pei-Dong Liu,Min Qian Pdf

This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of such systems, and can be viewed as an update of Kifer's book. An entropy formula of Pesin's type occupies the central part. The introduction of relation numbers (ch.2) is original and most methods involved in the book are canonical in dynamical systems or measure theory. The book is intended for people interested in noise-perturbed dynam- ical systems, and can pave the way to further study of the subject. Reasonable knowledge of differential geometry, measure theory, ergodic theory, dynamical systems and preferably random processes is assumed.

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 : Min Qian,Jian-Sheng Xie,Shu Zhu
Publisher : Springer
Page : 277 pages
File Size : 49,7 Mb
Release : 2009-07-07
Category : Mathematics
ISBN : 9783642019548

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Smooth Ergodic Theory for Endomorphisms by Min Qian,Jian-Sheng Xie,Shu Zhu Pdf

Ideal for researchers and graduate students, this volume sets out a general smooth ergodic theory for deterministic dynamical systems generated by non-invertible endomorphisms. Its focus is on the relations between entropy, Lyapunov exponents and dimensions.

Author : Pei-Dong Liu,Min Qian
Publisher : Unknown
Page : 240 pages
File Size : 49,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662200198

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Smooth Ergodic Theory of Random Dynamical Systems by Pei-Dong Liu,Min Qian Pdf

Author : Peter Walters
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 46,5 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 : Jon Aaronson
Publisher : American Mathematical Soc.
Page : 298 pages
File Size : 49,7 Mb
Release : 1997
Category : Ergodic theory
ISBN : 9780821804940

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An Introduction to Infinite Ergodic Theory by Jon Aaronson Pdf

Infinite ergodic theory is the study of measure preserving transformations of infinite measure spaces. The book focuses on properties specific to infinite measure preserving transformations. The work begins with an introduction to basic nonsingular ergodic theory, including recurrence behaviour, existence of invariant measures, ergodic theorems, and spectral theory. A wide range of possible "ergodic behaviour" is catalogued in the third chapter mainly according to the yardsticks of intrinsic normalizing constants, laws of large numbers, and return sequences. The rest of the book consists of illustrations of these phenomena, including Markov maps, inner functions, and cocycles and skew products. One chapter presents a start on the classification theory.

Author : Luis Barreira,Yakov Pesin
Publisher : Unknown
Page : 128 pages
File Size : 44,5 Mb
Release : 2014-02-19
Category : Electronic
ISBN : 1299707300

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Nonuniform Hyperbolicity by Luis Barreira,Yakov Pesin Pdf

A self-contained, comprehensive account of modern smooth ergodic theory, the mathematical foundation of deterministic chaos.

Author : Ricardo Mane
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 55,9 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 : Boris Hasselblatt
Publisher : Springer
Page : 334 pages
File Size : 40,6 Mb
Release : 2017-12-15
Category : Mathematics
ISBN : 9783319430591

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Ergodic Theory and Negative Curvature by Boris Hasselblatt Pdf

Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.

Author : Hillel Furstenberg
Publisher : American Mathematical Society
Page : 69 pages
File Size : 53,5 Mb
Release : 2014-08-08
Category : Mathematics
ISBN : 9781470410346

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Ergodic Theory and Fractal Geometry by Hillel Furstenberg Pdf

Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics. A co-publication of the AMS and CBMS.

Author : I. P. Cornfeld,S. V. Fomin,Y. G. Sinai
Publisher : Springer Science & Business Media
Page : 487 pages
File Size : 49,8 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.