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Ergodic Theory and Negative Curvature

Boris Hasselblatt
Ergodic Theory and Negative Curvature by Boris Hasselblatt PDF

Author : Boris Hasselblatt
Publisher : Springer
Page : 334 pages
File Size : 45,11 MB
Release : 2017-12-15
Category : Mathematics
ISBN : 3319430599

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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.

Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

Françoise Dal'Bo
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo PDF

Author : Françoise Dal'Bo
Publisher : Springer
Page : 148 pages
File Size : 47,78 MB
Release : 2014-07-17
Category : Mathematics
ISBN : 3319048074

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The work consists of two introductory courses, developing different points of view on the study of the asymptotic behaviour of the geodesic flow, namely: the probabilistic approach via martingales and mixing (by Stéphane Le Borgne); the semi-classical approach, by operator theory and resonances (by Frédéric Faure and Masato Tsujii). The contributions aim to give a self-contained introduction to the ideas behind the three different approaches to the investigation of hyperbolic dynamics. The first contribution focus on the convergence towards a Gaussian law of suitably normalized ergodic sums (Central Limit Theorem). The second one deals with Transfer Operators and the structure of their spectrum (Ruelle-Pollicott resonances), explaining the relation with the asymptotics of time correlation function and the periodic orbits of the dynamics.

Ergodic Theory and Dynamical Systems

Yves Coudène
Ergodic Theory and Dynamical Systems by Yves Coudène PDF

Author : Yves Coudène
Publisher : Springer
Page : 192 pages
File Size : 29,51 MB
Release : 2016-11-10
Category : Mathematics
ISBN : 1447172876

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This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics. This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors. Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader.

Geometric and Ergodic Aspects of Group Actions

S. G. Dani
Geometric and Ergodic Aspects of Group Actions by S. G. Dani PDF

Author : S. G. Dani
Publisher : Springer Nature
Page : 176 pages
File Size : 41,99 MB
Release : 2020-01-13
Category : Mathematics
ISBN : 9811506833

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This book gathers papers on recent advances in the ergodic theory of group actions on homogeneous spaces and on geometrically finite hyperbolic manifolds presented at the workshop “Geometric and Ergodic Aspects of Group Actions,” organized by the Tata Institute of Fundamental Research, Mumbai, India, in 2018. Written by eminent scientists, and providing clear, detailed accounts of various topics at the interface of ergodic theory, the theory of homogeneous dynamics, and the geometry of hyperbolic surfaces, the book is a valuable resource for researchers and advanced graduate students in mathematics.

The Ergodic Theory of Discrete Groups

Peter J. Nicholls
The Ergodic Theory of Discrete Groups by Peter J. Nicholls PDF

Author : Peter J. Nicholls
Publisher : Cambridge University Press
Page : 237 pages
File Size : 22,33 MB
Release : 1989-08-17
Category : Mathematics
ISBN : 0521376742

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The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.

Introduction to Smooth Ergodic Theory

Luís Barreira
Introduction to Smooth Ergodic Theory by Luís Barreira PDF

Author : Luís Barreira
Publisher : American Mathematical Society
Page : 355 pages
File Size : 19,66 MB
Release : 2023-04-28
Category : Mathematics
ISBN : 1470473070

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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.