Ergodic Theory And Negative Curvature

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Ergodic Theory And Negative Curvature

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Author : Boris Hasselblatt
language : en
Publisher: Springer
Release Date : 2017-12-15

Ergodic Theory And Negative Curvature written by Boris Hasselblatt and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-15 with Mathematics categories.

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.

Lectures On Spaces Of Nonpositive Curvature

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Author : Werner Ballmann
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Lectures On Spaces Of Nonpositive Curvature written by Werner Ballmann and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.

Ergodic Theory Semisimple Lie Groups And Foliations By Manifolds Of Negative Curvature

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Author : Robert J. Zimmer
language : en
Publisher:
Release Date : 1982

Ergodic Theory Semisimple Lie Groups And Foliations By Manifolds Of Negative Curvature written by Robert J. Zimmer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with categories.

Ergodic Theory

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Author : Cesar E. Silva
language : en
Publisher: Springer Nature
Release Date : 2023-07-31

Ergodic Theory written by Cesar E. Silva and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-31 with Mathematics categories.

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

Ergodic Theory And Its Connection With Harmonic Analysis

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Author : Karl Endel Petersen
language : en
Publisher: Cambridge University Press
Release Date : 1995

Ergodic Theory And Its Connection With Harmonic Analysis written by Karl Endel Petersen and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.

Tutorial survey papers on important areas of ergodic theory, with related research papers.

Ergodic Theory And Topological Dynamics Of Group Actions On Homogeneous Spaces

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Author : M. Bachir Bekka
language : en
Publisher: Cambridge University Press
Release Date : 2000-05-11

Ergodic Theory And Topological Dynamics Of Group Actions On Homogeneous Spaces written by M. Bachir Bekka and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-05-11 with Mathematics categories.

This book, first published in 2000, focuses on developments in the study of geodesic flows on homogenous spaces.

Mathematics Of Complexity And Dynamical Systems

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Author : Robert A. Meyers
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-10-05

Mathematics Of Complexity And Dynamical Systems written by Robert A. Meyers and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-05 with Mathematics categories.

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.

Lectures On Ergodic Theory And Pesin Theory On Compact Manifolds

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Author : Mark Pollicott
language : en
Publisher: Cambridge University Press
Release Date : 1993-02-04

Lectures On Ergodic Theory And Pesin Theory On Compact Manifolds written by Mark Pollicott and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-02-04 with Mathematics categories.

These lecture notes provide a unique introduction to Pesin theory and its applications.

Ergodic Theory

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Author : Karl E. Petersen
language : en
Publisher: Cambridge University Press
Release Date : 1989-11-23

Ergodic Theory written by Karl E. Petersen and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989-11-23 with Mathematics categories.

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.

Ergodic Theory

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Author : I. P. Cornfeld
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Ergodic Theory written by I. P. Cornfeld and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.