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.
Analytic And Probabilistic Approaches To Dynamics In Negative Curvature
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Author : Françoise Dal'Bo
language : en
Publisher: Springer
Release Date : 2014-07-17
Analytic And Probabilistic Approaches To Dynamics In Negative Curvature written by Françoise Dal'Bo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-17 with Mathematics categories.
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.
Equidistribution And Counting Under Equilibrium States In Negative Curvature And Trees
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Author : Anne Broise-Alamichel
language : en
Publisher: Springer Nature
Release Date : 2019-12-16
Equidistribution And Counting Under Equilibrium States In Negative Curvature And Trees written by Anne Broise-Alamichel and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-16 with Mathematics categories.
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.
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.
Equilibrium States In Negative Curvature
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Author : Barbara Schapira
language : en
Publisher:
Release Date : 2015
Equilibrium States In Negative Curvature written by Barbara Schapira and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Curvature categories.
With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications to the variational principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers
Geodesic Flows On Closed Riemann Manifolds With Negative Curvature
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Author : D. V. Anosov
language : en
Publisher:
Release Date : 1969
Geodesic Flows On Closed Riemann Manifolds With Negative Curvature written by D. V. Anosov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Mathematics categories.
Geometric And Ergodic Aspects Of Group Actions
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Author : S. G. Dani
language : en
Publisher: Springer Nature
Release Date : 2020-01-13
Geometric And Ergodic Aspects Of Group Actions written by S. G. Dani and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-13 with Mathematics categories.
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.
Ergodic Theory With Applications To Dynamical Systems And Statistical Mechanics
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Author : I︠A︡kov Grigorʹevich Sinaĭ
language : en
Publisher: Springer
Release Date : 1989
Ergodic Theory With Applications To Dynamical Systems And Statistical Mechanics written by I︠A︡kov Grigorʹevich Sinaĭ and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Mathematics categories.
1. Ordinary differential equations and smooth dynamical systems by D.V. Anosov, V.I. Arnold (eds.). 2. Ergodic theory with applications to dynamical systems an d statistical mechanics by Ya. G. Sinai (ed.). 3. [without special title]. 4. S ymplectic geometry and its applications by V.I. Arnold, S.P. Novikov (eds.).
Lectures On Ergodic Theory
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Author : Paul R. Halmos
language : en
Publisher: Courier Dover Publications
Release Date : 2017-11-15
Lectures On Ergodic Theory written by Paul R. Halmos and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-15 with Mathematics categories.
This concise classic by a well-known master of mathematical exposition covers recurrence, ergodic theorems, ergodicity and mixing properties, and the relation between conjugacy and equivalence. 1956 edition.
Ergodic Theory And Dynamical Systems Ii
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Author : Katok
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11
Ergodic Theory And Dynamical Systems Ii written by Katok 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 2013-11-11 with Mathematics categories.