Analytic and probabilistic approaches to dynamics in negative curvature /

Analytic and probabilistic approaches to dynamics in negative curvature /

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Bibliographic Details
Imprint:	Cham : Springer, 2014.
Description:	1 online resource (xi, 138 pages) : illustrations (some color).
Language:	English
Series:	Springer INdAM Series, 2281-518X ; volume 9 Springer INdAM series ; v. 9.
Subject:	Geodesic flows. Central limit theorem. Ergodic theory. Central limit theorem. Ergodic theory. Geodesic flows.
Format:	E-Resource Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/11086633

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Other authors / contributors:	Dal'bo-Milonet, Françoise, editor. Peigné, Marc, editor. Sambusetti, Andrea, editor.
ISBN:	9783319048079 3319048074 3319048066 9783319048062 9783319048062
Notes:	Includes bibliographical references and index. Online resource; title from PDF title page (SpringerLink, viewed July 30, 2014).
Summary:	The work of E. Hopf and G.A. Hedlund, in the 1930s, on transitivity and ergodicity of the geodesic flow for hyperbolic surfaces, marked the beginning of the investigation of the statistical properties and stochastic behavior of the flow. The first central limit theorem for the geodesic flow was proved in the 1960s by Y. Sinai for compact hyperbolic manifolds. Since then, strong relationships have been found between the fields of ergodic theory, analysis, and geometry. Different approaches and new tools have been developed to study the geodesic flow, including measure theory, thermodynamic formalism, transfer operators, Laplace operators, and Brownian motion. All these different points of view have led to a deep understanding of more general dynamical systems, in particular the so-called Anosov systems, with applications to geometric problems such as counting, equirepartition, mixing, and recurrence properties of the orbits. This book comprises two independent texts that provide a self-contained introduction to two different approaches to the investigation of hyperbolic dynamics. The first text, by S. Le Borgne, explains the method of martingales for the central limit theorem. This approach can be used in several situations, even for weakly hyperbolic flows, and the author presents a good number of examples and applications to equirepartition and mixing. The second text, by F. Faure and M. Tsujii, concerns the semiclassical approach, by operator theory: chaotic dynamics is described through the spectrum of the associated transfer operator, with applications to the asymptotic counting of periodic orbits. The book will be of interest for a broad audience, from PhD and Post-Doc students to experts working on geometry and dynamics.
Other form:	Print version: Analytic and probabilistic approaches to dynamics in negative curvature. Cham, Switzerland : Springer, ©2014 xi, 138 pages Springer INdAM series ; Volume 9 2281-5198 9783319048062
Standard no.:	10.1007/978-3-319-04807-9

Description
Summary:	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);nbsp;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.
Physical Description:	1 online resource (xi, 138 pages) : illustrations (some color).
Bibliography:	Includes bibliographical references and index.
ISBN:	9783319048079 3319048074 3319048066 9783319048062
ISSN:	2281-518X ;