Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

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.

List of contents

1 S. Le Borgne: Martingales in Hyperbolic Geometry.- 2 F. Faure, M. Tsujii: Semi classical Approach for the Ruelle-Pollicott Spectrum of Hyperbolic Dynamics.

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); 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.

Product details

• Assisted by: Françoise Dal'Bo (Editor), Mar Peigné (Editor), Marc Peigné (Editor), Andrea Sambusetti (Editor)
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 17.01.2014
• EAN: 9783319048062
• ISBN: 978-3-31-904806-2
• No. of pages: 138
• Dimensions: 167 mm x 245 mm x 14 mm
• Weight: 344 g
• Illustrations: XI, 138 p. 32 illus., 12 illus. in color.
• Series: Springer INdAM Series; Springer INdAM Series
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; B, Dynamics, Mathematics and Statistics, Probability Theory and Stochastic Processes, Differential Geometry, Probability & statistics, Dynamical Systems and Ergodic Theory, Ergodic theory, Dynamical systems, Probabilities, Stochastics, Probability Theory, Functional analysis & transforms, Operator Theory, Differential & Riemannian geometry, Hyperbolic Geometry, Non-Euclidean geometry