Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

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Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
Springer, Jul 17, 2014 - Mathematics - 138 pages
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.

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Chapter 1 Martingales in Hyperbolic Geometry
Chapter 2 Semiclassical Approach for the RuellePollicott Spectrum of Hyperbolic Dynamics

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