Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

1. Front Matter

   Pages i-xi

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2. Martingales in Hyperbolic Geometry

   • Stéphane Le Borgne

   Pages 1-63

3. Semiclassical Approach for the Ruelle-Pollicott Spectrum of Hyperbolic Dynamics

   • Frédéric Faure, Masato Tsujii

   Pages 65-135

4. Back Matter

   Pages 137-138

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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.

• brownian motion
• central limit theorem
• fuchsian group
• hyperbolic dynamics
• ruelle-pollicott resonances

• IRMAR, Université de Rennes 1, Rennes, France

  Françoise Dal'Bo

• Lab. de Mathématiques et Physique, Université François Rabelais, Tours, France

  Marc Peigné

• Dipartimento di Matematica, Sapienza - Università di Roma, Roma, Italy

  Andrea Sambusetti

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