## Analytic and Probabilistic Approaches to Dynamics in Negative CurvatureThe 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
2 Semiclassical Approach for the RuellePollicott Spectrum of Hyperbolic Dynamics | 64 |

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ˇ ˇ ˇ algebra anisotropic Sobolev space Anosov diffeomorphism Anosov map asymptotic atoms bounded bundle cat map central limit theorem compact consider contact Anosov flow convergence in distribution correlation functions cotangent space decay of correlations decomposition deduce deﬁned Deﬁnition denote diffeomorphism differences of martingale discrete spectrum dynamical system eigenvalues ergodic ergodic sums example exists exponentially mixing f W M ﬁeld filtration ﬁnite finite volume ﬂow Fourier geodesic flow Hölder continuous horocycle Hyperbolic Dynamics integral Lemma linear Liverani manifold map f martingale Math matrix measure obtain orthogonal partition periodic orbits PkE.fjA0 prequantum map Proposition Qk.f E.fjA0 quantization quantum random variables Remark reversed martingale Ruelle spectrum Ruelle-Pollicott resonances Ruelle-Pollicott spectrum semiclassical analysis sequence of differences SL2R smooth Sobolev space Springer surface symbol symplectic trace formula transfer operator trapped set Tsujii unstable leaves Weyl zeta function