# Analytic and Probabilistic Approaches to Dynamics in by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

The paintings involves introductory classes, constructing diversified issues of view at the research of the asymptotic behaviour of the geodesic movement, specifically: the probabilistic method through martingales and combining (by Stéphane Le Borgne); the semi-classical strategy, by way of operator conception and resonances (by Frédéric Faure and Masato Tsujii). The contributions objective to offer a self-contained creation to the tips in the back of the 3 diversified ways to the research of hyperbolic dynamics. the 1st contribution specialize in the convergence in the direction of a Gaussian legislations of certainly normalized ergodic sums (Central restrict Theorem). the second bargains with move Operators and the constitution in their spectrum (Ruelle-Pollicott resonances), explaining the relation with the asymptotics of time correlation functionality and the periodic orbits of the dynamics.

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The operators ˝ being hermitian, cı is positive. One can show (cf. [6]) that there exists a scalar product Q such that cı D Q. ı C / Q. /. ˝ m /vk1 : cım One can chose m so large that the series X d2 ı cım ı2KO converges. For such a choice of m we have the following proposition. 4. Let ' and be two centered C 1 functions on G= . 3) Let us come back to the CLT. Here we consider the time one map T D g1 of the flow gt . The theorem for continuous time follows. 1 Martingales in Hyperbolic Geometry 39 Tn Fig.

K X ˇ ˇp ˇ P. k 1/˛ Putting together the preceding inequalities we get P. n C 1/2 Ä C ˛ 4 t u which tends to 0 as ˛ approaches to 0. The CLT and the invariance principle are the only examples of limit theorem for which we will give complete proofs. Nevertheless similar computations can provide other properties. Let us mention a couple of examples. 2 The CLT for Vector Valued Functions Consider the case where ' has values in R2 (the d -dimensional case is treated in the same manner). '1 ; '2 / generates a sequence of differences of a martingale if '1 and '2 do.

Id; /F . v; t; k; w/ 7 ! 1; d /= . Under the action of T n this set dilates in the Â C direction, coils in X (see Fig. 2) and, as we will see, spreads itself uniformly in X . y/ Z T nU D Œ e n ;e n d 1 Œ ; BK . d 1/t dvdtdkdÂ C : 40 S. d 1U ; 'i Œ e n ;e n d 1 Œ ; BK . Â C T n x/ dÂ C j Ä C. 4) We just have to bound the quantity jhT n 1U ; 'ij which is small because of the exponential decay of correlations. As 1U is not differentiable, one has to regularize it. For a real number larger than 1, we call -identity a sequence .