By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings contains introductory classes, constructing assorted issues of view at the examine of the asymptotic behaviour of the geodesic circulation, particularly: the probabilistic strategy through martingales and combining (by Stéphane Le Borgne); the semi-classical process, by way of operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions goal to offer a self-contained advent to the information at the back of the 3 varied ways to the research of hyperbolic dynamics. the 1st contribution specialize in the convergence in the direction of a Gaussian legislation 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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Extra resources for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
1. t1 ; t2 ; : : : ; tk / 2 Œ0; 1k . It is well explained in  how to get the convergence towards the Wiener measure (and Billingsley treat the case of the sequences of differences of a martingale; this is also done in the short paper ). One can proceed in two steps: first get the 1 Martingales in Hyperbolic Geometry 21 convergence of finite dimensional distributions, second prove that the sequence considered is tight. A sequence of probabilities . K/ > 1 . y/j < /: Let . Œ0; 1/. 1 n and, for every > 0, lim sup P.
Wt1 ; Wt2 W t1 ; : : : ; W tl Wtl 1 /: Making use of the theorem of Levy it reduces to show that, for all uj , 0 0 E @exp @i l X 11 uj . tj l Y 1 //AA ! j D1 exp. tj tj 1 /=2/: j D1 One more time, we consider just the bounded case. i p /A n 1 u2j Yk2 u j Yk u j Yk C '. 1 C i p / exp. exp. n n j D1 Œnt j X kDŒntj 1 11 u Y j k Yk2 C '. p //AA : n Birkhoff theorem implies that u2j n Œnt j X kDŒntj Yk2 ! t/ Ä C t 3 , the Lebesgue theorem implies that the difference between 24 S. exp. n n j D1 Œnt j X kDŒntj 1 11 u Y j k Yk2 C '.
1/. The operators ˝ being hermitian, cı is positive. One can show (cf. ) 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.
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti