By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings includes introductory classes, constructing diverse issues of view at the research of the asymptotic behaviour of the geodesic circulation, particularly: the probabilistic strategy through martingales and combining (by Stéphane Le Borgne); the semi-classical method, by means of operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions goal to offer a self-contained advent to the guidelines at the back of the 3 varied ways to the research of hyperbolic dynamics. the 1st contribution specialise in the convergence in the direction of a Gaussian legislations of certainly normalized ergodic sums (Central restrict Theorem). the second offers 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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Additional info for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
We will use the following property of equidistribution of unstable leaves for the geodesic flow, due to Burger . 4. 9. y/ˇˇ Ä CT ˛ jjf jjH 3 ; 2 where jjf jjH 3 refers to some Sobolev norm. 2 This enables us to construct an adequate filtration. v; t; w/ 2 Œ r; r3 g do not coincide with X (it is the case if r is small enough). The set P0 defines a partition of X : P0 and its complementary set. They both have a non-empty interior. hut /t 2R do cross P0 and c P0 . lx ; ux /g: This is a partition up to the null set F0 .
Let l be a positive integer, t1 ; : : : ; tl , l numbers in Œ0; 1. We want to show that 1 Martingales in Hyperbolic Geometry . Wt1 ; : : : ; Wtl /: It is equivalent to the following convergence (we just apply a linear transformation) which gives simpler computations: . 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.
F jA p2Z2 jp1 cp j converges. f jAk / jj1 < 1 k 0 are thus convergent in this case. 1 The Donsker Invariance Principle We are interested now in another limit theorem. This is also a convergence in distribution but for probabilities defined on the space of continuous functions on Œ0; 1. Œ0; 1/; kk1 /. 5. Œ0; 1/ a unique measure P for which the following properties hold: (i) For every t, the law of the random variable Wt W f 7! t/ is the centered gaussian law with variance t. (ii) For every t; s with s < t, the random variables Wt Ws and Ws are independent.