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

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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We will use the following property of equidistribution of unstable leaves for the geodesic flow, due to Burger [10]. 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; r3 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.

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