By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

ISBN-10: 3319048066

ISBN-13: 9783319048062

ISBN-10: 3319048074

ISBN-13: 9783319048079

The paintings comprises introductory classes, constructing diverse issues of view at the examine of the asymptotic behaviour of the geodesic circulation, specifically: the probabilistic strategy through martingales and combining (by Stéphane Le Borgne); the semi-classical method, through operator conception and resonances (by Frédéric Faure and Masato Tsujii). The contributions objective to provide a self-contained advent to the tips at the back of the 3 diversified 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 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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**Example text**

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. 2 Coiling of the image of a “cube” in X As already said our proof of the CLT will rely on a quantitative information of the repartition of the unstable leaves of T in G= .

V; t; k; w/ 7! v; t; k; w/ 7! 2r0 / for r0 small enough. v; t; k; w/ 7! y/ D fÂuC y = u 2 d Y 1 . T 2 x/ \ : : : This is the coarser partition among those which are finer than Q and that are included in their image by T 1 . For every n 2 Z let us call An the -algebra the atoms of which are the sets T n Q01 . An / is an increasing filtration. x/ shows that the atoms of A0 are unions of atoms of A1 . er/. T k x/g: 36 S. x/: /x On the other hand, as the boundaries of Q are regular, there exists C > 0 such that, for every > 0, we have fx 2 G= C W .

It is similar to the idea of integrations by parts that give information on the decay of the Fourier coefficients of a several times differentiable periodic function. This idea still works in a more general setting, but one has to replace the imaginary exponential functions by what is called Km -finite functions (where Km is a maximal 1 Martingales in Hyperbolic Geometry 37 compact subgroup of G), and the differentiation by the application of the Casimir operator. G= /. It is proved in [3] that does not weakly contain the identity.

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

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