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 comprises introductory classes, constructing diversified issues of view at the examine of the asymptotic behaviour of the geodesic stream, particularly: the probabilistic technique through martingales and combining (by Stéphane Le Borgne); the semi-classical procedure, through operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions goal to offer a self-contained advent to the tips in the back of the 3 various methods to the research of hyperbolic dynamics. the 1st contribution specialize in the convergence in the direction of a Gaussian legislations of definitely 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

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F . 1 X G n //: If ' is Hölder continuous or the characteristic function of a set with smooth boundary then it is well approximated by the regularized functions and we apply the preceding reasoning by regularizing not only 1U but also '. 2 Example 6: The Geodesic Flow on a Surface with Constant Curvature of Finite Volume Let X be the unit tangent bundle of a connected finite volume hyperbolic surface (of constant curvature). We can describe the structure of X as follows: there exists a compact subset X0 of X such that X n X0 is a finite union of cusps.

Let ı be an element of KO m and let ı be the corresponding dominant weight. The Weyl formula gives the dimension dı of the irreducible representation associated to ı as a function of : dı D Y h˛; ı C i ; h˛; i ˛2R C where RC is the set of positive roots and the half sum of the positive roots. For every ı 2 KO m , let ı be the character of ı (the function on Km defined as the trace of the action of the elements of Km on the irreducible representation of Km associated to ı), ı D dı ı , and Z Pı D U.

The integrals on connected components may be different only on pieces containing a point of the boundary of T n FT n x (on entire “slices” the integrals of ' and are equal by definition of ). T n FT u Z n / . ÂuC ; ÂuC0 / Ä r0 . Cr 0 e n /. Cr 0 e n /T n . T n FT u Z n / .

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