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
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 / .