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

By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

ISBN-10: 3319048066

ISBN-13: 9783319048062

ISBN-10: 3319048074

ISBN-13: 9783319048079

The paintings contains introductory classes, constructing diverse issues of view at the examine of the asymptotic behaviour of the geodesic circulation, particularly: the probabilistic procedure through martingales and combining (by Stéphane Le Borgne); the semi-classical procedure, by way of operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions goal to provide a self-contained advent to the information at the back of the 3 diverse ways to the research of hyperbolic dynamics. the 1st contribution concentrate on 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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For surfaces obtained by gluing copies of one finite volume surface cut along three (or more) periodic 50 S. Le Borgne geodesic orbits, the multidimensional Donsker invariant principle holds for ' the finite valued function that describes the displacements in the fibers. We deduce from the preceding theorem that almost surely the proportion of the time which the geodesic flow spends in a given cone oscillates infinitely often between 0 and 1. 1 Mixing and Directional Regularity The content of this section has been essentially given in [35].

1; d /. 1; d /= . 1; d /= is the action on the so called frame bundle. 1; d /= ! 1; d /= W x 7 ! gt x: As in Sect. G= ; T D g1 ; / and we treat only the case of C 1 functions. C1 1 k' C T ' C : : : C T n 1 'k22 : n 34 S. Le Borgne Let Sn ' be the ' C T ' C : : : C T n 1 '. w2 ; : : : ; wd / 0 ÂvC B B B B B B DB B B B B @ 1C jvj2 2 jvj2 2 1 v2 v3 :: : :: : vd jvj2 2 jvj2 2 v2 v3 :: : :: : vd v2 v2 1 0 :: : :: : 0 1 vd C vd C C 0C C 0C C C 0C C C 0A 1 v3 : : : vd 1 v3 : : : vd 1 0 ::: 0 1 0 :: 0 : :: : 1 0 ::: 0 and Âw D t ÂwC the transpose of ÂwC .

2r C 0/ dÂuC0 0/ Z ˘0; 1 . k1 Âs1 x; k2 Âs2 x/Á CC k'k1 mu . 6. 5. r0 /x with T n FT n . 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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Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

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