By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings involves introductory classes, constructing diversified issues of view at the learn of the asymptotic behaviour of the geodesic circulation, specifically: the probabilistic procedure through martingales and combining (by Stéphane Le Borgne); the semi-classical method, through operator idea and resonances (by Frédéric Faure and Masato Tsujii). The contributions target to provide a self-contained creation to the tips in the back of the 3 diversified 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 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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Additional info for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
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. In the upper-half plane model of the hyperbolic space, a cusp is described as I c; C1Œ with I a compact subset of R. The preceding technique still works in this new situation. But of course here it is no longer true that every image of a large stable cube in X is well distributed.
X/ D 0: Let us come back to Zd -fibered hyperbolic surfaces (d 3). 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; 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 . d 1/ and Â C (resp. Â ) the groups isomorphic to Rd 1 of matrices ÂvC (resp.
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti