By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti

The paintings includes introductory classes, constructing diversified issues of view at the examine of the asymptotic behaviour of the geodesic circulate, specifically: the probabilistic process through martingales and combining (by Stéphane Le Borgne); the semi-classical technique, by means of operator conception and resonances (by Frédéric Faure and Masato Tsujii). The contributions target to offer a self-contained advent to the guidelines at the back of the 3 varied techniques 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**

**Example text**

We will briefly show it for the surfaces of constant negative curvature that are fibered above a finite volume with Zd -fibers [12, 23, 45]. Once again we study the time-one map associated. It can be represented as a skew-product over the time one geodesic flow in the finite volume case: T' W X Zd ! x; y/ 7! x//; where T is the time-one map of the flow defined on the base (of finite volume) and ' is a function with values in Zd describing the displacement in the fibers. Sn ' 2 B/ (where B is a ball) is equivalent to cn d=2 then the cocycle is recurrent for (and only for) d Ä 2.

If ' satisfies the CLT for the subsequences then the cocycle Sn ' is recurrent. X; T; / is a K-system, then recurrence may imply the ergodicity. It is thus possible to deduce that the flow with fibers Zd is ergodic if and only if d Ä 2 (see [23] for more details). 2; R/= 0 the unit tangent bundle of a finite volume hyperbolic surface. Cutting the surface along a periodic geodesic or two and gluing together copies along the chosen geodesic(s) define new surfaces of infinite volume (cp. Fig. 2; R/= with 0 = D Z or 0 = D Z2 .

Cauchy-Schwarz inequality shows that this is the case only when '1 and '2 are proportional that is when ' takes its values in a line of R2 . If we apply the Gordin method to a regular function with values in R2 then we obtain that the normalized ergodic sums tend to a non degenerated gaussian vector if ' is not cohomologous to a function taking its values in a line of R2 . We also have a multidimensional version of the Donsker invariance principle that asserts the convergence in distribution of the interpolated lines defined by the ergodic sums of Rd -valued regular functions to a Brownian motion in Rd .