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

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

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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 .

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