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

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 movement, specifically: the probabilistic procedure through martingales and combining (by Stéphane Le Borgne); the semi-classical process, via operator conception and resonances (by Frédéric Faure and Masato Tsujii). The contributions objective to provide a self-contained advent to the tips at the back of the 3 assorted techniques to the research of hyperbolic dynamics. the 1st contribution specialise in the convergence in the direction of a Gaussian legislation 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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4. 9. y/ˇˇ Ä CT ˛ jjf jjH 3 ; 2 where jjf jjH 3 refers to some Sobolev norm. 2 This enables us to construct an adequate filtration. v; t; w/ 2 Œ r; r3 g do not coincide with X (it is the case if r is small enough). The set P0 defines a partition of X : P0 and its complementary set. They both have a non-empty interior. hut /t 2R do cross P0 and c P0 . lx ; ux /g: This is a partition up to the null set F0 . hut /t 2R . • The image by T of an atom of Q01 is a union of atoms. Let A0 be the -algebra of Borel sets that are saturated for the equivalence relation defined by Q01 .

6 For more details see [34]. 48 S. Le Borgne Fig. 12. X; T; / be a dynamical system and ' a function with values in R2 . 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.

T k x/ define a vectorial process. s. to 1 in the direction of the mean ' d . Consider the case when ' is centered. A question is: in which directions at infinity the ergodic sums are they going? When 1 Martingales in Hyperbolic Geometry 49 ' satisfies a CLT, one can think that the sums behave analogously to a Brownian motion. Bt / denote the standard Brownian motion in Rd , for d 2. 13. Let C be a cone with non empty interior and non empty exterior and boundary @C of measure 0. X; T; / be an ergodic dynamical system and ' be a measurable function on X with values in Rd , d 2.

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