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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Extra info for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
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; r3 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 . 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  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.