By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings comprises introductory classes, constructing assorted issues of view at the research of the asymptotic behaviour of the geodesic move, specifically: the probabilistic method through martingales and combining (by Stéphane Le Borgne); the semi-classical strategy, through operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions target to offer a self-contained creation to the guidelines in the back of the 3 diversified methods 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 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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Extra info for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
For every ı 2 KO m , let ı be the character of ı (the function on Km defined as the trace of the action of the elements of Km on the irreducible representation of Km associated to ı), ı D dı ı , and Z Pı D U. 1) Km Remark that kPı 'k1 Ä dı2 k'k1 . The operator Pı is the projection of L2 . L2 . //. We have the decomposition L2 . / D M ı2KO m Fı : 38 S. Le Borgne For a given vector v in L2 . / let vı WD Pı v. 2) One says that v is C 1 if the map k 7! k/v is C 1 . One defines the derived representation of U on the space of C 1 elements; it is a representation of the Lie algebra Km of Km and that one can be extended to a representation of the universal enveloping algebra of Km .
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 . The second of the preceding properties shows that T A0 is a sub- -algebra of A0 . hut /t 2R . We now want a control of the lengths of the atoms of A0 . x/? T k x/ for every k 0, that is when T k fhut x = t 2 Œ ; g D fhue kt T k x=t 2Œ ; g does not hit F0 . x/, if for every k 0, T k x does not belongs to the set fhut y = y 2 F0 ; t 2 Œ e k ; e k g.
T k x/ for every k 0, that is when T k fhut x = t 2 Œ ; g D fhue kt T k x=t 2Œ ; g does not hit F0 . x/, if for every k 0, T k x does not belongs to the set fhut y = y 2 F0 ; t 2 Œ e k ; e k g. These sets have a measure bounded by Ce k . 2. x/ has measure larger than 1 C . 3. lx1 ; u1 x /g. f jAn / jj2 30 S. Le Borgne for a C 1 function. The second convergence is an obvious consequence of the mean value theorem: the size of the atoms of An are uniformly exponentially small for n tending toward infinity.