By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings includes introductory classes, constructing diverse issues of view at the research of the asymptotic behaviour of the geodesic stream, specifically: the probabilistic process through martingales and combining (by Stéphane Le Borgne); the semi-classical method, through operator thought and resonances (by Frédéric Faure and Masato Tsujii). The contributions goal to provide a self-contained advent to the tips at the back of the 3 various ways to the research of hyperbolic dynamics. the 1st contribution specialize in 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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Proof. 'jAn / jj2 < 1: n<0 The partitions Qn1 D T n Q01 are measurable in the sense of Rokhlin. 0;T n x/T n For large n > 0 these quantities are near to the value of ' at x. 0;T n x/T n Let ' be a C 1 function. We take a point x 2 T n Wn . T n x/. ˇ n /T n x 42 S. 0; T Ä C. @U. @U. x/j Ä C. x/ Ä C n C Cˇ n jj'jj21 : This proves the convergence of the first series. The convergence of the second one is trivial. x; y/Á The preceding theorem is still true if ' is Á-Hölder continuous or the characteristic function of a set with smooth boundary (and such a function is never a coboundary).
0; 1/. 1 n and, for every > 0, lim sup P. y/j jx yj<˛ > / D 0: The tightness of a normalized sequence defined by a bounded stationary ergodic martingale is a consequence of the following maximal inequality. 6 (“Doob inequality”2 ). Fk /k 0 . Then we have ! P sup j Sk4 j ˛ Ä 0ÄkÄn E Sn4 : ˛ Proof. Let Aj be the set Aj D sup Sk4 < ˛: 0ÄkÄj The sequence of sets obtained is decreasing and their intersection is An the complementary set of which is B D sup0ÄkÄn Sk4 ˛. Let us consider the sum 2 This inequality is true for non negative submartingales.
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 . 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 .
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti