By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings comprises introductory classes, constructing varied issues of view at the research of the asymptotic behaviour of the geodesic move, particularly: the probabilistic method through martingales and combining (by Stéphane Le Borgne); the semi-classical strategy, through operator concept and resonances (by Frédéric Faure and Masato Tsujii). The contributions objective to provide a self-contained advent to the information at the back of the 3 diverse techniques to the research of hyperbolic dynamics. the 1st contribution concentrate on 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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Additional resources for Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
Xg 1 / dg: n n G G We identify the Lie-algebra of G and the set of right-invariant vector field on G. Then, for every n, the functions n are C 1 and for every differential operator ˝ of the universal enveloping algebra of G, ˝. ˝ n / . If ' is an integrable function on G= , it defines a function on G that is invariant. One check that n ' is right -invariant. ˇ/ be the set of points of G= at distance less than ˇ from the boundary of U . Let ' be a C 1 centered function on G= . 3) applied to n . / sequence n insure the existence of a constant C > 0 such that: 1 Martingales in Hyperbolic Geometry ˇ ˇhT n ˇ 1U ; 'iˇ Ä C ˝ m .
Yk / be a sequence of random variables. Œ0; 1/: D 1 p for t 2 Œ kn ; kC1 n , k D 0; : : : ; n 1. 7. Let Yk a sequence of differences of martingale (or of reversed martingale) with variance 1. L W: Proof. We begin by the convergence of finite dimensional distributions. Let l be a positive integer, t1 ; : : : ; tl , l numbers in Œ0; 1. We want to show that 1 Martingales in Hyperbolic Geometry . Wt1 ; : : : ; Wtl /: It is equivalent to the following convergence (we just apply a linear transformation) which gives simpler computations: .
P 3C 5 2 ). x C tv/dt: lk;x For a fixed k, these segments may have two different lengths: the numbers uk;x lk;x belongs to a pair fa k ; b k g (where a and b are the two numbers necessary 1 Martingales in Hyperbolic Geometry 19 to the description of the two length of the unstable side of the rectangles of the Markov partition). 2i t. uk;x X p2Z2 jcp j lk;x / j2i . p 5 1 p1 2 1 p j 52 1 p1 C p2 j : C p2 /j 20 S. x/j Ä C k cjp1 j 1 . f jA p2Z2 jp1 cp j converges. f jAk / jj1 < 1 k 0 are thus convergent in this case.
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti