By Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti
The paintings comprises introductory classes, constructing various issues of view at the examine of the asymptotic behaviour of the geodesic move, particularly: the probabilistic process through martingales and combining (by Stéphane Le Borgne); the semi-classical strategy, through operator idea and resonances (by Frédéric Faure and Masato Tsujii). The contributions objective 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 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
3 The CLT Along Subsequences We consider ergodic sums at random times with the hypothesis that these times define a sequence of positive density in N. 8 (“CLT along subsequences”). ˝; B; P/ and n W ˝ ! 1 a: k Then n. / 1 1 X p Yj . 2; R/ by a cocompact lattice of the one parameter group gt W G= ! G= 1 Martingales in Hyperbolic Geometry 27 Â y7 ! gt /dt by T . To construct a filtration as explained before, it is convenient to be in a discrete setting. Let T denotes the translation by g1 to the left: T W G= !
Y/ Z T nU D Œ e n ;e n d 1 Œ ; BK . d 1/t dvdtdkdÂ C : 40 S. d 1U ; 'i Œ e n ;e n d 1 Œ ; BK . Â C T n x/ dÂ C j Ä C. 4) We just have to bound the quantity jhT n 1U ; 'ij which is small because of the exponential decay of correlations. As 1U is not differentiable, one has to regularize it. For a real number larger than 1, we call -identity a sequence . Id; n /. • jj˝ m n jj1 Ä C Cn . • n is Lipschitz-continuous with Lipschitz constant Cn . Such sequences do exist. xg 1 / dg: n n G G We identify the Lie-algebra of G and the set of right-invariant vector field on G.
By k 7 ! k 1 x/ which can be decomposed as a sum of irreducible representations. Let KO m be the set of the equivalence classes of irreducible representations of Km and ı be an element of KO m . Let us fix a base R of the root system of Km the Lie algebra of Km . Let us call W the associated Weyl chamber. To each irreducible representation of Km is uniquely associated a linear form belonging to a lattice in W : the dominant weight of the representation. Let ı be an element of KO m and let ı be the corresponding dominant weight.
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature by Françoise Dal'Bo, Marc Peigné, Andrea Sambusetti