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Martingale method and geodesic flow on a surface of constant negative curvature. (Méthode de martingales et flot géodésique sur une surface de courbure constante négative.) (French) Zbl 0983.37034

Authors’ abstract: Let \(({\mathcal T}^1S, m,(T^t)_{t\in\mathbb{R}})\) be the geodesic flow on the unit tangent bundle of a surface \(S\) of negative constant curvature and finite volume. We show that every Hölder function on \({\mathcal T}^1S\) is, for the discrete time action of the geodesic flow, homologous to a martingale increment. From this representation follow the central limit theorem and its improvements, and a characterization of Hölder functions which are coboundaries in the class of measurable functions.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
58J65 Diffusion processes and stochastic analysis on manifolds
60G42 Martingales with discrete parameter
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