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Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions. (English) Zbl 0784.60076

From the introduction: Here we present, in a very specific case, a result analogous to Spitzer’s theorem which describes the asymptotic law of the windings of a two-dimensional Brownian motion around a point. The spaces we consider are modular surfaces obtained as quotients of the hyperbolic plane by normal subgroups of finite index in the modular group SL\(_ 2(\mathbb{Z})\). Such a modular surface has finite hyperbolic area and is naturally compactified in a compact Riemann surface of genus \(g\) by adding \(c\) points (cusps). Then, roughly speaking, our main result says that the normalized homological winding of the geodesic flow converges toward the product of the two non-degenerate probobability laws. The first one is a \(2g\)-dimensional Gaussian law associated with the compactification; the second one is a \((c-1)\)-dimensional Cauchy law which is itself the convolution of \(c\) elementary Cauchy laws corresponding to the cusps.

MSC:

60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
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