## 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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### References:

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