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Growth of the number of simple closed geodesics on hyperbolic surfaces. (English) Zbl 1177.37036
Let $$X$$ be a complete hyperbolic Riemann surface of finite area, genus $$g$$, with $$n$$ cusps, and let $$\gamma_0$$ be a simple closed geodesic on $$X$$. Let then $$s(L)$$ (respectively, $$s_X(L,\gamma_0)$$) denote the number of simple closed geodesics $$\gamma$$ of length $$\leq L$$ (respectively, having the same type as $$\gamma_0$$, i.e. such that $$X-\gamma$$ be homeomorphic to $$X\setminus\gamma_0$$).
We call a multi-curve on $$X$$ any formal linear combination $$\gamma:= \sum^k_{i=1} a_i\gamma_i$$, with positive rational coefficients $$a_i$$, of disjoint, essential, non-peripheral, simple closed curves $$\gamma_i$$, which belong to distinct homotopy classes. The length of such $$\gamma$$ is defined by $\ell_X(\gamma):= \sum^k_{i=1} a_i\ell_X(\gamma_i).$ Then the main result is that, for any multicurve $$\gamma$$, $$\lim_{L\to\infty} s_X(L,\gamma)/L^{6g- 6+2n}= N_\gamma(X)$$, where $$N_\gamma$$ is a positive continuous proper function on the moduli space of complete hyperbolic Riemann surfaces of genus $$g$$ with $$n$$ cusps. Moreover, the contributions of $$\gamma$$ and $$X$$ is $$N_\gamma(X)$$ separate into: $$N_\gamma(X)= c(\gamma)\times\beta(X)$$, where $$c(\gamma)$$ is a positive rational depending only on the type $$\gamma$$, and $$\beta(X)$$ is a normalized measure of the set of multi-curves on $$X$$ having integral coefficients and length $$\leq 1$$.
Central ingredients of the proof are:
– The Weil-Petersson volume form on the moduli space, and the integral of $$s_X(L,\gamma)$$ with respect to it;
– the ergodic action of the mapping class group on the space of measured geodesic laminations (endowed with the Thurston measure), over a closed surface of genus $$g$$ with $$n$$ boundary components.

##### MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C22 Geodesics in global differential geometry 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 37J50 Action-minimizing orbits and measures (MSC2010)
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