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.


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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