Counting Mapping Class group orbits on hyperbolic surfaces

Let $S_{g,n}$ be a surface of genus $g$ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[γ]$ of a closed curve $γ\in π_{1}(S_{g,n})$ contains a unique closed geodesic on $X$. Let $\ell_γ(X)$ denote the hyperbolic length of the geodesic representative of $γ$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length $\leq L$ on $X$ with at most $k$ self-intersections. We also discuss properties of random pants decomposition of large length on $X$. Both these results are based on ergodic properties of the earthquake flow on a natural bundle over the moduli space $\mathcal{M}_{g,n}$ of hyperbolic surfaces of genus $g$ with $n$ cusps.

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