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 $[\gamma ]$ of a closed curve $\gamma \in {\pi }_1(S_{g,n})$ contains a unique closed geodesic on $X$. Let ${\mathrm{\ell }}_{\gamma }(X)$ denote the hyperbolic length of the geodesic representative of $\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}\mathrm{.}$ As an application, one can obtain the asymptotics of the growth of $s_X^k(L)$, the number of closed curves of length $\le L$ 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.