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 \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} 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.