Orbits of non-simple closed curves on a surface

Abstract: The mapping class group of a surface $\S$ acts on the set of closed geodesics on $\S$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits that contain geodesics of length at most $L$, for each $L >0$. This result complements a recent result of Mirzakhani, which gives the asymptotic growth of the number of closed geodesics of length at most $L$ in a single mapping class group orbit. Furthermore, we develop a new, combinatorial approach to studying geodesics on surfaces, which should be of independent interest.