Counting closed geodesics on a hyperbolic surface

We discuss the asymptotic behavior of the number of closed geodesics of a given combinatorial type on a hyperbolic surface (the closed curve can be disconnected or have self intersections).

We will see that this question is closely related to the distribution of lengths of closed curves on a random pants decomposition of a hyperbolic surface. We use the ergodicity of the earthquake flow to study this problem.

More generally, we consider the action of mapping class group on the space of geodesic currents, and discuss the growth of the orbit of an arbitrary point. We discuss both topological and geometric versions of this problem.

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