Geodesic Currents and Counting Problems

For every positive, continuous and homogeneous function f on the space of currents on a compact surface Σ ¯ , and for every compactly supported filling current α , we compute as L → ∞ , the number of mapping classes ϕ so that f ( ϕ ( α ) ) ≤ L . As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmüller space equipped with the Thurston metric.