Geodesics Currents and Counting Problems

Abstract: For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overline{\Sigma}$, and for every compactly supported filling current $\alpha$, we compute as $L \to \infty$, the number of mapping classes $\phi$ so that $f(\phi(\alpha))\leq L$. As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichm\"uller space equipped with the Thurston metric.