Geodesics Currents and Counting Problems

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\bar{\mathrm{\Sigma }}$, and for every compactly supported filling current $\alpha$, we compute as $L\to \mathrm{\infty }$, the number of mapping classes $\varphi$ so that $f(\varphi (\alpha ))\le L$. 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.