Measures on the space of geodesic currents
Let S be a compact (hyperbolic) surface. The space of geodesic currents C(S) can be viewed as the completion of the set of all weighted closed geodesics on S, the same way as the space of measured laminations ML(S) is the completion of all weighted simple closed geodesics. In particular, ML(S) can naturally be identified with a subset of C(S). In this talk we look at mapping class group invariant ergodic measures on C(S) and extend the Lindenstrauss-Mirzakhani and Hamenstädt classification of such measures on ML(S) to the space of currents. Essentially, any such measure not supported on ML(S) must be atomic. This is joint work with Gabrielle Mondello.