Ergodic invariant measures on the space of geodesic currents

Abstract

Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenst\"adt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations $\mathcal{M}\mathcal{L}(S)$ to the space of geodesic currents $\mathcal{C}(S)$, and we discuss the homogeneous case. Moreover, we extend Lindenstrauss-Mirzakhani's classification of orbit closures to $\mathcal{C}(S)$. Our argument relies on their results and on the decomposition of a current into a sum of three currents with isotopically disjoint supports: a measured lamination without closed leaves, a simple multi-curve and a current that binds its hull.