ERGODIC INVARIANT MEASURES ON THE SPACE OF GEODESIC CURRENTS

Let S be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss–Mirzakhani’s and Hamenstädt’s classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations ML(S) to the space of geodesic currents C(S), and we discuss the homogeneous case. Moreover, we extend Lindenstrauss–Mirzakhani’s classification of orbit closures to 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.