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 Hamenstädt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations $\mathcal{ML}(S)$ 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.

arXiv comment48 pages. V2: We added an almost complete classification of homogeneous (locally finite, ergodic, invariant) measures. V4: Final version, to appear in Annales l'Institut Fourier; proof of Proposition 7.4 changed, an appendix added, and minor revisions