Ergodicity of the Horocycle Flow

We prove that ergodicity of the horocycle ow on a surface of constant negative curvature is equivalent to ergodicity of the associated boundary action. As a corollary we obtain ergodicity of the horocycle ow on several large classes of covering surfaces. There are two natural \geometric ows" on (the unitary tangent bundle of) an arbitrary surface of constant negative curvature: the geodesic and the horocycle ows. Both of them preserve the Liouville measure. The study of the ergodic properties of these ows for compact surfaces goes back to the works of Hopf and Hedlund in the 30's. Later ergodicity of the geodesic ow was profoundly studied in the non-compact case as well. The main result here is the Hopf{Tsuji{Sullivan theorem, see Sullivan 13 , according to which the geodesic ow on a surface M is ergodic with respect to the Liouville measure ii the Brownian motion on M is recurrent. In particular, it implies that for regular Z d-covers of compact surfaces the geodesic ow is ergodic ii d 2. However, until very recently little was known about ergodicity of the horocycle ow in the non-compact case. In 1995 Starkov 12 conjectured that it is equivalent to ergodicity of the boundary action of the associated Fuchsian group. Inspired by this conjecture, Babillot and Ledrappier 4 proved that the horocycle ow is ergodic on all Z d-covers of compact surfaces (independently of the degree of the cover, in contrast with the geodesic ow case). In the present paper we prove Starkov's conjecture (Corollary 5). In fact, we give a complete description of the ergodic components of the horocycle ow in terms of the ergodic decomposition of the associated boundary action (Theorem 4). Since ergodicity of the boundary action is equivalent to absence of non-constant bounded harmonic functions on the surface, ergodicity of the horo-cycle ow admits a characterization in terms of the stochastic properties of the Brownian motion (Theorem 16) which complements the analogous result for the geodesic ow. As a corollary we obtain several criteria for ergodic-ity of the horocycle ow on covering surfaces (Theorem 18). In particular, the horocycle ow is ergodic on polycyclic covers of compact surfaces and on nilpotent covers of surfaces with ergodic geodesic ow (with recurrent Brownian motion).