Ergodic Properties of Horocycle Flows for Surfaces of Negative Curvature

Abstract

Extensive work has been done in understanding the ergodic properties of the classical horocycle flow for a compact connected, orientable surface of constant negative curvature ([9], [10], [11], [15], [23]). Such a flow may be viewed as a one-parameter subgroup action of a compact homogeneous space. Many of the results are based on this observation; in particular the Lie group structure and group representations play an important role. The basic goal of our approach is to give completely different proofs (of these results) which are valid in the variable negative curvature case as well and are more dynamic in nature. Apparently, in this case one does not have the Lie group structure and cannot use representation theory. The spirit of our approach (which goes back to Hedlund) is to view horocycle flows as flows-i.e., as continuous flows whose orbits are permuted and expanded by another flow; for horocycle flows, the other flow is the geodesic flow. In all of our proofs, we exploit the interdependence between these two flows. We first prove (in Section 3) that every continuous parametrization of a horocycle flow is topologically mixing; for this, we use the well-known minimality of the horocycle flow. Next (in Section 4) we prove that for every horocycle flow there is a large class of continuous reparametrizations which are measure theoretically mixing, with respect to the unique invariant Borel probability measure (uniqueness was established earlier ([9], [21])). For this, we use the same idea as in proving topological mixing, but instead of minimality exploit unique ergodicity of horocycle flows.* The class of parametrizations, for which we are able to prove measure theoretic mixing, includes the classical horocycle parametrizations, unit speed parametrizations and special uniformly expanding parametrizations (whose existence was proved earlier ([21])). With the latter parametrization the geodesic and horocycle flows satisfy a simple commutation relation; and the proof of An abstract of a preliminary version of this paper was published in AMS Notices, October, 1975 # 728-614. * Ergodicity would suffice, but using only this one does not get the stronger conclusionLemma 4.5.