Ergodic Theory and Dynamical Systems

Research Article

Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature

J. Feldmana1 and D. Ornsteina2

a1 Department of Mathematics, University of California, Berkeley, CA 94720, USA

a2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Abstract

Let g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: gg′. Furthermore, any homeomorphic conjugacy gg′; must be a C1 diffeomorphism.

(Received November 05 1985)

(Revised April 18 1986)