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

  • J. Feldman (a1) and D. Ornstein (a2)
  • DOI:
  • Published online: 01 September 2008

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.

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[B]M. Berger (A. L. Besse ). Manifolds All of Whose Geodesies Are Closed. Springer-Verlag, 1978.

[D]S. G. Dani . Dynamics of the horospherical flow. Bull. Amer. Math. Soc. 3 (1980), 10371039.

[G-K]V. Guillemin & D. Kazhdan . Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301302.

[M1]B. Marcus . Unique ergodicity of the horocycle flow: variable negative curvature case. Israel J. Math. 21 (1975), 133144.

[M2]B. Marcus . Topological conjugacy of horocycle flows. Amer. J. Math. (1983), 623632.

[Ma]G. A. Margulis . Certain measures associated with U-flows on compact manifolds. Funct. Anal. Appl. 4 (1970), 5567.

[R1]M. Ratner . Rigidity of horocycle flows. Ann. Math. 115 (1982), 587614.

[R3]M. Ratner . Horocycle flows; joinings and rigidity of products. Ann. Math. 118 (1983), 277313.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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