Horocycle flows on certain surfaces without conjugate points

Author:
Patrick Eberlein

Journal:
Trans. Amer. Math. Soc. **233** (1977), 1-36

MSC:
Primary 58F15; Secondary 34C35, 53C20

DOI:
https://doi.org/10.1090/S0002-9947-1977-0516501-3

MathSciNet review:
0516501

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Abstract: We study the topological but not ergodic properties of the horocycle flow in the unit tangent bundle *SM* of a complete two dimensional Riemannian manifold *M* without conjugate points that satisfies the ``uniform Visibility'' axiom. This axiom is implied by the curvature condition but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set for is classified into four types depending upon the fundamental group of *M*. The extremes that be a minimal set for and that admit periodic orbits are related to the existence or nonexistence of compact ``totally convex'' sets in *M*. Periodic points are dense in if they exist at all. The only compact minimal sets in are periodic orbits if *M* is noncompact The flow is minimal in *SM* if and only if *M* is compact. In general is topologically transitive in and the vectors in with dense orbits are classified. If the fundamental group of *M* is finitely generated and then is topologically mixing in *SM*.

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0516501-3

Article copyright:
© Copyright 1977
American Mathematical Society