Horocycle flows on certain surfaces without conjugate points

Abstract:

We study the topological but not ergodic properties of the horocycle flow $\{ {h_t}\}$ 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 $K \leqslant c < 0$ 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 ${\Omega _h} \subseteq SM$ for $\{ {h_t}\}$ is classified into four types depending upon the fundamental group of M. The extremes that ${\Omega _h}$ be a minimal set for $\{ {h_t}\}$ and that ${\Omega _h}$ admit periodic orbits are related to the existence or nonexistence of compact “totally convex” sets in M. Periodic points are dense in ${\Omega _h}$ if they exist at all. The only compact minimal sets in ${\Omega _h}$ are periodic orbits if M is noncompact The flow $\{ {h_t}\}$ is minimal in SM if and only if M is compact. In general $\{ {h_t}\}$ is topologically transitive in ${\Omega _h}$ and the vectors in ${\Omega _h}$ with dense orbits are classified. If the fundamental group of M is finitely generated and ${\Omega _h} = SM$ then $\{ {h_t}\}$ is topologically mixing in SM.