Some Dynamical Properties on Manifolds with no Conjugate Points

TL;DR

This paper investigates the dynamics of geodesic flows on non-compact Riemannian manifolds without conjugate points, establishing key properties like the Anosov Closing Lemma and transitivity under certain conditions.

Contribution

It proves fundamental dynamical properties for geodesic flows on manifolds with no conjugate points, extending understanding beyond compact cases.

Findings

Proved the Anosov Closing Lemma for these manifolds

Established local product structure of geodesic flows

Demonstrated transitivity of flows under bounded asymptote and visibility

Abstract

In this article, we study the dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. We prove the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows on $\Omega_1$ under the conditions of bounded asymptote and uniform visibility. As an application, we further discuss about some generic properties of the set of invariant probability measures