Abstract
We study the C 2-structural stability conjecture from Mane's viewpoint for geodesics flows of compact manifolds without conjugate points. The structural stability conjecture is an open problem in the category of geodesic flows because the C 1 closing lemma is not known in this context. Without the C 1 closing lemma, we combine the geometry of manifolds without conjugate points and a recent version of Franks' Lemma from Mane's viewpoint to prove the conjecture for compact surfaces, for compact three dimensional manifolds with quasi-convex universal coverings where geodesic rays diverge, and for n-dimensional, generalized rank one manifolds.
Highlights
The motivations for the main results in this article come from two sources
The question of how far we can go in proving the C1 stability conjecture for geodesic flows without a C1 closing lemma is an interesting, appealing problem in Riemannian geometry and dynamical systems
Some important results about the topological dynamics of the geodesic flow of compact manifolds without conjugate points and hyperbolic global geometry are known for nonpositive curvature manifolds
Summary
The motivations for the main results in this article come from two sources. First of all, the challenging problem of the C1 closing lemma for geodesic flows that remains an open, very difficult problem. Some important results about the topological dynamics of the geodesic flow of compact manifolds without conjugate points and hyperbolic global geometry are known for nonpositive curvature manifolds. The density of the set of periodic orbits, another important feature of topological dynamics, is known for visibility manifolds with nonpositive sectional curvatures (see for instance [Bal95]). Notice that by one of the main results of [LRR16], the C2 structural stability of the geodesic flow of a compact manifold from Mañé’s viewpoint implies the hyperbolicity of the closure of the set of periodic orbits. The goal of the paper is to deal with the stability conjecture of geodesic flows of compact manifolds without conjugate points, a geometric condition that is much weaker than nonpositive curvature but still ensures many important properties for the topological dynamics of the geodesic flow.
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