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April  2006, 14(2): 365-383. doi: 10.3934/dcds.2006.14.365

Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry

Rafael O. Ruggiero 1,

1. 

Departamento de Matemática, Pontificia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brazil

Received  June 2004 Revised  June 2005 Published  November 2005

We extend some previous results concerning the relationship between weak stability properties of the geodesic flow of manifolds without conjugate points and the global geometry of the manifold. We focus on the study of geodesic flows of compact manifolds without conjugate points satisfying either the shadowing property or topological stability, and we prove for three dimensional manifolds that under these assumptions the fundamental groups of certain quasi-convex manifolds have the Preissmann's property. This result generalizes a similar one obtained for manifolds with bounded asymptote.
Keywords: hyperbolic spaces, topological stability, homoclinic orbits., Conjugate points, Preissmann’s property, shadowing property.
Mathematics Subject Classification: 37D40, 37J45, 34D3.
Citation: Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
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