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A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

  • Mário BessaEmail author
Article

Abstract

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if the closure of the set of periodic orbits of \({\varphi ^{t}_{g}}\) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C2-residual \(\mathscr{R}\) such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if \({\varphi ^{t}_{g}}\) is an Anosov flow.

Keywords

Expansiveness Residual sets Anosov Geodesic flows 

Mathematics Subject Classification (2010)

37C20 37D40 53D25 

Notes

Acknowledgements

The author was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. The author would like to thank the reviewers for their careful reading of the manuscript and the helpful comments, CMUP for providing the necessary conditions in which this work was developed, and to Maria Joana Torres for conversations about this subject.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

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