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.
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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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Bessa, M. A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows. Math Phys Anal Geom 21, 14 (2018). https://doi.org/10.1007/s11040-018-9271-7
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DOI: https://doi.org/10.1007/s11040-018-9271-7