Abstract
We prove thatC 1-persistently expansive geodesic flows of compact, boundaryless Riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set. In the case of compact surfaces we deduce that the geodesic flow isC 1-persistently expansive if and only if it is an Anosov flow.
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References
Anosov, D.: Geodesic flow on closed Riemannian manifolds of negative curvature. Trudy Math. Inst. Steklov90, (1967)
Eberlein, P.: When is a geodesic flow of Anosov type I. J. Diff. Geom.8, 437–463 (1973)
Klingenberg, W.: Lectures on closed geodesics. Berlin, Heidelberg, New York: Springer 1978
Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann.197, 323–334 (1972)
Lewowicz, J.: Expansive homeomorphisms of surface. Preprint.
Mañe, R.: Quasi-Anosov diffeomorphisms. Lecture Notes in Math. Vol. 468, pp. 27–29. Berlin, Heidelberg, New York: Springer 1974
Mañe, R.: An ergodic closing lemma. Ann. Math.116, 503–540 (1982)
Moser, J.: Proof of a generalised form of a fixed point theorem due to G. D. Birkhoff. Lecture Notes in Math. Vol. 597, pp. 464–549. Berlin, Heidelberg, New York: Springer 1977
Pugh, C.: An improved closing lemma and a general density theorem. Am. J. Math.89, 1010–1021 (1967)
Pugh, C., Shub, M.: The Ω-stability theorem for flows. Invent. Math.11, 150–158 (1970)
Paternain, M.: Expansive flows and the fundamental group. Tese de Doutorado. IMPA (1990)
Smale, S.: The Ω-stability theorem. Proc. Symp. Pure Math.14 (1970). Am. Math. Soc., pp. 289–298
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Communicated by J.-P. Eckmann
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Ruggiero, R.O. Persistently expansive geodesic flows. Commun.Math. Phys. 140, 203–215 (1991). https://doi.org/10.1007/BF02099298
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DOI: https://doi.org/10.1007/BF02099298