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The accessibility property of expansive geodesic flows without conjugate points

Published online by Cambridge University Press:  01 February 2008

RAFAEL OSWALDO RUGGIERO
RAFAEL OSWALDO RUGGIERO*
Affiliation:
Pontificia Universidade Católica do Rio de Janeiro, PUC-Rio, Dep. de Matemática, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brazil (email: rorr@mat.puc-rio.br)
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Abstract

Let (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two points θ1, θ2 in the unit tangent bundle, there exists a continuous path joining θ1, θ2 formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 229 - 244
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Non-positive Curvature. Birkhäuser, Boston, 1985.CrossRefGoogle Scholar
[2]Brin, M.. Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9 (1975), 9–19.CrossRefGoogle Scholar
[3]Brin, M.. The topology of group extensions of C systems. Mat. Zametki 18 (1975), 453–465.Google Scholar
[4]Brin, M. and Gromov, M.. On the ergodicity of frame flows. Invent. Math. 60 (1980), 1–7.CrossRefGoogle Scholar
[5]Brin, M. and Karcher, H.. Frame flows on manifolds with pinched negative curvature. Compositio Math. 52 (1984), 275–297.Google Scholar
[6]Brin, M. and Pesin, Ya.. Flows of frames on manifolds of negative curvature. Uspekhi Mat. Nauk. 28 (1973), 169–170.Google Scholar
[7]Burns, K., Pugh, C. and Wilkinson, A.. Stable ergodicity and Anosov flows. Topology 39 (2000), 149–159.CrossRefGoogle Scholar
[8]Burns, K., Pugh, C., Shub, M. and Wilkinson, A.. Recent results about stable ergodicity. Smooth Ergodic Theory and Its Applications (Proc. AMS Summer Research Institute, Seattle, WA, 26 July–13 August 1999) (Proceedings of Symposia in Pure Mathematics, 69). Eds. A. Katok et al. American Mathematical Society, Providence, RI, 2001, pp. 327–366.CrossRefGoogle Scholar
[9]Burns, K. and Wilkinson, A.. Stable ergodicity of skew products. Ann. Sci. École Norm. Sup. 32 (1999), 859–889.CrossRefGoogle Scholar
[10]Busemann, H.. The Geometry of Geodesics. Academic Press, New York, 1955.Google Scholar
[11]Croke, C. and Schroeder, V.. The fundamental group of manifolds without conjugate points. Comment. Math. Helv. 61 (1986), 161–175.CrossRefGoogle Scholar
[12]Dolgopyat, D. and Wilkinson, A.. Stability accessibility in C 1 dense. Geometric Methods in Dynamics (II). Volume in Honor of J. Palis. Eds. W. de Melo, M. Viana and J.-C. Yoccoz. Astérisque 287 (2003), 33–60. American Mathematical Society, Providence, RI.Google Scholar
[13]Katok, A. and Kononenko, A.. Cocycles’ stability for partially hyperbolic systems. Math. Res. Lett. 3 (1996), 191–210.CrossRefGoogle Scholar
[14]Eberlein, P.. Geodesic flows in certain manifolds with no conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
[15]Ghys, E. and De La Harpe, P.. Sur les Groupes Hyperboliques d’après M. Gromov (Progress in Mathematics, 83). Birkhäuser, Boston, 1990.CrossRefGoogle Scholar
[16]Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140 (1994), 295–329.CrossRefGoogle Scholar
[17]Gromov, M.. Hyperbolic Groups (Essays in Group Theory, 8). Ed. S. M. Gersten. Springer, Berlin, pp. 75–264.CrossRefGoogle Scholar
[18]Morse, J. M.. A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Amer. Math. Soc. 26 (1924), 25–60.CrossRefGoogle Scholar
[19]Niticaˇ, V. and Török, A.. An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one. Topology 40 (2001), 259–278.CrossRefGoogle Scholar
[20]Paternain, M.. Expansive geodesic flows on surfaces. Ergod. Th. & Dynam. Sys. 13(1) (1993), 153–165.CrossRefGoogle Scholar
[21]Pesin, Ja. B.. Geodesic flows on closed Riemannian manifolds without focal points. Math. USSR Izvestija 11(6) (1977), 1195–1228.CrossRefGoogle Scholar
[22]Pugh, C. and Shub, M.. Stable ergodicity and julienne quasiconformality. J. Eur. Math. Soc. 1(2) (2000), 1–52.CrossRefGoogle Scholar
[23]Ruggiero, R.. Expansive dynamics and hyperbolic geometry. Bol. Soc. Brasil. Mat. 25 (1994), 139–172.CrossRefGoogle Scholar
[24]Ruggiero, R.. Expansive geodesic flows in manifolds without conjugate points. Ergod. Th. & Dynam. Sys. 17 (1997), 211–225.CrossRefGoogle Scholar
[25]Ruggiero, R.. On the divergence of geodesic rays in manifolds without conjugate points, dynamics of the geodesic flow and global geometry. Geometric Methods in Dynamics (II). Volume in Honor of Jacob Palis. Eds. W. de Melo, M. Viana and J.-C. Yoccoz. Astérisque 287 (2003), 231–250. American Mathematical Society, Providence, RI.Google Scholar
[26]Shub, M. and Wilkinson, A.. Stably ergodic approximation: two examples. Ergod. Th. & Dynam. Sys. 20(3) (2000), 875–893.CrossRefGoogle Scholar