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A new proof of Franks' lemma for geodesic flows
1. | Department of Mathematics, University of Michigan, Ann Arbor, MI, United States |
References:
[1] |
H. N. Alishah and J. Lopes Diaz, Realization of tangent perturbations in discrete and continuous time conservative systems, preprint, arXiv:1310.1063. |
[2] |
M.-C. Arnaud, The generic symplectic $C^1$-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Ergod. Th. & Dynam. Sys., 22 (2002), 1621-1639.
doi: 10.1017/S0143385702000706. |
[3] |
M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows, J. Diff. Equations, 245 (2008), 3127-3143.
doi: 10.1016/j.jde.2008.02.045. |
[4] |
C. Bonatti, L. Diaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[5] |
C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergod. Th. & Dynam. Sys., 26 (2006), 1307-1337.
doi: 10.1017/S0143385706000253. |
[6] |
G. Contreras, Geodesic flows with positive topological entropy, twist maps and hyperbolicity, Ann. of Math., 172 (2010), 761-808.
doi: 10.4007/annals.2010.172.761. |
[7] |
G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$, J. Diff. Geom., 61 (2002), 1-49. |
[8] |
J-H. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeitschrift, 153 (1977), 237-251.
doi: 10.1007/BF01214477. |
[9] |
J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. A.M.S., 158 (1971), 301-308.
doi: 10.1090/S0002-9947-1971-0283812-3. |
[10] |
V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. I.H. Poicaré, 23 (2006), 641-661.
doi: 10.1016/j.anihpc.2005.06.002. |
[11] |
W. Klingenberg, Lectures on Closed Geodesics, Grundleheren Math. Wiss. 230, Springer-Verlag, New York, 1978. |
[12] |
F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342.
doi: 10.1007/BF00181572. |
[13] |
C. Morales, M. J. Pacifico and E. Pujals, Robust transitive singular sets for $3$-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160 (2004), 375-432.
doi: 10.4007/annals.2004.160.375. |
[14] |
G. Paternain, Geodesic Flows, Progress in Math. Vol. 180, Birkhäuser, 1999.
doi: 10.1007/978-1-4612-1600-1. |
[15] |
T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov, Institut de Mathématiques de Bourgogne, Dijon Preprint 412 (2005). |
show all references
References:
[1] |
H. N. Alishah and J. Lopes Diaz, Realization of tangent perturbations in discrete and continuous time conservative systems, preprint, arXiv:1310.1063. |
[2] |
M.-C. Arnaud, The generic symplectic $C^1$-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Ergod. Th. & Dynam. Sys., 22 (2002), 1621-1639.
doi: 10.1017/S0143385702000706. |
[3] |
M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows, J. Diff. Equations, 245 (2008), 3127-3143.
doi: 10.1016/j.jde.2008.02.045. |
[4] |
C. Bonatti, L. Diaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[5] |
C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergod. Th. & Dynam. Sys., 26 (2006), 1307-1337.
doi: 10.1017/S0143385706000253. |
[6] |
G. Contreras, Geodesic flows with positive topological entropy, twist maps and hyperbolicity, Ann. of Math., 172 (2010), 761-808.
doi: 10.4007/annals.2010.172.761. |
[7] |
G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$, J. Diff. Geom., 61 (2002), 1-49. |
[8] |
J-H. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeitschrift, 153 (1977), 237-251.
doi: 10.1007/BF01214477. |
[9] |
J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. A.M.S., 158 (1971), 301-308.
doi: 10.1090/S0002-9947-1971-0283812-3. |
[10] |
V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. I.H. Poicaré, 23 (2006), 641-661.
doi: 10.1016/j.anihpc.2005.06.002. |
[11] |
W. Klingenberg, Lectures on Closed Geodesics, Grundleheren Math. Wiss. 230, Springer-Verlag, New York, 1978. |
[12] |
F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342.
doi: 10.1007/BF00181572. |
[13] |
C. Morales, M. J. Pacifico and E. Pujals, Robust transitive singular sets for $3$-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160 (2004), 375-432.
doi: 10.4007/annals.2004.160.375. |
[14] |
G. Paternain, Geodesic Flows, Progress in Math. Vol. 180, Birkhäuser, 1999.
doi: 10.1007/978-1-4612-1600-1. |
[15] |
T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov, Institut de Mathématiques de Bourgogne, Dijon Preprint 412 (2005). |
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