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November  2014, 34(11): 4875-4895. doi: 10.3934/dcds.2014.34.4875

A new proof of Franks' lemma for geodesic flows

Daniel Visscher 1,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI, United States

Received  November 2013 Revised  February 2014 Published  May 2014

Given a Riemannian manifold $(M,g)$ and a geodesic $\gamma$, the perpendicular part of the derivative of the geodesic flow $\phi_g^t: SM \rightarrow SM$ along $\gamma$ is a linear symplectic map. The present paper gives a new proof of the following Franks' lemma, originally found in [7] and [6]: this map can be perturbed freely within a neighborhood in $Sp(n)$ by a $C^2$-small perturbation of the metric $g$ that keeps $\gamma$ a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifold. When $\dim M \geq 3$, the original metric must belong to a $C^2$--open and dense subset of metrics.
Keywords: linear Poincaré map, Jacobi fields, Franks' lemma, perturbation., geodesic flow.
Mathematics Subject Classification: 37C10, 53D25, 34D1.
Citation: Daniel Visscher. A new proof of Franks' lemma for geodesic flows. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4875-4895. doi: 10.3934/dcds.2014.34.4875
References:
[1]

H. N. Alishah and J. Lopes Diaz, Realization of tangent perturbations in discrete and continuous time conservative systems,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$, J. Diff. Geom., 61 (2002), 1-49.  Google Scholar

[8]

J-H. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeitschrift, 153 (1977), 237-251. doi: 10.1007/BF01214477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Klingenberg, Lectures on Closed Geodesics, Grundleheren Math. Wiss. 230, Springer-Verlag, New York, 1978.  Google Scholar

[12]

F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342. doi: 10.1007/BF00181572.  Google Scholar

[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.  Google Scholar

[14]

G. Paternain, Geodesic Flows, Progress in Math. Vol. 180, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[15]

T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov, Institut de Mathématiques de Bourgogne, Dijon Preprint 412 (2005). Google Scholar

show all references

References:
[1]

H. N. Alishah and J. Lopes Diaz, Realization of tangent perturbations in discrete and continuous time conservative systems,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$, J. Diff. Geom., 61 (2002), 1-49.  Google Scholar

[8]

J-H. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeitschrift, 153 (1977), 237-251. doi: 10.1007/BF01214477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Klingenberg, Lectures on Closed Geodesics, Grundleheren Math. Wiss. 230, Springer-Verlag, New York, 1978.  Google Scholar

[12]

F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342. doi: 10.1007/BF00181572.  Google Scholar

[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.  Google Scholar

[14]

G. Paternain, Geodesic Flows, Progress in Math. Vol. 180, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[15]

T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov, Institut de Mathématiques de Bourgogne, Dijon Preprint 412 (2005). Google Scholar

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