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

A new proof of Franks' lemma for geodesic flows

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.
Citation: Daniel Visscher. A new proof of Franks' lemma for geodesic flows. Discrete & Continuous Dynamical Systems - A, 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.  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.  doi: 10.1016/j.jde.2008.02.045.  Google Scholar

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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.  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.  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.  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.   Google Scholar

[8]

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

[9]

J. Franks, Necessary conditions for the stability of diffeomorphisms,, Trans. A.M.S., 158 (1971), 301.  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.  doi: 10.1016/j.anihpc.2005.06.002.  Google Scholar

[11]

W. Klingenberg, Lectures on Closed Geodesics,, Grundleheren Math. Wiss. 230, (1978).   Google Scholar

[12]

F. Klok, Generic singularities of the exponential map on Riemannian manifolds,, Geom. Dedicata, 14 (1983), 317.  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.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[14]

G. Paternain, Geodesic Flows,, Progress in Math. Vol. 180, (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, (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.  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.  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.  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.  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.  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.   Google Scholar

[8]

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

[9]

J. Franks, Necessary conditions for the stability of diffeomorphisms,, Trans. A.M.S., 158 (1971), 301.  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.  doi: 10.1016/j.anihpc.2005.06.002.  Google Scholar

[11]

W. Klingenberg, Lectures on Closed Geodesics,, Grundleheren Math. Wiss. 230, (1978).   Google Scholar

[12]

F. Klok, Generic singularities of the exponential map on Riemannian manifolds,, Geom. Dedicata, 14 (1983), 317.  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.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[14]

G. Paternain, Geodesic Flows,, Progress in Math. Vol. 180, (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, (2005).   Google Scholar

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