A new proof of Franks' lemma for geodesic flows

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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

Abstract
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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 and 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, 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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