Article Contents
Article Contents

# A new proof of Franks' lemma for geodesic flows

• 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.
Mathematics Subject Classification: 37C10, 53D25, 34D10.

 Citation:

•  [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).