# American Institute of Mathematical Sciences

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, (). [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. [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. [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. [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. [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. [7] G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$,, J. Diff. Geom., 61 (2002), 1. [8] J-H. Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Zeitschrift, 153 (1977), 237. doi: 10.1007/BF01214477. [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. [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. [11] W. Klingenberg, Lectures on Closed Geodesics,, Grundleheren Math. Wiss. 230, (1978). [12] F. Klok, Generic singularities of the exponential map on Riemannian manifolds,, Geom. Dedicata, 14 (1983), 317. 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. doi: 10.4007/annals.2004.160.375. [14] G. Paternain, Geodesic Flows,, Progress in Math. Vol. 180, (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, (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, (). [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. [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. [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. [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. [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. [7] G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$,, J. Diff. Geom., 61 (2002), 1. [8] J-H. Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Zeitschrift, 153 (1977), 237. doi: 10.1007/BF01214477. [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. [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. [11] W. Klingenberg, Lectures on Closed Geodesics,, Grundleheren Math. Wiss. 230, (1978). [12] F. Klok, Generic singularities of the exponential map on Riemannian manifolds,, Geom. Dedicata, 14 (1983), 317. 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. doi: 10.4007/annals.2004.160.375. [14] G. Paternain, Geodesic Flows,, Progress in Math. Vol. 180, (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, (2005).
 [1] Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473 [2] Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239 [3] Ayadi Lazrag, Ludovic Rifford, Rafael O. Ruggiero. Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence. Journal of Modern Dynamics, 2016, 10: 379-411. doi: 10.3934/jmd.2016.10.379 [4] Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089 [5] Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669 [6] Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581 [7] Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 [8] Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453 [9] D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495 [10] Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715 [11] José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213 [12] Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173 [13] Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104. [14] Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933 [15] Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643 [16] Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148 [17] Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365 [18] Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 [19] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [20] Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015

2017 Impact Factor: 1.179