American Institute of Mathematical Sciences

Advanced
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 - 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

[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

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

[1]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[2]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[3]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[4]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[5]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[6]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141
[7]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[8]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[9]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[10]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[11]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[12]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[13]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[14]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[15]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[16]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[17]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[18]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[19]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[20]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences