2016, 10: 379-411. doi: 10.3934/jmd.2016.10.379

Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence

1. 

Université Nice Sophia Antipolis, CNRS, LJAD, UMR, 7351, 06100 Nice, France

2. 

Université Nice Sophia Antipolis, Institut Universitaire de France, CNRS, LJAD, UMR 7351, 06100 Nice, France

3. 

PUC-Rio, Departamento de Matemática, Rua Marqués de São Vicente 225, Gávea, 22450-150, Rio de Janeiro, Brazil

Received  April 2014 Revised  May 2016 Published  September 2016

We prove a uniform Franks' lemma at second order for geodesic flows on a compact Riemannian manifold and apply the result in persistence theory. Our approach, which relies on techniques from geometric control theory, allows us to show that Mañé (i.e., conformal) perturbations of the metric are sufficient to achieve the result.
Citation: 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
References:
[1]

A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004). doi: 10.1007/978-3-662-06404-7.

[2]

A. Agrachev and P. Lee, Optimal transportation under nonholonomic constraints,, Trans. Amer. Math. Soc., 361 (2009), 6019. doi: 10.1090/S0002-9947-09-04813-2.

[3]

D. V. Anosov, Generic properties of closed geodesics,, Izv. Akad. Nauk. SSSR Ser. Mat., 46 (1982), 675.

[4]

J. Bocknak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Ergebnisse des Mathematik und ihrer Grenzgebiete (3), (1998). doi: 10.1007/978-3-662-03718-8.

[5]

C. M. Carballo and J. A. G. Miranda, Jets of closed orbits of Mañé's generic Hamiltonian flows,, Bull. Braz. Math. Soc. (N. S.), 44 (2013), 219. doi: 10.1007/s00574-013-0010-1.

[6]

G. Contreras, Partially hyperbolic geodesic flows are Anosov,, C. R. Math. Acad. Sci. Paris, 334 (2002), 585. doi: 10.1016/S1631-073X(02)02196-9.

[7]

G. Contreras, Geodesic flows with positive topological entropy, twist maps and hyperbolicity,, Ann. of Math. (2), 172 (2010), 761. doi: 10.4007/annals.2010.172.761.

[8]

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points,, Ergodic Theory Dynam. Systems, 19 (1999), 901. doi: 10.1017/S014338579913387X.

[9]

J.-M. Coron, Control and nonlinearity,, Mathematical Surveys and Monographs, (2007).

[10]

M. Coste, Ensembles semi-algébriques,, in Real Algebraic Geometry and Quadratic Forms (Rennes, (1981), 109.

[11]

M. P. do Carmo, Riemannian Geometry,, Birkhäuser Boston, (1992).

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry-Methods and Applications, Part I. The Geometry of Surfaces, Transformation Groups, and Fields,, Second edition, (1992).

[13]

A. Figalli and L. Rifford, Closing Aubry sets I,, Comm. Pure Appl. Math., 68 (2015), 210. doi: 10.1002/cpa.21511.

[14]

A. Figalli and L. Rifford, Closing Aubry sets II,, Comm. Pure Appl. Math., 68 (2015), 345. doi: 10.1002/cpa.21512.

[15]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. doi: 10.1090/S0002-9947-1971-0283812-3.

[16]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).

[17]

V. Jurjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).

[18]

W. Klingenberg and F. Takens, Generic properties of geodesic flows,, Math. Ann., 197 (1972), 323. doi: 10.1007/BF01428204.

[19]

R. Kulkarni, Curvature structures and conformal transformations,, J. Diff. Geom., 4 (1970), 425.

[20]

A. Lazrag, Control Theory and Dynamical Systems,, Thesis, (2014).

[21]

A. Lazrag, A geometric control proof of linear Franks' lemma for geodesic flows,, preprint, (2014).

[22]

R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021.

[23]

R. Mañé, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161.

[24]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems,, Nonlinearity, 5 (1992), 623. doi: 10.1088/0951-7715/5/3/001.

[25]

R. Mañé, Global Variational Methods in Conservative Dynamics,, IMPA, (1993).

[26]

J. Moser, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff,, in Geometry and Topology (Proc. III Latin Amer. School of Math., (1976), 464.

[27]

S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems,, Amer. J. Math., 99 (1977), 1061. doi: 10.2307/2374000.

[28]

E. Oliveira, Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem,, Discrete Contin. Dyn. Syst., 21 (2008), 551. doi: 10.3934/dcds.2008.21.551.

[29]

M. Paternain, Expansive geodesic flows on surfaces,, Ergodic Theory Dynam. Systems, 13 (1993), 153. doi: 10.1017/S0143385700007264.

[30]

C. C. Pugh, The closing lemma,, Amer. J. Math., 89 (1967), 956. doi: 10.2307/2373413.

[31]

C. C. Pugh, An improved closing lemma and a general density theorem,, Amer. J. Math., 89 (1967), 1010. doi: 10.2307/2373414.

[32]

C. Pugh and C. Robinson, The $C^{1}$ closing lemma, including Hamiltonians,, Ergodic Theory Dynam. Systems, 3 (1983), 261. doi: 10.1017/S0143385700001978.

[33]

L. Rifford, Sub-Riemannian Geometry and Optimal Transport,, Springer Briefs in Mathematics, (2014). doi: 10.1007/978-3-319-04804-8.

[34]

L. Rifford and R. Ruggiero, Generic properties of closed orbits of Hamiltonian flows from Mañé's viewpoint,, Int. Math. Res. Not. IMRN, 22 (2012), 5246.

[35]

C. Robinson, Generic properties of conservative systems I and II,, Amer. J. Math., 92 (1970), 562. doi: 10.2307/2373361.

[36]

R. Ruggiero, Persistently expansive geodesic flows,, Comm. Math. Phys., 140 (1991), 203. doi: 10.1007/BF02099298.

[37]

R. Ruggiero, On the creation of conjugate points,, Math. Z., 208 (1991), 41. doi: 10.1007/BF02571508.

[38]

T. Sakai, Riemannian Geometry,, Translated from the 1992 Japanese original by the author, (1992).

[39]

C. Villani, Optimal transport. Old and new,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9.

[40]

D. Visscher, A new proof of Franks' lemma for geodesic flows,, Discrete Contin. Dyn. Syst., 34 (2014), 4875. doi: 10.3934/dcds.2014.34.4875.

[41]

T. Vivier, Robustly Transitive 3-Dimensional Regular Energy Surfaces are Anosov,, Ph.D. Thesis, (2005).

show all references

References:
[1]

A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004). doi: 10.1007/978-3-662-06404-7.

[2]

A. Agrachev and P. Lee, Optimal transportation under nonholonomic constraints,, Trans. Amer. Math. Soc., 361 (2009), 6019. doi: 10.1090/S0002-9947-09-04813-2.

[3]

D. V. Anosov, Generic properties of closed geodesics,, Izv. Akad. Nauk. SSSR Ser. Mat., 46 (1982), 675.

[4]

J. Bocknak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Ergebnisse des Mathematik und ihrer Grenzgebiete (3), (1998). doi: 10.1007/978-3-662-03718-8.

[5]

C. M. Carballo and J. A. G. Miranda, Jets of closed orbits of Mañé's generic Hamiltonian flows,, Bull. Braz. Math. Soc. (N. S.), 44 (2013), 219. doi: 10.1007/s00574-013-0010-1.

[6]

G. Contreras, Partially hyperbolic geodesic flows are Anosov,, C. R. Math. Acad. Sci. Paris, 334 (2002), 585. doi: 10.1016/S1631-073X(02)02196-9.

[7]

G. Contreras, Geodesic flows with positive topological entropy, twist maps and hyperbolicity,, Ann. of Math. (2), 172 (2010), 761. doi: 10.4007/annals.2010.172.761.

[8]

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points,, Ergodic Theory Dynam. Systems, 19 (1999), 901. doi: 10.1017/S014338579913387X.

[9]

J.-M. Coron, Control and nonlinearity,, Mathematical Surveys and Monographs, (2007).

[10]

M. Coste, Ensembles semi-algébriques,, in Real Algebraic Geometry and Quadratic Forms (Rennes, (1981), 109.

[11]

M. P. do Carmo, Riemannian Geometry,, Birkhäuser Boston, (1992).

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry-Methods and Applications, Part I. The Geometry of Surfaces, Transformation Groups, and Fields,, Second edition, (1992).

[13]

A. Figalli and L. Rifford, Closing Aubry sets I,, Comm. Pure Appl. Math., 68 (2015), 210. doi: 10.1002/cpa.21511.

[14]

A. Figalli and L. Rifford, Closing Aubry sets II,, Comm. Pure Appl. Math., 68 (2015), 345. doi: 10.1002/cpa.21512.

[15]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. doi: 10.1090/S0002-9947-1971-0283812-3.

[16]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).

[17]

V. Jurjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).

[18]

W. Klingenberg and F. Takens, Generic properties of geodesic flows,, Math. Ann., 197 (1972), 323. doi: 10.1007/BF01428204.

[19]

R. Kulkarni, Curvature structures and conformal transformations,, J. Diff. Geom., 4 (1970), 425.

[20]

A. Lazrag, Control Theory and Dynamical Systems,, Thesis, (2014).

[21]

A. Lazrag, A geometric control proof of linear Franks' lemma for geodesic flows,, preprint, (2014).

[22]

R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021.

[23]

R. Mañé, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161.

[24]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems,, Nonlinearity, 5 (1992), 623. doi: 10.1088/0951-7715/5/3/001.

[25]

R. Mañé, Global Variational Methods in Conservative Dynamics,, IMPA, (1993).

[26]

J. Moser, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff,, in Geometry and Topology (Proc. III Latin Amer. School of Math., (1976), 464.

[27]

S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems,, Amer. J. Math., 99 (1977), 1061. doi: 10.2307/2374000.

[28]

E. Oliveira, Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem,, Discrete Contin. Dyn. Syst., 21 (2008), 551. doi: 10.3934/dcds.2008.21.551.

[29]

M. Paternain, Expansive geodesic flows on surfaces,, Ergodic Theory Dynam. Systems, 13 (1993), 153. doi: 10.1017/S0143385700007264.

[30]

C. C. Pugh, The closing lemma,, Amer. J. Math., 89 (1967), 956. doi: 10.2307/2373413.

[31]

C. C. Pugh, An improved closing lemma and a general density theorem,, Amer. J. Math., 89 (1967), 1010. doi: 10.2307/2373414.

[32]

C. Pugh and C. Robinson, The $C^{1}$ closing lemma, including Hamiltonians,, Ergodic Theory Dynam. Systems, 3 (1983), 261. doi: 10.1017/S0143385700001978.

[33]

L. Rifford, Sub-Riemannian Geometry and Optimal Transport,, Springer Briefs in Mathematics, (2014). doi: 10.1007/978-3-319-04804-8.

[34]

L. Rifford and R. Ruggiero, Generic properties of closed orbits of Hamiltonian flows from Mañé's viewpoint,, Int. Math. Res. Not. IMRN, 22 (2012), 5246.

[35]

C. Robinson, Generic properties of conservative systems I and II,, Amer. J. Math., 92 (1970), 562. doi: 10.2307/2373361.

[36]

R. Ruggiero, Persistently expansive geodesic flows,, Comm. Math. Phys., 140 (1991), 203. doi: 10.1007/BF02099298.

[37]

R. Ruggiero, On the creation of conjugate points,, Math. Z., 208 (1991), 41. doi: 10.1007/BF02571508.

[38]

T. Sakai, Riemannian Geometry,, Translated from the 1992 Japanese original by the author, (1992).

[39]

C. Villani, Optimal transport. Old and new,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9.

[40]

D. Visscher, A new proof of Franks' lemma for geodesic flows,, Discrete Contin. Dyn. Syst., 34 (2014), 4875. doi: 10.3934/dcds.2014.34.4875.

[41]

T. Vivier, Robustly Transitive 3-Dimensional Regular Energy Surfaces are Anosov,, Ph.D. Thesis, (2005).

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