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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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). Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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. Google Scholar

[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. Google Scholar

[41]

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

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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). Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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. Google Scholar

[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. Google Scholar

[41]

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

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