December  2014, 34(12): 5359-5374. doi: 10.3934/dcds.2014.34.5359

Realization of tangent perturbations in discrete and continuous time conservative systems

1. 

CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal

2. 

Departamento de Matemática ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal

Received  October 2013 Revised  March 2014 Published  June 2014

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.
Citation: Hassan Najafi Alishah, João Lopes Dias. Realization of tangent perturbations in discrete and continuous time conservative systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5359-5374. doi: 10.3934/dcds.2014.34.5359
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).   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. Lopes Dias, Generic dynamics of 4-dimensional $C^2$ Hamiltonian systems,, Comm. Math. Phys., 281 (2008), 597.  doi: 10.1007/s00220-008-0500-y.  Google Scholar

[4]

M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows,, J. Differential Equations, 245 (2008), 3127.  doi: 10.1016/j.jde.2008.02.045.  Google Scholar

[5]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.  doi: 10.1007/s002200050412.  Google Scholar

[6]

C. Bonatti, L. J. Díaz and E. R. 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

[7]

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

[8]

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

[9]

G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$,, J. Diff. Geom., 61 (2002), 1.   Google Scholar

[10]

M. Crainic and R. Loja Fernandes, Integrability of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[11]

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

[12]

B. Hernández-Bermejo, New solutions of the Jacobi equations for three-dimensional Poisson structures,, J. Math. Phys., 42 (2001), 4984.  doi: 10.1063/1.1402174.  Google Scholar

[13]

V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms,, Ann. I. H. Poincaré - AN, 23 (2006), 641.  doi: 10.1016/j.anihpc.2005.06.002.  Google Scholar

[14]

C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures,, Springer, (2013).  doi: 10.1007/978-3-642-31090-4.  Google Scholar

[15]

J. M. Lee, Introduction to Smooth Manifolds,, $2^{nd}$ edition, (2013).   Google Scholar

[16]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Reidel Publishing Co., (1987).  doi: 10.1007/978-94-009-3807-6.  Google Scholar

[17]

C. Martin, The Poisson structure of the mean-field equations in the $\Phi^4$ theory,, Ann. Physics, 271 (1999), 294.  doi: 10.1006/aphy.1998.5875.  Google Scholar

[18]

C. Morales, M. J. Pacífico and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers,, Ann. Math., 160 (2004), 375.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[19]

P. J. Morrison, Hamiltonian description of the ideal fluid,, Rev. Modern Phys., 70 (1998), 467.  doi: 10.1103/RevModPhys.70.467.  Google Scholar

[20]

J. Moser, On the volume elements on a manifold,, Trans. Amer. Math. Soc., 120 (1965), 286.  doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[21]

P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[22]

G. Picard and T. W. Johnston, Instability cascades, Lotka-Volterra population equations, and Hamiltonian chaos,, Phys. Rev. Lett., 48 (1982), 1610.  doi: 10.1103/PhysRevLett.48.1610.  Google Scholar

[23]

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting,, Ann. of Math. (2), 169 (2009), 675.  doi: 10.4007/annals.2009.169.675.  Google Scholar

[24]

R. Saghin and Z. Xia, Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms,, Trans. Amer. Math. Soc., 358 (2006), 5119.  doi: 10.1090/S0002-9947-06-04171-7.  Google Scholar

[25]

D. A. Visscher, Franks' Lemma in Geometric Contexts,, PhD dissertation, (2012).   Google Scholar

[26]

T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov,, Preprint Dijon, (2005).   Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).   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. Lopes Dias, Generic dynamics of 4-dimensional $C^2$ Hamiltonian systems,, Comm. Math. Phys., 281 (2008), 597.  doi: 10.1007/s00220-008-0500-y.  Google Scholar

[4]

M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows,, J. Differential Equations, 245 (2008), 3127.  doi: 10.1016/j.jde.2008.02.045.  Google Scholar

[5]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19.  doi: 10.1007/s002200050412.  Google Scholar

[6]

C. Bonatti, L. J. Díaz and E. R. 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

[7]

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

[8]

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

[9]

G. Contreras and G. Paternain, Genericity of geodesic flows with positive topological entropy on $S^2$,, J. Diff. Geom., 61 (2002), 1.   Google Scholar

[10]

M. Crainic and R. Loja Fernandes, Integrability of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[11]

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

[12]

B. Hernández-Bermejo, New solutions of the Jacobi equations for three-dimensional Poisson structures,, J. Math. Phys., 42 (2001), 4984.  doi: 10.1063/1.1402174.  Google Scholar

[13]

V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms,, Ann. I. H. Poincaré - AN, 23 (2006), 641.  doi: 10.1016/j.anihpc.2005.06.002.  Google Scholar

[14]

C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures,, Springer, (2013).  doi: 10.1007/978-3-642-31090-4.  Google Scholar

[15]

J. M. Lee, Introduction to Smooth Manifolds,, $2^{nd}$ edition, (2013).   Google Scholar

[16]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Reidel Publishing Co., (1987).  doi: 10.1007/978-94-009-3807-6.  Google Scholar

[17]

C. Martin, The Poisson structure of the mean-field equations in the $\Phi^4$ theory,, Ann. Physics, 271 (1999), 294.  doi: 10.1006/aphy.1998.5875.  Google Scholar

[18]

C. Morales, M. J. Pacífico and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers,, Ann. Math., 160 (2004), 375.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[19]

P. J. Morrison, Hamiltonian description of the ideal fluid,, Rev. Modern Phys., 70 (1998), 467.  doi: 10.1103/RevModPhys.70.467.  Google Scholar

[20]

J. Moser, On the volume elements on a manifold,, Trans. Amer. Math. Soc., 120 (1965), 286.  doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[21]

P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[22]

G. Picard and T. W. Johnston, Instability cascades, Lotka-Volterra population equations, and Hamiltonian chaos,, Phys. Rev. Lett., 48 (1982), 1610.  doi: 10.1103/PhysRevLett.48.1610.  Google Scholar

[23]

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting,, Ann. of Math. (2), 169 (2009), 675.  doi: 10.4007/annals.2009.169.675.  Google Scholar

[24]

R. Saghin and Z. Xia, Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms,, Trans. Amer. Math. Soc., 358 (2006), 5119.  doi: 10.1090/S0002-9947-06-04171-7.  Google Scholar

[25]

D. A. Visscher, Franks' Lemma in Geometric Contexts,, PhD dissertation, (2012).   Google Scholar

[26]

T. Vivier, Robustly transitive $3$-dimensional regular energy surfaces are Anosov,, Preprint Dijon, (2005).   Google Scholar

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