May  2014, 34(5): 1811-1827. doi: 10.3934/dcds.2014.34.1811

When are the invariant submanifolds of symplectic dynamics Lagrangian?

1. 

Avignon University, LMA EA 2151, F-84000, Avignon, France

Received  March 2013 Revised  July 2013 Published  October 2013

Let $\mathcal{L}$ be a $D$-dimensional submanifold of a $2D$ dimensional exact symplectic manifold $(M, \omega)$ and let $f: M\rightarrow M$ be a symplectic diffeomorphism. In this article, we deal with the link between the dynamics $f_{|\mathcal{L}}$ restricted to $\mathcal{L}$ and the geometry of $\mathcal{L}$ (is $\mathcal{L}$ Lagrangian, is it smooth, is it a graph … ?).
    We prove different kinds of results.
    1. for $D=3$, we prove that is $\mathcal{L}$ if a torus that carries some characteristic loop, then either $\mathcal{L}$ is Lagrangian or $f_{|\mathcal{L}}$ can not be minimal (i.e. all the orbits are dense) with $(f^k_{|\mathcal{L}})$ equilipschitz;
    2. for a Tonelli Hamiltonian of $T^*\mathbb{T}^3$, we give an example of an invariant submanifold $\mathcal{L}$ with no conjugate points that is not Lagrangian and such that for every $f:T^*\mathbb{T}^3\rightarrow T^*\mathbb{T}^3$ symplectic, if $f(\mathcal{L})=\mathcal{L}$, then $\mathcal{L}$ is not minimal;
    3. with some hypothesis for the restricted dynamics, we prove that some invariant Lipschitz $D$-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian, $C^1$ and graphs;
    4.we give similar results for $C^1$ submanifolds with weaker dynamical assumptions.
Citation: Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811
References:
[1]

M.-C. Arnaud, Fibrés de Green et régularité des graphes $C^0$-lagrangiens invariants par un flot de Tonelli, (French) [Green fibrations and regularity of $C^0$-Lagrangian graphs invariant under a Tonelli flow], Ann. Henri Poincaré, 9 (2008), 881-926. doi: 10.1007/s00023-008-0375-7.

[2]

M.-C. Arnaud, On a theorem due to Birkhoff, Geometric and Functional Analysis, 20 (2010), 1307-1316. doi: 10.1007/s00039-010-0091-6.

[3]

V. Arnol'd and A. Avez, Ergodic problems of classical mechanics, Translated from the French by A. Avez. W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[4]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.

[5]

P. Bernard and J. dos Santos, A geometric definition of the Ma-Mather set and a theorem of Marie-Claude Arnaud, Math. Proc. Cambridge Philos. Soc., 152 (2012), 167-178. doi: 10.1017/S0305004111000685.

[6]

M. Bialy, Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus, Comm. Math. Phys., 126 (1989), 13-24. doi: 10.1007/BF02124329.

[7]

M. Bialy and L. Polterovich, Hamiltonian diffeomorphisms and Lagrangian distributions, Geom. Funct. Anal., 2 (1992), 173-210. doi: 10.1007/BF01896972.

[8]

M. Bialy and L. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom, Invent. Math., 97 (1989), 291-303. doi: 10.1007/BF01389043.

[9]

M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627. doi: 10.1007/BF01444639.

[10]

J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel,…), (French) [Invariant tori of Hamiltonian dynamical systems (following Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel,…)] Seminar Bourbaki, Vol. 1984/85. Astérisque No., 133-134 (1986), 113-157.

[11]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, (French)Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233.

[12]

M. Herman, Inégalités "a priori''pour des tores lagrangiens invariants par des difféomorphismes symplectiques, (French) [A priori inequalities for Lagrangian tori invariant under symplectic diffeomorphisms] Inst. Hautes Études Sci. Publ. Math. No., 70 (1989), 47-101 (1990). doi: 10.1007/BF02698874.

[13]

J. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver. Revised reprint of the 1965 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, (1977). ii+149 pp

[15]

A. Weinstein, Lectures on symplectic manifolds, Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8-12, 1976. Regional Conference Series in Mathematics, No. 29. American Mathematical Society, Providence, R.I., (1977). iv+48 pp.

show all references

References:
[1]

M.-C. Arnaud, Fibrés de Green et régularité des graphes $C^0$-lagrangiens invariants par un flot de Tonelli, (French) [Green fibrations and regularity of $C^0$-Lagrangian graphs invariant under a Tonelli flow], Ann. Henri Poincaré, 9 (2008), 881-926. doi: 10.1007/s00023-008-0375-7.

[2]

M.-C. Arnaud, On a theorem due to Birkhoff, Geometric and Functional Analysis, 20 (2010), 1307-1316. doi: 10.1007/s00039-010-0091-6.

[3]

V. Arnol'd and A. Avez, Ergodic problems of classical mechanics, Translated from the French by A. Avez. W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[4]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.

[5]

P. Bernard and J. dos Santos, A geometric definition of the Ma-Mather set and a theorem of Marie-Claude Arnaud, Math. Proc. Cambridge Philos. Soc., 152 (2012), 167-178. doi: 10.1017/S0305004111000685.

[6]

M. Bialy, Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus, Comm. Math. Phys., 126 (1989), 13-24. doi: 10.1007/BF02124329.

[7]

M. Bialy and L. Polterovich, Hamiltonian diffeomorphisms and Lagrangian distributions, Geom. Funct. Anal., 2 (1992), 173-210. doi: 10.1007/BF01896972.

[8]

M. Bialy and L. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom, Invent. Math., 97 (1989), 291-303. doi: 10.1007/BF01389043.

[9]

M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627. doi: 10.1007/BF01444639.

[10]

J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel,…), (French) [Invariant tori of Hamiltonian dynamical systems (following Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel,…)] Seminar Bourbaki, Vol. 1984/85. Astérisque No., 133-134 (1986), 113-157.

[11]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, (French)Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233.

[12]

M. Herman, Inégalités "a priori''pour des tores lagrangiens invariants par des difféomorphismes symplectiques, (French) [A priori inequalities for Lagrangian tori invariant under symplectic diffeomorphisms] Inst. Hautes Études Sci. Publ. Math. No., 70 (1989), 47-101 (1990). doi: 10.1007/BF02698874.

[13]

J. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver. Revised reprint of the 1965 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, (1977). ii+149 pp

[15]

A. Weinstein, Lectures on symplectic manifolds, Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8-12, 1976. Regional Conference Series in Mathematics, No. 29. American Mathematical Society, Providence, R.I., (1977). iv+48 pp.

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