# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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], 9 (2008), 881. doi: 10.1007/s00023-008-0375-7. Google Scholar [2] M.-C. Arnaud, On a theorem due to Birkhoff,, Geometric and Functional Analysis, 20 (2010), 1307. doi: 10.1007/s00039-010-0091-6. Google Scholar [3] V. Arnol'd and A. Avez, Ergodic problems of classical mechanics,, Translated from the French by A. Avez. W. A. Benjamin, (1968). Google Scholar [4] P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615. doi: 10.1090/S0894-0347-08-00591-2. Google Scholar [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. doi: 10.1017/S0305004111000685. Google Scholar [6] M. Bialy, Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus,, Comm. Math. Phys., 126 (1989), 13. doi: 10.1007/BF02124329. Google Scholar [7] M. Bialy and L. Polterovich, Hamiltonian diffeomorphisms and Lagrangian distributions,, Geom. Funct. Anal., 2 (1992), 173. doi: 10.1007/BF01896972. Google Scholar [8] M. Bialy and L. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom,, Invent. Math., 97 (1989), 291. doi: 10.1007/BF01389043. Google Scholar [9] M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem,, Math. Ann., 292 (1992), 619. doi: 10.1007/BF01444639. Google Scholar [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, 133-134 (1986), 133. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF02698874. Google Scholar [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, (1965). Google Scholar [14] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [15] A. Weinstein, Lectures on symplectic manifolds,, Expository lectures from the CBMS Regional Conference held at the University of North Carolina, (1977), 8. Google Scholar

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##### 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], 9 (2008), 881. doi: 10.1007/s00023-008-0375-7. Google Scholar [2] M.-C. Arnaud, On a theorem due to Birkhoff,, Geometric and Functional Analysis, 20 (2010), 1307. doi: 10.1007/s00039-010-0091-6. Google Scholar [3] V. Arnol'd and A. Avez, Ergodic problems of classical mechanics,, Translated from the French by A. Avez. W. A. Benjamin, (1968). Google Scholar [4] P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems,, J. Amer. Math. Soc., 21 (2008), 615. doi: 10.1090/S0894-0347-08-00591-2. Google Scholar [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. doi: 10.1017/S0305004111000685. Google Scholar [6] M. Bialy, Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus,, Comm. Math. Phys., 126 (1989), 13. doi: 10.1007/BF02124329. Google Scholar [7] M. Bialy and L. Polterovich, Hamiltonian diffeomorphisms and Lagrangian distributions,, Geom. Funct. Anal., 2 (1992), 173. doi: 10.1007/BF01896972. Google Scholar [8] M. Bialy and L. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom,, Invent. Math., 97 (1989), 291. doi: 10.1007/BF01389043. Google Scholar [9] M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff's theorem,, Math. Ann., 292 (1992), 619. doi: 10.1007/BF01444639. Google Scholar [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, 133-134 (1986), 133. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF02698874. Google Scholar [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, (1965). Google Scholar [14] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [15] A. Weinstein, Lectures on symplectic manifolds,, Expository lectures from the CBMS Regional Conference held at the University of North Carolina, (1977), 8. Google Scholar
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