# When are the invariant submanifolds of symplectic dynamics Lagrangian?

Abstract : Let L be a D-dimensional submanifold of a 2D-dimensional exact symplectic manifold (M, w) and let f be a symplectic diffeomorphism onf M. In this article, we deal with the link between the dynamics of f restricted to L and the geometry of L (is L Lagrangian, is it smooth, is it a graph...?). We prove different kinds of results. - for D=3, we prove that if a torus that carries some characteristic loop, then either L is Lagrangian or the restricted dynamics g of f to L can not be minimal (i.e. all the orbits are dense) with (g^k) equilipschitz; - for a Tonelli Hamiltonian of the cotangent bundle M of the 3-dimenional torus, we give an example of an invariant submanifold L with no conjugate points that is not Lagrangian and such that for every symplectic diffeomorphism f of M, if $f(L)=L$, then $L$ is not minimal; - 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; -we give similar results for C^1 submanifolds with weaker dynamical assumptions.
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Cited literature [14 references]

https://hal-univ-avignon.archives-ouvertes.fr/hal-01065122
Contributor : Marie-Claude Arnaud <>
Submitted on : Wednesday, September 17, 2014 - 10:03:17 PM
Last modification on : Tuesday, January 14, 2020 - 10:38:15 AM
Long-term archiving on: : Thursday, December 18, 2014 - 12:05:50 PM

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• HAL Id : hal-01065122, version 1
• ARXIV : 1409.5204

### Citation

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. MR3124714 Reviewed Arnaud, Marie-Claude When are the invariant submanifolds of symplectic dynamics Lagrangian? Discrete Contin. Dyn. Syst., 2014, 34 (5), pp.1811-1827. ⟨hal-01065122⟩

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