When are the invariant submanifolds of symplectic    dynamics Lagrangian?

Marie-Claude Arnaud

doi:10.3934/dcds.2014.34.1811

Article Contents

2014, Volume 34, Issue 5: 1811-1827. Doi: 10.3934/dcds.2014.34.1811

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When are the invariant submanifolds of symplectic dynamics Lagrangian?

Marie-Claude Arnaud^1,

1.
Avignon University, LMA EA 2151, F-84000, Avignon

Received: February 28, 2013

Revised: June 30, 2013

Published: April 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 37J05; Secondary: 70H03, 70H05, 37J50, 37J10.

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References

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