When are the invariant submanifolds of symplectic dynamics Lagrangian?
Marie-Claude Arnaud
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1811-1827 doi: 10.3934/dcds.2014.34.1811
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.
keywords: Symplectic dynamics minimizing submanifolds. invariant submanifolds Lagrangian dynamics Lagrangian submanifolds
A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map
Marie-Claude Arnaud
Journal of Modern Dynamics 2011, 5(3): 583-591 doi: 10.3934/jmd.2011.5.583
We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that:
    •$\Gamma$ is not differentiable;
    •$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.
keywords: invariant curves Twist maps irregularity of invariant curves

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