When are the invariant submanifolds of symplectic dynamics Lagrangian?

TL;DR

This paper investigates the conditions under which invariant submanifolds in symplectic dynamics are Lagrangian, providing new results linking geometric properties with dynamical behavior in various settings.

Contribution

It establishes new criteria connecting the Lagrangian property of invariant submanifolds with their dynamical features in symplectic and Hamiltonian systems.

Findings

For D=3, Lagrangian or non-minimal dynamics are linked to characteristic loops.

Constructs an example of a non-Lagrangian invariant submanifold with no conjugate points.

Proves invariant Lipschitz submanifolds are Lagrangian, C^1, and graphs under certain conditions.

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.