Mather theory and symplectic rigidity

,D-Math, Eth Zürich, Rämistrasse 101, 8092 Zürich, Switzerland ,Mads R Bisgaard

doi:10.3934/jmd.2019018

Abstract

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Polterovich we also detect interesting invariant measures for $\phi_H$ by studying a generalization of the symplectic shape of sublevel sets of $H$. This approach differs from the first one in that it works also for $(M,\omega)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $\mathbb{R}^{2n}$ and twisted cotangent bundles.

Highlights

We present applications to Hamiltonian systems on R2 and twisted cotangent bundles
Mather [32] discovered that one can loosen the regularity assumptions and the Diophantine condition in the KAM theorem and still find plenty of interesting invariant sets which persist perturbations of, if one pays the price of replacing the non-degeneracy condition with a stronger convexity assumption
One of the aims of the current paper is to study, using methods coming from symplectic topology, what happens when one relaxes the convexity assumption

Summary

MATHER THEORY AND SYMPLECTIC RIGIDITY

We consider a closed monotone Lagrangian submanifold ⊂ ( , ) which is non-narrow in the sense that its quantum homology ∗( ; Z2) doesn’t vanish (see Section 5) In this setting Leclercq-Zapolsky [31] recently developed a theory of Lagrangian spectral invariants. We will denote by M ∶ ⊂ ( ) the set of measures ∈ ( ) with ( ) ∈ ∶ (0) arising as convex combinations of weak∗-limits of sequences ( ) given by (2), such that ( ) and ( ) satisfy the criteria mentioned above This notation is justified by results, first due to Viterbo [53, Proposition 13.3] and later Monzner-Vichery-Zapolsky [34, Theorem 1.11], saying that ∶ coincides with Mather’s -function when ⊂ ∗ is the zero-section and is Tonelli.

In order to define

In our example we can take

Fix now

In particular we can estimate

Since h is a Dini subdifferential of at and