Symplectic and contact properties of the Ma\~n\'e critical value of the universal cover

TL;DR

This paper explores symplectic properties of the Mañé critical value in the universal cover of Tonelli Hamiltonians, revealing invariance properties and counterexamples to classical theorems in symplectic and dynamical systems.

Contribution

It demonstrates that the critical energy level is never of virtual contact type in higher dimensions and shows symplectic invariance of certain dynamical invariants, providing new counterexamples.

Findings

Critical energy level not of virtual contact type in dimensions ≥ 3

Finiteness of Peierls barrier and Aubry set is symplectically invariant

Counterexamples to Mather's graph theorem with zero homotopy measures

Abstract

We discuss several symplectic aspects related to the Ma\~n\'e critical value c_u of the universal cover of a Tonelli Hamiltonian. In particular we show that the critical energy level is never of virtual contact type for manifolds of dimension greater than or equal to three. We also show the symplectic invariance of the finiteness of the Peierls barrier and the Aubry set of the universal cover. We also provide an example where c_u coincides with the infimum of Mather's \alpha -function but the Aubry set of the universal cover is empty and the Peierls barrier is finite. A second example exhibits all the ergodic invariant minimizing measures with zero homotopy, showing that, quite surprisingly, the union of their supports is not a graph, in contrast with Mather's celebrated graph theorem.