On commuting Tonelli Hamiltonians: Autonomous case

TL;DR

This paper demonstrates that key dynamical sets and functions are identical for two commuting autonomous Tonelli Hamiltonians, revealing their deep interconnectedness and establishing quasi-linearity and common subsolutions in Hamilton-Jacobi theory.

Contribution

It proves the equivalence of Aubry, Mather, and barrier sets for commuting Hamiltonians and shows the quasi-linearity of alpha functions, introducing new insights into their joint structure.

Findings

Aubry, Mather, and barrier sets coincide for commuting Hamiltonians

Alpha functions exhibit quasi-linearity in this context

Existence of common C^{1,1} critical subsolutions for Hamilton-Jacobi equations

Abstract

We show that the Aubry sets, the Ma\~{n}\'{e} sets, Mather's barrier functions are the same for two commuting autonomous Tonelli Hamiltonians. We also show the quasi-linearity of $\alpha$-functions from the dynamical point of view and the existence of common $C^{1,1}$ critical subsolution for their associated Hamilton-Jacobi equations.