On the tangent cones of Aubry sets

TL;DR

This paper investigates the geometric structure of Aubry sets in Tonelli Hamiltonian systems, demonstrating that their tangent cones are contained within cones bounded by Green bundles, thus refining previous understanding.

Contribution

It improves earlier results by showing the tangent cones of Aubry sets are contained within cones bounded by Green bundles in Tonelli Hamiltonian systems.

Findings

Tangent cones of Aubry sets are contained in cones bounded by Green bundles.

The result refines previous understanding of Aubry set geometry.

Provides a geometric inclusion relation for Aubry sets in Hamiltonian dynamics.

Abstract

We show that the paratingent cone of the Aubry set of the Tonelli Hamiltonian is contained in a cone bounded by the Green bundles. Our result improves the earlier result of M.-C. Arnaud on tangent cones of the Aubry sets.