Frequency locking for Tonelli Lagrangians

TL;DR

This paper proves that for a generic class of Tonelli Lagrangians in two-dimensional spaces, most cohomology classes have Aubry sets consisting of a single hyperbolic periodic orbit, highlighting a generic dynamical property.

Contribution

It establishes a generic property of Tonelli Lagrangians in two dimensions where Aubry sets are typically single hyperbolic periodic orbits for a dense subset of cohomology classes.

Findings

Existence of an open dense subset of cohomology classes with single hyperbolic periodic Aubry sets.

Genericity result for Tonelli Lagrangians in two-dimensional configuration spaces.

Characterization of Aubry sets in the generic case.

Abstract

We prove that for a generic Tonelli Lagrangian on a configuration space of dimension two, there exists an open dense subset of cohomology classes, whose Aubry set consists of exactly one hyperbolic periodic orbit.