Homoclinic orbits and critical points of barrier functions

TL;DR

This paper explores the relationship between critical points of Mather's barrier functions and minimal homoclinic orbits, providing new theoretical insights and proving the existence of such orbits on two-dimensional tori.

Contribution

It introduces a critical point theorem for barrier functions and establishes a connection between these points and homoclinic orbits on a7^n, with applications to a7^2.

Findings

Established a link between critical points of barrier functions and homoclinic orbits.

Proved a critical point theorem for barrier functions.

Demonstrated the existence of homoclinic orbits on a7^2.

Abstract

We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on $\mathbb{T}^n$. We also prove a critical point theorem for barrier functions, and the existence of such homoclinic orbits on $\mathbb{T}^2$ as an application.