Ghost circles in lattice Aubry-Mather theory

TL;DR

This paper extends Aubry-Mather theory for lattice recurrence relations by introducing ghost circles, which interpolate Aubry-Mather sets via gradient flow, providing new insights into the structure of solutions with gaps.

Contribution

It proves the existence of ghost circles for all rotation vectors and actions, linking Aubry-Mather sets with continuous gradient-flow families, and offers a new proof regarding gaps in these sets.

Findings

Ghost circles exist for all rotation vectors and actions.

A compactness theorem for ghost circles is established.

Gaps in Aubry-Mather sets are characterized by minimizers or non-minimizing solutions.

Abstract

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the gradient flow of the formal action function and we prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called "ghost circle". The existence of ghost circles is first proved for rational rotation vectors and Morse action functions. The main technical result is a compactness theorem for ghost circles, based on a parabolic Harnack inequality for the gradient flow, which implies the existence of ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be parametrized by minimizers, or contain a non-minimizing solution.