Classification of discrete weak KAM solutions on linearly repetitive quasi-periodic sets

TL;DR

This paper establishes the existence and classification of discrete weak KAM solutions for certain Hamilton-Jacobi equations on linearly repetitive quasi-periodic sets, advancing understanding in almost periodic settings.

Contribution

It proves the existence of weak KAM solutions in non-degenerate, weakly twist discrete systems and classifies all solution types under linearly repetitive quasi-periodic conditions.

Findings

Existence of weak KAM solutions in non-degenerate, weakly twist interactions.

Complete classification of weak KAM solutions on linearly repetitive quasi-periodic sets.

Insights into the structure of solutions in almost periodic environments.

Abstract

In discrete schemes, weak KAM solutions may be interpreted as approximations of correctors for some Hamilton-Jacobi equations in the periodic setting. It is known that correctors may not exist in the almost periodic setting. We show the existence of discrete weak KAM solutions for non-degenerate and weakly twist interactions in general. Furthermore, assuming equivariance with respect to a linearly repetitive quasi-periodic set, we completely classify all possible types of weak KAM solutions.