Percival Lagrangian approach to Aubry-Mather theory

TL;DR

This paper offers streamlined proofs of core Aubry-Mather theory results using hull functions, applicable in any dimension, and compares these with other formalisms to enhance understanding.

Contribution

It introduces a simplified proof framework for Aubry-Mather theory results based on hull functions, extending to arbitrary dimensions and comparing with existing methods.

Findings

Existence of quasi-periodic minimizers confirmed.

Multiplicity results for minimizers with gaps.

Comparison of hull function approach with other formalisms.

Abstract

We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions We also compare the proofs and results with those obtained in other formalisms.