Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps

TL;DR

This paper introduces discrete weak KAM theory through examples, illustrating its connection to classical weak KAM theory and applying it to twist maps, highlighting new results and differences between discrete and continuous cases.

Contribution

It provides a self-contained introduction to discrete weak KAM theory, including new results on convergence of solutions and comparisons with classical theory.

Findings

Discrete weak KAM theory serves as a simplified model for classical theory.

New results on convergence of solutions of discounted equations for degenerate perturbations.

Clarifies differences and relations between discrete and classical weak KAM theories.

Abstract

The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear, but where central ideas and results persist. It can therefore serve as a good introduction to (continuous) weak KAM theory. After a general exposition of the general abstract theory, several examples are studied. The last section is devoted to the historical problem of conservative twist maps of the annulus. At the end of the first three Chapters, the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory are discussed. Some key differences are also highlighted between the discrete and classical theory. Those results are new. The text also contains other results never published before, such as the convergence of solutions of discounted equations for degenerate perturbations.