2023/08/11 by Maxime Zavidovique, Zavidovique, Maxime · 1 citation
Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Optimization and Control (math.OC) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2308.06356
openalex publication_date 2023/08/11 · openalex created_date 2023/08/16 · openalex updated_date 2026/07/28
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.