An introduction to KAM theory I: the basics

Abstract

The KAM (Kolmogorov Arnold and Moser) theory studies the persistence of quasi-periodic solutions under perturbations. It started from a basic set of theorems and it has grown into a systematic theory that settles many questions. The basic theorem is rather surprising since it involves delicate regularity properties of the functions considered, rather subtle number theoretic properties of the frequency as well as geometric properties of the dynamical systems considered. In these lectures, we plan to cover a complete proof of a particularly representative theorem in KAM theory. In the first lecture we will cover all the prerequisites (analysis, number theory and geometry). In the second lecture we will present a complete proof of Moser's twist map theorem (indeed a generalization to more dimensions). The proof also lends itself to very efficient numerical algorithms.