KAM Theory. Part II. Kolmogorov spaces

TL;DR

This paper develops a functional calculus for Kolmogorov spaces, establishing exponential maps, Lie group actions, and normal form theorems within an infinite-dimensional KAM theory framework.

Contribution

It introduces functorial analysis and local operators to extend Lie theory and normal form results to Kolmogorov spaces in infinite dimensions.

Findings

Defined a functional calculus based on local operators

Proved a fixed point theorem for Kolmogorov spaces

Established general normal form theorems

Abstract

This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define the exponential and therefore relation between Lie algebra and Lie group actions in the infinite dimensional context. Then we introduce a notion of finite dimensional reduction and use it to prove a fixed point theorem for Kolmogorov spaces. We conclude by proving general normal theorems.