Rotation domains for maps of bounded type

TL;DR

This paper introduces a new method to derive KAM linearization theorems using stable foliations of renormalization operators, with applications in complex dynamics including Herman rings.

Contribution

It provides a direct approach to KAM theorems from stable foliation existence, with novel applications in complex dynamical systems.

Findings

Derived KAM linearization theorems from stable foliations

Applied method to persistence of Herman rings

Extended classical results to multi-dimensional maps

Abstract

We present a novel approach for deriving KAM-type linearization theorems directly -- and almost immediately -- from the existence of the stable foliation for a renormalization operator. We give a few illustrations in dynamics in one and several complex variables, starting with a version of the classical theorem of Arnol'd and ending with a result on persistence of Herman rings in families of two-dimensional maps.