Finite dimensional invariant KAM tori for tame vector fields

TL;DR

This paper develops a Nash-Moser/KAM algorithm for constructing invariant tori in tame vector fields, applicable in both analytic and Sobolev contexts, with minimal hypotheses for convergence.

Contribution

It introduces a formal, minimal-hypotheses Nash-Moser/KAM algorithm for invariant tori in tame vector fields, applicable in finite and infinite dimensions.

Findings

Algorithm works in analytic and Sobolev classes.

Reduces problem to solving linear forced equations.

Provides minimal conditions for convergence.

Abstract

We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows to reduce to solving only linear forced equations. We remark that our algorithm works at the same time in analytic and Sobolev class.