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Fixed point theorems for small divisors problems

Mauricio Garay, Duco van Straten

TL;DR

The paper develops a framework of fixed-point theorems for Banach scales of holomorphic functions to address small divisor problems in dynamical systems. It first establishes an analytic inverse-function theory (including Hamilton's and Zehnder's results) using Newton-type iterations and Bruno-sequence control, avoiding smoothing operators in the analytic setting. It then builds a contraction-mapping approach (Morse lemma, perturbative factors) and extends to KAM-type iterations with tameness and small-denominator analysis, culminating in a KAM fixed-point theorem. Together, these results provide a cohesive toolkit for proving the existence of invariant structures (e.g., tori) in nearly integrable systems and offer pathways to normal-form results and conjectures such as Herman’s.

Abstract

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems.

Fixed point theorems for small divisors problems

TL;DR

The paper develops a framework of fixed-point theorems for Banach scales of holomorphic functions to address small divisor problems in dynamical systems. It first establishes an analytic inverse-function theory (including Hamilton's and Zehnder's results) using Newton-type iterations and Bruno-sequence control, avoiding smoothing operators in the analytic setting. It then builds a contraction-mapping approach (Morse lemma, perturbative factors) and extends to KAM-type iterations with tameness and small-denominator analysis, culminating in a KAM fixed-point theorem. Together, these results provide a cohesive toolkit for proving the existence of invariant structures (e.g., tori) in nearly integrable systems and offer pathways to normal-form results and conjectures such as Herman’s.

Abstract

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems.
Paper Structure (20 sections, 11 theorems, 83 equations)

This paper contains 20 sections, 11 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. An analytic inverse function theorem
  3. The Hamilton inverse function theorem
  4. The analytic case
  5. What loss of regularity do we expect?
  6. Local maps
  7. The classical Newton iteration
  8. Bruno sequences
  9. Analytic version of the Hamilton theorem
  10. The Zehnder implicit function theorem
  11. A contraction mapping theorem
  12. The Morse lemma
  13. Perturbative factors
  14. Contraction mapping theorem
  15. KAM iterations
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $E$ and $F$ be tame Fréchet spaces, let $U\subset F$ be an open subset, and let $f:U\longrightarrow G$ be a smooth tame map. Suppose that for each $x \in U$ the linearization $Df(x):F\longrightarrow G$ is invertible, and the family of inverses, as a map $U\times G\longrightarrow F$ is smooth t

Theorems & Definitions (25)

  • Theorem 2.1: Hamilton_implicit
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4: Bruno
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 15 more