Complete integrability of diffeomorphisms and their local normal forms

TL;DR

This paper studies the conditions under which a family of commuting diffeomorphisms near a fixed point can be simplified to a normal form through analytic or smooth transformations, enhancing understanding of their integrability and local structure.

Contribution

It introduces a notion of integrability for families of diffeomorphisms and provides sufficient conditions for their normalization via analytic or smooth transformations.

Findings

Sufficient conditions for normal form transformation.

Definition of integrability for diffeomorphism families.

Normal forms achieved through analytic or smooth transformations.

Abstract

In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of $\mathbb{K}^n$ ($\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp. a smooth) transformation if the initial diffeomorphisms are analytic (resp. smooth).