Elliptic fixed points with an invariant foliation: Some facts and more questions

TL;DR

This paper investigates the analytic properties of formal conjugacies and normal forms for non-resonant elliptic fixed points of analytic diffeomorphisms that preserve a circular foliation.

Contribution

It provides insights into the relationship between invariant foliations and the analyticity of conjugacies and normal forms near elliptic fixed points.

Findings

Conditions under which the conjugacy nd normal form are analytic.

Examples illustrating when analyticity fails or holds.

New results on the structure of invariant foliations near elliptic fixed points.

Abstract

We address the following question: let F:(R^2,0)->(R^2,0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let \Phi be a formal conjugacy to a normal form N. Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of \Phi and N?