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

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 Φbe 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 Φand N?

28 pages