Geometric normalization

TL;DR

This paper introduces geometric normalization for planar diffeomorphisms with irrational elliptic fixed points, linking formal conjugacies to invariant foliations and demonstrating generic divergence results.

Contribution

It defines geometric normalization as a new concept extending classical normalizations, connecting formal conjugacies to invariant foliations in a novel way.

Findings

Geometric normalizations correspond to a unique formal invariant foliation.

Generic divergence results show non-existence of analytic invariant foliations.

Includes classical formal normalizations as a special case.

Abstract

For a local analytic diffeomorphism of the plane with an irrational elliptic fixed point at 0, we introduce the notion of ``geometric normalization'', which includes the classical formal normalizations as a special case: it is a formal conjugacy to a formal diffeomorphism which preserves the foliation by circles centered at 0. We show that geometric normalizations, despite of non-uniqueness, correspond in a natural way to a unique formal invariant foliation. We show, in various contexts, generic results of divergence for the geometric normalizations, which amount to the generic non-existence of any analytic invariant foliation.