Formal Poincare'-Dulac renormalization for holomorphic germs

TL;DR

This paper introduces a new formal renormalization method that simplifies the Poincaré-Dulac normal form for holomorphic germs, providing a complete classification for certain bi-dimensional cases and exploring resonance phenomena.

Contribution

It develops a general renormalization procedure that yields simpler formal normal forms than classical methods, with a complete classification for bi-dimensional superattracting germs.

Findings

Complete list of normal forms for bi-dimensional superattracting germs

Simplification of normal forms beyond classical Poincaré-Dulac forms

Examples illustrating second-order resonance phenomena

Abstract

In this revised version, applying a general renormalization procedure for formal self-maps, producing a formal normal form simpler than the classical Poincar\'e-Dulac normal form, we shall give a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang, Zheng and Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.