A Poincar\'e-Dulac renormalization theorem for attracting rigid germs in $\mathbb{C}^d$

TL;DR

This paper extends the Poincaré-Dulac renormalization theory to attracting rigid germs in complex dimensions three and higher, revealing finite principal resonances and aiding classification of such dynamical systems.

Contribution

It introduces the concept of principal resonances for attracting rigid germs and proves a Poincaré-Dulac renormalization theorem in this context.

Findings

Finite number of principal resonances for attracting rigid germs.

Existence of a Poincaré-Dulac renormalization theorem in this setting.

Application to classification of specific attracting rigid germs in any dimension.

Abstract

Studying the dynamics of attracting rigid germs $f:(\mathbb{C}^d, 0) \rightarrow (\mathbb{C}^d, 0)$ in dimension $d \geq 3$, a new phenomenon arise: principal resonances. The resonances of the classic Poincar\'e-Dulac theory are given by (multiplicative) relations between the eigenvalues of $df_0$; principal resonances arise as (multiplicative) relations between the non-null eigenvalues of $df_0$, and the "leading term" for the superattracting part of $f$. We shall prove that for attracting rigid germs there are only finitely-many principal resonances, and a Poincar\'e-Dulac renormalization theorem in this case. We shall conclude with some considerations on the classification of a special class of attracting rigid germs in any dimension, and we specialize the result to the 3-dimensional case.