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

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.