Matteo Ruggiero — Page 3

arXiv2011

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

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 Poincaré-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 Poincaré-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.

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arXiv2010

Rigidification of holomorphic germs with non-invertible differential

We study holomorphic germs $f:(\mathbb{C}^2, 0) \rightarrow (\mathbb{C}^2,0) with non-invertible differential $df_0$. In order to do this, we search for a modification $π:X \rightarrow (\mathbb{C}^2,0)$ (i.e., a composition of point blow-ups over the origin), and an infinitely near point $p \in π^{-1}(0)$, such that the germ $f$ lifts to a holomorphic germ $\hat{f}:(X,p) \rightarrow (X,p)$ which is rigid (i.e., the generalized critical set of $\hat{f}$ is totally invariant and has normal crossings at $p$). We extend a previous result for superattracting germs to the general case, and deal with the uniqueness of this process in the semi-superattracting case ($\operatorname{Spec}(df_0)=\{0, λ\}$ with $λ\neq 0$). We specify holomorphic normal forms for the nilpotent case and for the type $(0,\mathbb{D})$, that is $\operatorname{Spec}(df_0)=\{0, λ\}$ with $λ$ in the unitary disk $\mathbb{D} \subset \mathbb{C}$, and formal normal forms for the type $(0, \mathbb{C} \setminus \mathbb{D})$.

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