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 $\pi:X \rightarrow (\mathbb{ C}^2,0)$ (i.e., a composition of point blow-ups over the origin), and an infinitely near point $p \in \pi^{ -1}(0)$, 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 ($\operatorname{ Spec}(df_0)=\{ 0, \lambda\}$ with $\lambda \neq 0$). We specify holomorphic normal forms for the nilpotent case and for the type $(0,\mathbb{ D})$, that is $\operatorname{ Spec}(df_0)=\{ 0, \lambda\}$ with $\lambda$ in the unitary disk $\mathbb{ D} \subset \mathbb{ C}$, and formal normal forms for the type $(0, \mathbb{ C} \setminus \mathbb{ D})$.
Categories: math.DS