2009/11/20 by Matteo Ruggiero, Ruggiero, Matteo · 1 citation
Mathematics · #32H50 (primary) 37F99 (secondary) #37G05 #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.0911.4023
openalex publication_date 2009/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study holomorphic germs f:(ℂ2, 0) → (ℂ2,0) with non-invertible differential df0. In order to do this, we search for a modification π:X → (ℂ2,0) (i.e., a composition of point blow-ups over the origin), and an infinitely near point p ∈ π-1(0), such that the germ f lifts to a holomorphic germ f:(X,p) → (X,p) which is rigid (i.e., the generalized critical set of 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 (Spec(df0)=\0, λ\ with λ≠ 0). We specify holomorphic normal forms for the nilpotent case and for the type (0,\mathbbD), that is Spec(df0)=\0, λ\ with λ in the unitary disk \mathbbD ⊂ ℂ, and formal normal forms for the type (0, ℂ ∖ \mathbbD).