Dynamics of one-resonant biholomorphisms

Abstract

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in ${\mathbb{C}}^n$ whose differentials have one-dimensional family of resonances in the first $m$ eigenvalues, $m\le n$ (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.