Dynamics of multi-resonant biholomorphisms

TL;DR

This paper investigates the dynamics of multi-resonant holomorphic diffeomorphisms in complex space, establishing conditions for basins of attraction and generalizing classical theorems to describe local behavior near fixed points.

Contribution

It introduces sharp criteria for basins of attraction and extends the Leau-Fatou flower theorem to multi-resonant cases in Poincare'-Dulac normal form.

Findings

Established conditions for existence of basins of attraction.

Generalized Leau-Fatou flower theorem for multi-resonant germs.

Provided a full local dynamical description near the origin.

Abstract

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in C^n such that the resonances among the first 1<= r<= n eigenvalues of the differential are generated over N by a finite number of Q-linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. Furthermore, we obtain a generalization of the Leau-Fatou flower theorem, providing a complete description of the dynamics in a full neighborhood of the origin for 1-resonant parabolically attracting holomorphic germs in Poincare'-Dulac normal form.