Attracting domains of maps tangent to the identity whose only characteristic direction is non-degenerate

TL;DR

This paper proves that certain complex two-variable maps tangent to the identity with a unique non-degenerate characteristic direction have domains where they behave like translations, including special cases that form Fatou-Bieberbach domains.

Contribution

It establishes the existence of translation-like domains for holomorphic fixed point germs tangent to the identity with a single non-degenerate characteristic direction.

Findings

Existence of attracting domains conjugate to translations

Identification of Fatou-Bieberbach domains in the automorphism case

Characterization of dynamics near fixed points

Abstract

We prove that a holomorphic fixed point germ in two complex variables, tangent to the identity, and whose only characteristic direction is non-degenerate, has a domain of attraction on which the map is conjugate to a translation. In the case of a global automorphism, the corresponding domain of attraction is a Fatou-Bieberbach domain.