Domain of attraction for maps tangent to the identity in $\mathbb{C}^2$ with characteristic direction of higher degree

TL;DR

This paper investigates the behavior of holomorphic fixed point germs in two complex variables tangent to the identity, identifying conditions under which points are attracted to the origin along specific characteristic directions.

Contribution

It establishes new conditions for the existence of attraction domains for maps tangent to the identity with higher degree characteristic directions in $\

Findings

Existence of attraction domains under certain higher degree characteristic directions

Conditions for convergence of points to the origin along characteristic directions

Extension of attraction theory to degenerate and higher degree cases

Abstract

We study holomorphic fixed point germs in two complex variables that are tangent to the identity and have a degenerate characteristic direction. We show that if that characteristic direction is also a characteristic direction for higher degree terms, is non-degenerate for a higher degree term, and satisfies some additional properties, then there is a domain of attraction on which points converge to the origin along that direction.