Interesting examples in $\mathbb{C}^2$ of maps tangent to the identity without domains of attraction

TL;DR

This paper presents a complex two-dimensional map tangent to the identity with no domain of attraction, illustrating nuanced dynamical behaviors and how higher degree terms influence attraction properties.

Contribution

It provides a novel example of such maps with multiple characteristic directions and analyzes how adding higher degree terms affects their attraction domains.

Findings

Map has three characteristic directions, with only one attracting points but not the origin.

Adding higher degree terms can create or destroy domains of attraction.

The example highlights complex dynamics in tangent-to-identity maps in $\

Abstract

We give an interesting example of a map in $\mathbb{C}^2$ that is tangent to the identity, but that does not have a domain of attraction along any of its characteristic direction. This map has three characteristic directions, two of which are not attracting while the third attracts points to that direction, but not to the origin. In addition, we show that if we add higher degree terms to this map, sometimes a domain of attraction along one of its characteristic directions will exist and sometimes one will not.