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

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.