On the Existence of Attracting Domains for Maps Tangent to the Identity.

Abstract

The main result we discuss is for germs of holomorphic self-maps in two complex variables that have a unique characteristic direction and this direction is non-degenerate. We prove that there exists a domain of attraction whose points converge to the origin along the characteristic direction and on which the map is conjugate to translation. In the case of a global automorphism, the corresponding invariant attracting domain is a Fatou-Bieberbach domain. In addition, we discuss other types of germs of holomorphic self-maps in two or more complex variables that fix the origin, in each case determining whether or not there exists an invariant attracting domain whose points converge to the origin tangentially to the same direction.