Dynamics of semigroups of entire maps of ${\mathbb{C}}^k$

Abstract

The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of ${\mathbb{C}}^k,k\ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of ${\mathbb{C}}^k,k\ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in ${\mathbb{C}}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of ${\mathbb{C}}^k.$ This generalizes a theorem of Fornaess-Sibony. Secondly, we define recurrent domains for semigroups and provide a description of such domains under some conditions.

arXiv comment14 pages