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. Second, we define recurrent domains for semigroups and provide a description of such domains under some conditions.