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

TL;DR

This paper investigates the properties of Fatou and Julia sets for semigroups of holomorphic endomorphisms in complex spaces, establishing conditions under which isolated points in Julia sets imply the presence of conjugate automorphisms.

Contribution

It generalizes a theorem of Fornaess-Sibony by linking isolated Julia set points to conjugate automorphisms in semigroups of entire maps.

Findings

Julia set of certain semigroups can contain isolated points

Presence of isolated points implies existence of conjugate automorphisms

Recurrent domains are characterized under specific conditions

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.