Dynamics of semigroups of entire maps of ℂ^k

2015/04/13 by Bera, Sayani, Pal, Ratna
#32H02 #32H50 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.03094

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 ℂ^k, k ≥ 2. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of ℂ^k, k ≥ 2. We prove that if the Julia set of a semigroup G which is generated by endomorphisms of maximal generic rank k in ℂ^k contains an isolated point, then G must contain an element that is conjugate to an upper triangular automorphism of 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.

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