Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

TL;DR

This paper studies the structure and dynamics of Julia sets in polynomial semigroups with bounded critical values, revealing how Fatou components are separated and constructing hyperbolic semigroups with specific properties.

Contribution

It provides new insights into the topology of Julia sets in polynomial semigroups and demonstrates methods to generate hyperbolic semigroups with postcritical boundedness.

Findings

Boundaries of certain Fatou components are separated by Cantor sets of quasicircles.

The distribution of Julia sets of individual maps within the semigroup's Julia set is characterized.

Methods for constructing hyperbolic semigroups with postcritical boundedness are developed.

Abstract

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.