Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

TL;DR

This paper classifies semi-hyperbolic polynomial semigroups with bounded postcritical sets and explores the complex dynamics of their Julia sets, revealing phenomena like Jordan curve Julia sets that are not quasicircles, which are absent in single polynomial iteration.

Contribution

It provides a classification of semi-hyperbolic polynomial semigroups and analyzes the properties of their Julia sets and Fatou components, especially in the context of random polynomial dynamics.

Findings

Julia sets are Jordan curves but not quasicircles in certain semigroup dynamics

Unbounded Fatou components are John domains, bounded ones are not

New phenomena in polynomial semigroup dynamics not seen in single polynomial iteration

Abstract

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $\Gamma $, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $\gamma \in \Gamma ^{\Bbb{N}}$, the Julia set $J_{\gamma}$ of $\gamma $ is a Jordan curve but not a quasicircle, the unbounded component of the Fatou set $F_{\gamma}$ of $\gamma$ is a John domain, and the bounded component of $F_{\gamma}$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups $G$ such that the planar postcritical set of $G$ is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.