Thick-Thin non-Autonomous Julia Sets

TL;DR

This paper constructs non-autonomous Julia sets that are neither uniformly perfect nor hereditarily non uniformly perfect, and demonstrates their decomposition into 'thick' and 'thin' parts with specific topological properties.

Contribution

It introduces a new class of non-autonomous Julia sets with intermediate properties and describes their 'thick-thin' decomposition, expanding understanding of Julia set structures.

Findings

Julia sets are not uniformly perfect but also not HNUP.

These Julia sets can be decomposed into 'thick' and 'thin' parts.

The 'thick' part is a countable union of uniformly perfect sets, while the 'thin' part is HNUP.

Abstract

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. For non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary, the Julia set is uniformly perfect for all sequences with suitably bounded coefficients, while Comerford, Stankewitz and Sumi showed in \cite{CSS} that for certain sequences of polynomials with unbounded coefficients, it is possible to have Julia sets which are HNUP. In this manuscript we give an example of a non-autonomous polynomial sequences whose Julia sets lie in between these two extremes in that they are not uniformly perfect, but also not HNUP. In addition we show that these Julia sets can be expressed as a `thick-thin' decomposition consisting of a ${\mathrm F}_\sigma$ subset which is a countable union of uniformly perfect sets and a ${\mathrm G}_\delta$ subset which is HNUP.