Hereditarily non Uniformly Perfect non-Autonomous Julia Sets

TL;DR

This paper studies a class of non-autonomous Julia sets, providing criteria for hereditary non-uniform perfectness and demonstrating that these sets can attain the maximum Hausdorff dimension of 1.

Contribution

It introduces a sharp criterion for hereditary non-uniform perfectness in non-autonomous Julia sets and shows these sets can have Hausdorff dimension 1.

Findings

Established a criterion for HNUP in non-autonomous Julia sets

Constructed examples with Hausdorff dimension 1

Used non-autonomous conformal iterated function systems and Bowen's formula

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. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urb\'{a}nski \cite{RU}.