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arXiv · 1810.04229

Hereditarily non Uniformly Perfect non-Autonomous Julia Sets

Mark Comerford·Rich Stankewitz·Hiroki Sumi
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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}.

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Mark Comerford, Rich Stankewitz, Hiroki Sumi. 2018-10-09. Hereditarily non Uniformly Perfect non-Autonomous Julia Sets. https://arxiv.org/abs/1810.04229

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