Hereditarily non uniformly perfect non-autonomous Julia sets

Abstract

<p style='text-indent:20px;'>Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [<xref ref-type="bibr" rid="b19">19</xref>] 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 <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> 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 Urbánski [<xref ref-type="bibr" rid="b15">15</xref>].</p>