Hereditarily non Uniformly Perfect non-Autonomous Julia Sets
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 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}.
Cite
@article{arxiv.1810.04229,
title = {Hereditarily non Uniformly Perfect non-Autonomous Julia Sets},
author = {Mark Comerford and Rich Stankewitz and Hiroki Sumi},
journal= {arXiv preprint arXiv:1810.04229},
year = {2019}
}
Comments
19 pages, 2 figures To appear in Discrete and Continuous Dynamical Systems