Hereditarily non uniformly perfect non-autonomous Julia sets

Mark Comerford,Rich Stankewitz,Hiroki Sumi

doi:10.3934/dcds.2020002

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>

Highlights

Our paper is concerned with non-autonomous iteration of complex polynomials
There is an extensive literature in the real variables case which is mainly focused on topological dynamics, chaos, and difference equations, e.g. [2, 3]
In our case we will show that the iterated Julia sets for suitably chosen polynomial sequences are Hereditarily non uniformly perfect (HNUP) by showing they satisfy the stronger property of pointwise thinness

Summary

Introduction

Our paper is concerned with non-autonomous iteration of complex polynomials. This subject was started by Fornaess and Sibony [9] in 1991 and by Sester, Sumi and others who were working in the closely related area of skew-products [16, 20, 21, 22, 23]. In our case we will show that the iterated Julia sets for suitably chosen polynomial sequences are HNUP by showing they satisfy the stronger property of pointwise thinness.

Results

Conclusion