# American Institute of Mathematical Sciences

January  2020, 40(1): 33-46. doi: 10.3934/dcds.2020002

## Hereditarily non uniformly perfect non-autonomous Julia sets

 1 Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA 2 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA 3 Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan

Received  October 2018 Revised  June 2019 Published  October 2019

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] 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 Urbánski [15].

Citation: Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002
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How the survival sets ${\mathcal S}_k$ are nested. The pictures show preimages of $\overline {\mathrm D}(0,2)$ at stages $M_k$ (in red) and $M_{k-1}$ (in blue) with $m_k = 3$. The dashed blue circle is ${\mathrm C}(0,2)$ while the unit circle is shown in black. Observe how $Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1)$ as in Remark 1(c) is shown in red at Stage $M_{k-1}$
Schematic for the proof of Theorem 1.6 in the case where $\limsup |c_k| = +\infty$. Note how the round annulus ${\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|})$ at stage $M_{k-1} + m_k$ (in this case $M_1+m_2$) is pulled back conformally first by the preimage branches of $Q_{M_{k-1},M_{k-1}+m_k}$ to form half the members of the collection $\mathcal C$ at Stage $M_1$. Then the preimage branches of $Q_{M_{k-1}}$ pull back the annuli in $\mathcal C$ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $\mathcal{S}_k$ at stage $0$
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