Hereditarily non uniformly perfect sets

Rich Stankewitz; Toshiyuki Sugawa; Hiroki Sumi

doi:10.3934/dcdss.2019150

Article Contents

2019, Volume 12, Issue 8: 2391-2402. Doi: 10.3934/dcdss.2019150

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Hereditarily non uniformly perfect sets

1.
Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA
2.
Graduate School of Information Sciences, Tohoku University, Sendai 980-8578, Japan
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: July 31, 2016

Revised: January 31, 2017

Published: December 2019

This work was partially supported by a grant from the Simons Foundation (#318239 to Rich Stankewitz). The research of the third author was partially supported by JSPS KAKENHI 24540211, 15K04899. The authors would also like to thank the referees for their helpful comments that improved the presentation of this paper

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Abstract

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give a detailed construction of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

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Mathematics Subject Classification: Primary: 31A15, 30C85, 37F35.

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Table 1. Does $ X $ imply $ Y $ when $ E \subset {\mathbb C} $ is a compact set?

	$ \dim_H E=0$	Cap $ E = 0$	$ E$ is HNUP	$ m_2(E)=0$	$ E$ is porous
$ \dim_H E=0$	$ \ast$	$ yes^1$	$ no^2$	$ no^3$	$ no^4$
Cap $ E = 0$	$ no^5$	$ \ast$	$ no^6$	$ no^7$	$ no^8$
$ E$ is HNUP	$ yes^9$	$ yes^{10}$	$ \ast$	$ no^{11}$	$ no^{12}$
$ m_2(E)=0$	$ yes^{13}$	$ yes^{14}$	$ no^{15}$	$ \ast$	$ yes^{16}$
$ E$ is porous	$ no^{17}$	$ no^{18}$	$ no^{19}$	$ no^{20}$	$ \ast$

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