TY - JOUR
T1 - Hereditarily non uniformly perfect sets
AU - Stankewitz, Rich
AU - Sugawa, Toshiyuki
AU - Sumi, Hiroki
N1 - Funding Information:
2010 Mathematics Subject Classification. Primary: 31A15, 30C85, 37F35. Key words and phrases. Hausdorff dimension, uniformly perfect sets, capacity, porous sets. 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.
Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Capacity
KW - Hausdorff dimension
KW - Porous sets
KW - Uniformly perfect sets
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U2 - 10.3934/dcdss.2019150
DO - 10.3934/dcdss.2019150
M3 - Article
AN - SCOPUS:85074911833
SN - 1937-1632
VL - 13
SP - 2391
EP - 2402
JO - Discrete and Continuous Dynamical Systems - Series S
JF - Discrete and Continuous Dynamical Systems - Series S
IS - 2
ER -