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December  2019, 12(8): 2391-2402. doi: 10.3934/dcdss.2019150

Hereditarily non uniformly perfect sets

Rich Stankewitz 1, , Toshiyuki Sugawa 2, and  Hiroki Sumi 3,

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  August 2016 Revised  February 2017 Published  January 2019

Fund Project: 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.

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.

Keywords: Hausdorff dimension, uniformly perfect sets, capacity, porous sets.
Mathematics Subject Classification: Primary: 31A15, 30C85, 37F35.
Citation: Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2391-2402. doi: 10.3934/dcdss.2019150
References:
[1]

L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973, McGraw-Hill Series in Higher Mathematics.  Google Scholar

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A. F. Beardon and C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2), 18 (1978), 475-483.  doi: 10.1112/jlms/s2-18.3.475.  Google Scholar

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R. Broderick, L. Fishman, D. Kleinbock, A. Reich and B. Weiss, The set of badly approximable vectors is strongly $ {$C^1$} $ incompressible, Math. Proc. Cambridge Philos. Soc., 153 (2012), 319-339.  doi: 10.1017/S0305004112000242.  Google Scholar

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K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications.  Google Scholar

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N. Falkner, Mathematical review of "Construction of measure by mass distribution", J. Yeh, Real Anal. Exchange, 35 (2010), 501-507. http://www.ams.org/mathscinet-getitem?mr=2683615.  Google Scholar

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S. D. Fisher, Function Theory on Planar Domains - A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983.  Google Scholar

[8]

L. Fishman, D. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, Mem. Amer. Math. Soc., 254 (2018), v+137 pp. doi: 10.1090/memo/1215.  Google Scholar

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P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. (2), 54 (1996), 515-529.  doi: 10.1112/jlms/54.3.515.  Google Scholar

[10]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[11]

C. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.  doi: 10.1007/BF01238490.  Google Scholar

[12]

T. Ransford, Potential Theory in the Complex Plane, vol. 28 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[13]

T. Sugawa, Uniformly perfect sets: Analytic and geometric aspects [translation of Sūgaku, 53 (2001), 387-402; mr1869018], Sugaku Expositions, 16 (2003), 225-242.  Google Scholar

[14]

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973, McGraw-Hill Series in Higher Mathematics.  Google Scholar

[2]

A. F. Beardon and C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2), 18 (1978), 475-483.  doi: 10.1112/jlms/s2-18.3.475.  Google Scholar

[3]

R. Broderick, L. Fishman, D. Kleinbock, A. Reich and B. Weiss, The set of badly approximable vectors is strongly $ {$C^1$} $ incompressible, Math. Proc. Cambridge Philos. Soc., 153 (2012), 319-339.  doi: 10.1017/S0305004112000242.  Google Scholar

[4]

K. J. Falconer, The Geometry of Fractal Sets, vol. 85 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986.  Google Scholar

[5]

K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications.  Google Scholar

[6]

N. Falkner, Mathematical review of "Construction of measure by mass distribution", J. Yeh, Real Anal. Exchange, 35 (2010), 501-507. http://www.ams.org/mathscinet-getitem?mr=2683615.  Google Scholar

[7]

S. D. Fisher, Function Theory on Planar Domains - A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983.  Google Scholar

[8]

L. Fishman, D. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, Mem. Amer. Math. Soc., 254 (2018), v+137 pp. doi: 10.1090/memo/1215.  Google Scholar

[9]

P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. (2), 54 (1996), 515-529.  doi: 10.1112/jlms/54.3.515.  Google Scholar

[10]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[11]

C. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.  doi: 10.1007/BF01238490.  Google Scholar

[12]

T. Ransford, Potential Theory in the Complex Plane, vol. 28 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[13]

T. Sugawa, Uniformly perfect sets: Analytic and geometric aspects [translation of Sūgaku, 53 (2001), 387-402; mr1869018], Sugaku Expositions, 16 (2003), 225-242.  Google Scholar

[14]

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.  Google Scholar

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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$ \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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