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Stable sets of planar homeomorphisms with translation pseudo-arcs
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 |
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.
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. |
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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. |
[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. |
[4] |
K. J. Falconer, The Geometry of Fractal Sets, vol. 85 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986. |
[5] |
K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications. |
[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. |
[7] |
S. D. Fisher, Function Theory on Planar Domains - A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983. |
[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. |
[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. |
[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. |
[11] |
C. Pommerenke,
Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.
doi: 10.1007/BF01238490. |
[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. |
[13] |
T. Sugawa, Uniformly perfect sets: Analytic and geometric aspects [translation of Sūgaku, 53 (2001), 387-402; mr1869018], Sugaku Expositions, 16 (2003), 225-242. |
[14] |
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. |
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. |
[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. |
[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. |
[4] |
K. J. Falconer, The Geometry of Fractal Sets, vol. 85 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986. |
[5] |
K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications. |
[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. |
[7] |
S. D. Fisher, Function Theory on Planar Domains - A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983. |
[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. |
[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. |
[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. |
[11] |
C. Pommerenke,
Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.
doi: 10.1007/BF01238490. |
[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. |
[13] |
T. Sugawa, Uniformly perfect sets: Analytic and geometric aspects [translation of Sūgaku, 53 (2001), 387-402; mr1869018], Sugaku Expositions, 16 (2003), 225-242. |
[14] |
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. |
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