Hereditarily non uniformly perfect sets. (English) Zbl 1436.31009
A compact set \(E\subset \mathbb{R}^n\) is said to be hereditarily non-uniformly perfect if no compact subset of it is uniformly perfect. The authors investigate various properties of hereditarily non-uniformly perfect sets 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, a planar compact set is explicitely constructed which has Hausdorff dimension 2 and is hereditarily non uniformly perfect.
Reviewer: Marius Ghergu (Dublin)
MSC:
| 31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |
| 30C85 | Capacity and harmonic measure in the complex plane |
| 37F35 | Conformal densities and Hausdorff dimension for holomorphic dynamical systems |
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. · Zbl 0272.30012 |
| [2] | A. F. Beardon; C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2), 18, 475-483 (1978) · Zbl 0399.30008 · doi:10.1112/jlms/s2-18.3.475 |
| [3] | R. Broderick; L. Fishman; D. Kleinbock; A. Reich; B. Weiss, The set of badly approximable vectors is strongly \(\begin{document} { C^1} \end{document}\) incompressible, Math. Proc. Cambridge Philos. Soc., 153, 319-339 (2012) · Zbl 1316.11064 · 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. · Zbl 0587.28004 |
| [5] | K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications. · Zbl 1285.28011 |
| [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. · Zbl 0511.30022 |
| [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. · Zbl 1442.11001 |
| [9] | P. Järvi; M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. (2), 54, 515-529 (1996) · Zbl 0872.30014 · doi:10.1112/jlms/54.3.515 |
| [10] | C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20, 726-740 (2010) · Zbl 1242.11054 · doi:10.1007/s00039-010-0078-3 |
| [11] | C. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math., 32, 192-199 (1979) · Zbl 0393.30005 · 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. · Zbl 0828.31001 |
| [13] | T. Sugawa, Uniformly perfect sets: Analytic and geometric aspects [translation of Sūgaku, 53 (2001), 387-402; mr1869018], Sugaku Expositions, 16 (2003), 225-242. · Zbl 1247.30040 |
| [14] | M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. · Zbl 0087.28401 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
