Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 3-51
Montana State University, Department of Mathematical Sciences
Bozeman MT 59717-2400, U.S.A.; jarek 'at' math.montana.edu
Abstract. We put forth the notion of p-resistance as a proxy for the combinatorial p-modulus and demonstrate its effectiveness by studying the (Ahlfors regular) conformal dimension of the Sierpinski carpet. Specifically, we construct large resistor network approximating the carpet, establish weak-sup and sub-multiplicativity of their p-resistances, identify the conformal dimension as the associated critical exponent, and provide numerical approximations and rigorous two-sided bounds. In particular, we prove that the conformal dimension of the carpet exceeds 1 + ln 2/ln 3, the Hausdorff dimension of the Cantor comb contained therein. A conjectural construction (and a numerical picture) of the quasi-symmetric uniformization of the carpet emerges as a byproduct.
2010 Mathematics Subject Classification: Primary 28A80, 30L10, 30C75, 31B15, 65E05.
Key words: Ahlfors regular, conformal dimension, p-resistance, p-extremal length, quasi-symmetric uniformization.
Reference to this article: J. Kwapisz: Conformal dimension via p-resistance: Sierpinski carpet. Ann. Acad. Sci. Fenn. Math. 45 (2020), 3-51.
https://doi.org/10.5186/aasfm.2020.4515
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