Conformal dimension via p-resistance: Sierpinski carpet

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 Sierpiński 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.