Resistance scaling on 4<i>N</i>-carpets

Abstract The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {4N} -carpet F , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo largeop="true" mathsize="160%" stretchy="false" symmetric="true">⋂</m:mo> <m:mi>n</m:mi> </m:msub> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> {\bigcap_{n}F_{n}=F} . On each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {F_{n}} , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">ℰ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>N</m:mi> </m:msub> </m:msub> <m:mrow> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mpadded width="+1.7pt"> <m:mi>v</m:mi> </m:mpadded> <m:mo>⁢</m:mo> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {u_{n}} be the unique harmonic function on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ρ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>N</m:mi> <m:mo stretchy="false">)</m: