Resistance Scaling on $4N$-Carpets

Abstract

The $4N$ carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a $4N$-carpet $F$, let $\{F_n{\}}_{n\ge 0}$ be the natural decreasing sequence of compact pre-fractal approximations with ${\cap }_n{F}_n=F$. On each $F_n$, let $\mathcal{E}(u,v)={\int }_{F_N}\nabla u\cdot \nabla vdx$ be the classical Dirichlet form and $u_n$ be the unique harmonic function on $F_n$ satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by Barlow and Bass (1990), we prove a resistance estimate of the following form: there is $\rho =\rho (N)>1$ such that $\mathcal{E}(u_n,u_n){\rho }^n$ is bounded above and below by positive constants independent of $n$. Such estimates have implications for the existence and scaling properties of Dirichlet forms on $F$.