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 {Fn} n≥0 be the natural decreasing sequence of compact pre-fractal approximations with ∩nFn = F . On each Fn, let E(u, v) = F N ∇u • ∇vdx be the classical Dirichlet form and un be the unique harmonic function on Fn 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 [2], we prove a resistance estimate of the following form: there is ρ = ρ(N ) > 1 such that E(un, un)ρ n is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Dirichlet forms on F .