Resistance scaling on 4N-carpets

Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with large group symmetries. For -carpet F , let { F n stretchy="false">} ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence compact pre-fractal approximations largeop="true" mathsize="160%" stretchy="false" symmetric="true">⋂ = {\bigcap_{n}F_{n}=F} . On each {F_{n}} ℰ stretchy="false">( u , v stretchy="false">) symmetric="true">∫ ∇ ⁡ ⋅ d x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} classical Dirichlet form and {u_{n}} unique harmonic function on satisfying mixed boundary value problem corresponding to assigning constant potential between two specific subsets boundary. Using method introduced by [M. T. Barlow R. F. Bass, resistance Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove estimate following form: there is ρ > 1 {\rho=\rho(N)>1} such that {\mathcal{E}(u_{n},u_{n})\rho^{n}} bounded above below constants independent n Such estimates have implications for existence scaling properties Brownian motion