Resistance Scaling on $4N$-Carpets

TL;DR

This paper establishes resistance scaling estimates for harmonic functions on $4N$-carpets, a class of fractals, revealing how energy scales with the fractal's level and impacting the understanding of Dirichlet forms on these structures.

Contribution

It introduces a resistance estimate for $4N$-carpets using a method by Barlow and Bass, advancing the analysis of energy scaling on these fractals.

Findings

Existence of a resistance scaling factor $ ho(N) > 1$

Energy of harmonic functions scales exponentially with level

Implications for Dirichlet form existence and properties

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 $\{F_n\}_{n \geq 0}$ be the natural decreasing sequence of compact pre-fractal approximations with $\cap_nF_n=F$. On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u \cdot \nabla v \, dx$ 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 $F$.