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arXiv · 2011.10662

Resistance Scaling on $4N$-Carpets

Claire Canner·Christopher Hayes·Shinyu Huang·Michael Orwin·Luke G. Rogers
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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 $F$.

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BibTeXRIS

Claire Canner, Christopher Hayes, Shinyu Huang, Michael Orwin, Luke G. Rogers. 2020-11-10. Resistance Scaling on $4N$-Carpets. https://arxiv.org/abs/2011.10662

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