Resistance Scaling on \(4N\)-Carpets

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)...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Authors: Canner, Claire, Hayes, Christopher, Huang, Shinyu, Orwin, Michael, Rogers, Luke G
Format: Article
Language:English
Subjects:
Boundary value problems
Carpets
Dirichlet problem
Fractals
Harmonic functions
Self-similarity
Online Access:Get full text
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Summary: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 \(F\).
ISSN:2331-8422