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Resistance scaling on 4N-carpets

Resistance scaling on 4N-carpets AbstractThe 4⁢N{4N}-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4⁢N{4N}-carpet F, let {Fn}n≥0{\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂nFn=F{\bigcap_{n}F_{n}=F}. On each Fn{F_{n}}, let ℰ⁢(u,v)=∫FN∇⁡u⋅∇⁡v⁢d⁢x{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and un{u_{n}} be the unique harmonic function on Fn{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 [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet,Proc. Roy. Soc. Lond. Ser. A431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ=ρ⁢(N)>1{\rho=\rho(N)>1} such that ℰ⁢(un,un)⁢ρn{\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Resistance scaling on 4N-carpets

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Publisher
de Gruyter
Copyright
© 2022 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2020-0330
Publisher site
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Abstract

AbstractThe 4⁢N{4N}-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4⁢N{4N}-carpet F, let {Fn}n≥0{\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂nFn=F{\bigcap_{n}F_{n}=F}. On each Fn{F_{n}}, let ℰ⁢(u,v)=∫FN∇⁡u⋅∇⁡v⁢d⁢x{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and un{u_{n}} be the unique harmonic function on Fn{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 [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet,Proc. Roy. Soc. Lond. Ser. A431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ=ρ⁢(N)>1{\rho=\rho(N)>1} such that ℰ⁢(un,un)⁢ρn{\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 2022

Keywords: Resistance; fractal; fractal carpet; Dirichlet form; walk dimension; spectral dimension; 28A80; 31C25; 31E05; 31C15; 60J65

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