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Resistance scaling on 4N-carpets

  • Claire Canner , Christopher Hayes ORCID logo EMAIL logo , Robin Huang , Michael Orwin and Luke G. Rogers
Published/Copyright: December 1, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 1

Abstract

The 4⁢N-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4⁢N-carpet F, let {Fn}n≥0 be the natural decreasing sequence of compact pre-fractal approximations with ⋂nFn=F. On each Fn, let ℰ⁢(u,v)=∫FN∇⁡u⋅∇⁡v⁢d⁢x be the classical Dirichlet form and un be the unique harmonic function on Fn 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 such that ℰ⁢(un,un)⁢ρn 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.

Keywords: Resistance; fractal; fractal carpet; Dirichlet form; walk dimension; spectral dimension
MSC 2010: 28A80; 31C25; 31E05; 31C15; 60J65

Communicated by Maria Gordina

Funding source: National Science Foundation

Award Identifier / Grant number: DMS REU 1659643

Funding statement: The research was supported by NSF DMS REU 1659643.

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Received: 2020-11-19
Revised: 2021-09-24
Published Online: 2021-12-01
Published in Print: 2022-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Contramodules over pro-perfect topological rings
  3. Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations
  4. Resistance scaling on 4N-carpets
  5. Newton polygons for L-functions of generalized Kloosterman sums
  6. Embeddings of locally compact abelian p-groups in Hawaiian groups
  7. A theorem of Roe and Strichartz on homogeneous trees
  8. Hankel determinants for starlike and convex functions associated with sigmoid functions
  9. Modular iterated integrals associated with cusp forms
  10. Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type
  11. Gompf connected sum for orbifolds and K-contact Smale–Barden manifolds
  12. Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces
https://doi.org/10.1515/forum-2020-0330
Keywords for this article
Resistance; fractal; fractal carpet; Dirichlet form; walk dimension; spectral dimension

Articles in the same Issue

  1. Frontmatter
  2. Contramodules over pro-perfect topological rings
  3. Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations
  4. Resistance scaling on 4N-carpets
  5. Newton polygons for L-functions of generalized Kloosterman sums
  6. Embeddings of locally compact abelian p-groups in Hawaiian groups
  7. A theorem of Roe and Strichartz on homogeneous trees
  8. Hankel determinants for starlike and convex functions associated with sigmoid functions
  9. Modular iterated integrals associated with cusp forms
  10. Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type
  11. Gompf connected sum for orbifolds and K-contact Smale–Barden manifolds
  12. Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces
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