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Convergence of resistances on generalized {S}ierpiński carpets

Shiping Cao, Zhen-Qing Chen

Abstract

We positively answer the open question of Barlow and Bass about the convergence of renormalized effective resistance between opposite faces of Euclidean domains approximating a generalized {S}ierpiński carpet.

Convergence of resistances on generalized {S}ierpiński carpets

Abstract

We positively answer the open question of Barlow and Bass about the convergence of renormalized effective resistance between opposite faces of Euclidean domains approximating a generalized {S}ierpiński carpet.
Paper Structure (17 sections, 37 theorems, 254 equations, 5 figures)

This paper contains 17 sections, 37 theorems, 254 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Convergence of functions and forms
  3. Objects related to GSC
  4. Convergence of functions
  5. Convergence of quadratic forms
  6. Dirichlet forms on GSCs
  7. Uniform elliptic Harnack inequality (EHI)
  8. Effective Resistances
  9. Mosco convergence
  10. Energy measures
  11. Harmonic functions with assigned mean boundary values
  12. Good approximation sequence of harmonic functions
  13. A finite dimensional kernel
  14. Proof of Proposition \ref{['prop41']}
  15. Convergence of resistances
  16. ...and 2 more sections

Key Result

Theorem 1.1

There is a constant $c_0>0$ so that for each $x\in F$ and $x_n\in F_n,n\geq 0$ such that $x_n\to x$ as $n\to\infty$, the law of $(X^{(F_n)}_t)_{t\geq 0}$ starting from $x_n$ converges weakly to some conservative continuous Markov process $(X^{(F)}_{t/c_0 })_{t\geq 0}$ starting from $x$ as $n\to \inf

Figures (5)

  • Figure 1: The standard Sierpiński carpet in ${\mathbbm R}^2$
  • Figure 2: Approximating domains $F_0$, $F_1$ and $F_2$ of the standard Sierpiński carpet
  • Figure 3: The Sierpiński sponge in ${\mathbbm R}^3$
  • Figure 4: $F_{\mathcal{B}_1(F)}$, $F_{\mathcal{B}_2(F)}$ and $F_{\mathcal{B}_3(F)}$ of the standard Sierpiński carpet $F$ in ${\mathbbm R}^2$
  • Figure 5: An illustration of $\operatorname{supp} [\bar{u}_A]$ for $A\in \eth_kF$ with $k=1,2,3$: $\operatorname{supp} [\bar{u}_A]$ is contained in the red area, while the blue area is $F_{\mathcal{B}_{k+2}(F)}\cup (F\setminus F_{\mathcal{B}_k-1(F)})$

Theorems & Definitions (83)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • ...and 73 more