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Dirichlet forms on unconstrained Sierpinski carpets

Shiping Cao, Hua Qiu

Abstract

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be a line segment of irrational length, and the non-diagonal assumption is dropped in this recurrent setting.

Dirichlet forms on unconstrained Sierpinski carpets

Abstract

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the grids. The intersection of two cells can be a line segment of irrational length, and the non-diagonal assumption is dropped in this recurrent setting.

Paper Structure

This paper contains 9 sections, 22 theorems, 164 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geometry of unconstrained Sierpinski carpets
  3. A limit form of Kusuoka and Zhou
  4. An extension algorithm and its application to (B)
  5. Building bricks
  6. Functions with linear boundary values
  7. Verification of the estimates
  8. Self-similar forms
  9. Proof of Proposition \ref{['prop32']}

Key Result

Theorem 1

Let $K$ be a $\mathcal{USC}$ and $\mu$ be the normalized Hausdorff measure on $K$. There is a strongly local, regular, irreducible, $D_4$-symmetric, self-similar Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(K,\mu)$.

Figures (6)

  • Figure 1: Unconstrained Sierpinski carpets ($\mathcal{USC}$).
  • Figure 2: The standard Sierpinski carpet.
  • Figure 3: More unconstrained Sierpinski carpets ($\mathcal{USC}$).
  • Figure 4: A $\mathcal{USC}$ with $k=7,N=32$ and $\Psi_{25}(x)=\frac{1}{7}x+(\frac{2}{7}+\sum\limits_{l=2}^\infty 7^{-\frac{l(l+1)}{2}},\frac{1}{7})$.
  • Figure 5: A finite decomposition of $T_{1,K}$ with $n=4$.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Theorem 1
  • Definition 2.1: Unconstrained Sierpinski carpets
  • Definition 2.2
  • proof : Proof of (A1)-(A4)
  • Definition 3.1: Poincare constants KZ
  • Proposition 3.2: KZ, Theorem 2.1
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Lemma 3.5
  • ...and 41 more