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Whether $p$-conductive homogeneity holds depends on $p$

Shiping Cao, Zhen-Qing Chen

Abstract

We introduce two fractals, in Euclidean spaces of dimension two and three respectively, such the $2$-conductive homogeneity holds but there is some $\eps \in (0, 1)$ so that the $p$-conductive homogeneity fails for every $p\in (1, 1+\eps)$. In addition, these two fractals have Ahlfors regular conformal dimension within the interval $(1, 2)$ and $(2, 3)$, respectively.

Whether $p$-conductive homogeneity holds depends on $p$

Abstract

We introduce two fractals, in Euclidean spaces of dimension two and three respectively, such the -conductive homogeneity holds but there is some so that the -conductive homogeneity fails for every . In addition, these two fractals have Ahlfors regular conformal dimension within the interval and , respectively.
Paper Structure (2 sections, 6 theorems, 37 equations, 4 figures)

This paper contains 2 sections, 6 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $p$-conductive homogeneity

Key Result

Lemma 1.1

For $p>1$, if $F^{(d)}$ is $p$-conductive homogeneous with respect to some covering system in the sense of Ki2, then the following holds. ($\textbf{A}_p$). There exist some positive constants $\sigma >0$ and $c_1, c_2 >0$ so that for each $n\geq 1,m\geq 0$ and $Q\in \mathcal{Q}^{(d)}_n(F^{(d)})$, where $\Gamma(Q):=\{Q'\in \mathcal{Q}^{(d)}_n(F^{(d)}):\ Q'\cap Q\neq\emptyset\}$ and $\Gamma(Q)^c:=\

Figures (4)

  • Figure 1: $F_1^{(2)}$ and $F^{(2)}$
  • Figure 2: The generalized carpet $F$ and its first level approximating $F_1$.
  • Figure 3: $Q_1$ marked red
  • Figure 4: $Q_2$ marked red

Theorems & Definitions (15)

  • Lemma 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7
  • Lemma 2.1
  • proof
  • ...and 5 more