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Asymptotic order of the quantization errors for self-affine measures on Bedford-McMullen carpets

Sanguo Zhu

Abstract

Let $E$ be a Bedford-McMullen carpet determined by a set of affine mappings $(f_{ij})_{(i,j)\in G}$ and $μ$ a self-affine measure on $E$ associated with a probability vector $(p_{ij})_{(i,j)\in G}$. We prove that, for every $r\in(0,\infty)$, the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension $s_r$. As a consequence, the $k$th quantization error for $μ$ of order $r$ is of the same order as $k^{-\frac{1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.

Asymptotic order of the quantization errors for self-affine measures on Bedford-McMullen carpets

Abstract

Let be a Bedford-McMullen carpet determined by a set of affine mappings and a self-affine measure on associated with a probability vector . We prove that, for every , the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension . As a consequence, the th quantization error for of order is of the same order as . In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.

Paper Structure

This paper contains 4 sections, 5 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The finiteness of the upper quantization coefficient for $\mu$
  4. The positivity of the lower quantization coefficient for $\mu$

Key Result

Theorem \oldthetheorem

Let $\mu$ be the self-affine measure as defined in (selfaffinemeas). Then for every $r\in(0,\infty)$ we have $0<\underline{Q}_{r}^{s_{r}}(\mu)\leq\overline{Q}_{r}^{s_{r}}(\mu)<\infty$.

Theorems & Definitions (11)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 1 more