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

Abstract

Let $E$ be a Bedford-McMullen carpet determined by a set of affine mappings $(f_{ij}{)}_{(i,j)\in G}$ and $\mu$ 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 $\mu$ 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.

References in corpus (2)

• Some recent developments in quantization of fractal measures
• On the quantization for self-affine measures on Bedford-McMullen carpets

Cited by in corpus (5)

• Quantization dimensions of compactly supported probability measures via Rényi dimensions
• Quantization dimensions of negative order
• Asymptotic order of the quantization error for a class of self-similar measures with overlaps
• Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets
• Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets