Asymptotic order of the quantization errors for a class of self-affine measures

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 $n$th quantization error for $\mu$ of order $r$ is of the same order as $n^{-\frac {1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.