Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets

Abstract

Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings $\{f_{ij}\}_{(i,j)\in G}$ and let $\mu$ be the self-affine measure associated with $\{f_{ij}\}_{(i,j)\in G}$ and a probability vector $(p_{ij})_{(i,j)\in G}$. We study the asymptotics of the geometric mean error in the quantization for $\mu$. Let $s_0$ be the Hausdorff dimension for $\mu$. Assuming a separation condition for $\{f_{ij}\}_{(i,j)\in G}$, we prove that the $n$th geometric error for $\mu$ is of the same order as $n^{-1/s_0}$.