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 $\hat{I}¼$ 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 $\hat{I}¼$. Let $s_0$ be the Hausdorff dimension for $\hat{I}¼$. Assuming a separation condition for $\{f_{ij}{\}}_{(i,j)\in G}$, we prove that the $n$th geometric error for $\hat{I}¼$ is of the same order as $n^{-1/s_0}$.