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)∈G and let µ be the self-affine measure associated with {f ij } (i,j)∈G and a probability vector (p ij ) (i,j)∈G . We study the asymptotics of the geometric mean error in the quantization for µ. Let s 0 be the Hausdorff dimension for µ. Assuming a separation condition for {f ij } (i,j)∈G , we prove that the nth geometric error for µ is of the same order as n −1/s 0 .2000 Mathematics Subject Classification. Primary 28A75, 28A80; Secondary 94A15.