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

Abstract

Let E E be a Bedford-McMullen carpet determined by a set of affine mappings ( f i j ) ( i , j ) ∈ G (f_{ij})_{(i,j)\in G} and μ \mu a self-affine measure on E E associated with a probability vector ( p i j ) ( i , j ) ∈ G (p_{ij})_{(i,j)\in G} . We prove that, for every r ∈ ( 0 , ∞ ) r\in (0,\infty ) , the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension s r s_r . As a consequence, the n n th quantization error for μ \mu of order r r is of the same order as n − 1 s r 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.