Asymptotics of the quantization errors for some Markov-type measures with complete overlaps

Abstract

Let G be a directed graph with 2N vertices 1,2,…,2N. Let T=(Ti,j)(i,j)∈G be a family of contractive similitudes on Rq. We assume that Ti˜,j˜=Ti,j for every (i˜,j˜) in {(i,j),(i,j+N),(i+N,j),(i+N,j+N)}∩G. We denote by K the Mauldin-Williams fractal determined by T. Let χ be a positive probability vector and P a 2N×2N row-stochastic matrix. We denote by ν the Markov-type measure associated with χ and P. Let μ be the image measure of ν under the natural projection, which is supported on K. We consider the following two cases: 1. G has two strongly connected components consisting of N vertices; 2. G is strongly connected. With some assumptions for G and T, for case 1, we determine the quantization dimension for μ in terms of the spectral radius of a related matrix; we prove that the lower quantization coefficient is always positive and establish a necessary and sufficient condition for the upper one to be finite. For case 2, we express the quantization dimension in terms of a pressure-like function and prove that the upper and lower quantization coefficient are always positive and finite.