Convergence order of the geometric mean errors for Markov-type measures

We study the quantization problem with respect to the geometric mean error for Markov-type measures μ on a class of fractal sets. Assuming the irreducibility of the corresponding transition matrix P, we determine the exact convergence order of the geometric mean errors of μ. In particular, we show that, the quantization dimension of order zero is independent of the initial probability vector when P is irreducible, while this is not true if P is reducible.