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 $\mu$ 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 $\mu$. 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.