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

Let $\mathcal{G}$ be a directed graph with vertices $1,2,\ldots, 2N$. Let $\mathcal{T}=(T_{i,j})_{(i,j)\in\mathcal{G}}$ be a family of contractive similitudes. For every $1\leq i\leq N$, let $i^+:=i+N$. For $1\leq i,j\leq N$, we define $\mathcal{M}_{i,j}=\{(i,j),(i,j^+),(i^+,j),(i^+,j^+)\}\cap\mathcal{G}$. We assume that $T_{\widetilde{i},\widetilde{j}}=T_{i,j}$ for every $(\widetilde{i},\widetilde{j})\in \mathcal{M}_{i,j}$. Let $K$ denote the Mauldin-Williams fractal determined by $\mathcal{T}$. Let $\chi=(\chi_i)_{i=1}^{2N}$ be a positive probability vector and $P$ a row-stochastic matrix which serves as an incidence matrix for $\mathcal{G}$. We denote by $\nu$ the Markov-type measure associated with $\chi$ and $P$. Let $\Omega=\{1,\ldots,2N\}$ and $G_\infty=\{\sigma\in\Omega^{\mathbb{N}}:(\sigma_i,\sigma_{i+1})\in\mathcal{G}, \;i\geq 1\}$. Let $\pi$ be the natural projection from $G_\infty$ to $K$ and $\mu=\nu\circ\pi^{-1}$. We consider the following two cases: 1. $\mathcal{G}$ has two strongly connected components consisting of $N$ vertices; 2. $\mathcal{G}$ is strongly connected. With some assumptions for $\mathcal{G}$ and $\mathcal{T}$, for case 1, we determine the exact value $s_r$ of the quantization dimension $D_r(\mu)$ for $\mu$ and prove that the $s_r$-dimensional lower quantization coefficient is always positive, but the upper one can be infinite; we establish a necessary and sufficient condition for the upper quantization coefficient for $\mu$ to be finite; for case 2, we determine $D_r(\mu)$ in terms of a pressure-like function and prove that $D_r(\mu)$-dimensional upper and lower quantization coefficient are both positive and finite.