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

Abstract

Let $\mathcal{G}$ be a directed graph with vertices $1,2,\dots ,2N$. Let $\mathcal{T}=(T_{i,j}{)}_{(i,j)\in \mathcal{G}}$ be a family of contractive similitudes. For every $1\le i\le N$, let $i^{+}:=i+N$. For $1\le i,j\le N$, we define ${\mathcal{M}}_{i,j}=\{(i,j),(i,j^{+}),(i^{+},j),(i^{+},j^{+})\}\cap \mathcal{G}$. We assume that $T_{\tilde{i},\tilde{j}}=T_{i,j}$ for every $(\tilde{i},\tilde{j})\in {\mathcal{M}}_{i,j}$. 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 $\Omega =\{1,\dots ,2N\}$ and $G_{\infty }=\{\sigma \in {\Omega }^{\mathbb{N}}:({\sigma }_i,{\sigma }_{i+1})\in \mathcal{G},i\ge 1\}$. 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.

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