Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems

Abstract

Let $\hat{I}½$ be a Borel probability measure on a $d$-dimensional Euclidean space ${\mathbb{R}}^d$, $d\ge 1$, with a compact support, and let $(p_0,p_1,p_2,\dots ,p_N)$ be a probability vector with $p_j>0$ for $0\le j\le N$. Let $\{S_j:1\le j\le N\}$ be a set of contractive mappings on ${\mathbb{R}}^d$. Then, a Borel probability measure $\hat{I}¼$ on ${\mathbb{R}}^d$ such that $\hat{I}¼={\sum }_{j=1}^N{p}_j\hat{I}¼\circ S_j^{-1}+p_0\hat{I}½$ is called an inhomogeneous measure, also known as a condensation measure on ${\mathbb{R}}^d$. For a given $r\in (0,+\infty )$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\hat{I}¼)$, of a Borel probability measure $\hat{I}¼$ on ${\mathbb{R}}^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $\hat{I}¼$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.