Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system

Abstract

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 how fast 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. If $D_r(\hat{I}¼)$ does not exists, we call ${\underline{D}}_r(\hat{I}¼)$ and ${\overline{D}}_r(\hat{I}¼)$, the lower and upper quantization dimensions of $\hat{I}¼$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i{\}}_{i=1}^N,(p_i{)}_{i=0}^N,\hat{I}½)$, where the mappings $f_i$ are bi-Lipschitz and the measure $\hat{I}½$ is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient.