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

TL;DR

The paper investigates the quantization behavior of condensation measures $\mu$ arising from bi-Lipschitz IFS condensations $(\{f_i\}_{i=1}^N, (p_i)_{0}^N, \nu)$, where $\nu$ is the image of an ergodic measure with bounded distortion on a conformal set. By establishing a framework that couples the condensation system with a driving conformal IFS and exploiting SSC/OSC conditions, the authors derive sharp bounds for the order-$r$ quantization dimensions in terms of auxiliary dimensions $d_r$ and $l_r$ defined via the Lipschitz constants $a_i$ and $b_i$ and weights $p_i$, together with the quantization dimension $D_r(\nu)$ of the driving measure. Since $\nu$ is an image of a bounded-distortion ergodic measure, $D_r(\nu)$ exists (equal to $k_r$), yielding explicit two-sided bounds $\max\{d_r, k_r\} \le \underline{D}_r(\mu) \le \overline{D}_r(\mu) \le \max\{l_r, k_r\}$ under SSC for the driving system; these results extend prior work by PRV. The paper also provides concrete analyses for infinite discrete distributions, proving cases where the quantization dimension vanishes or equals one, and demonstrates that the quantization coefficient can fail to exist even when the dimension exists, highlighting nuanced behavior beyond the dimension alone. These findings advance understanding of quantization in fractal and condensation contexts with complex base dynamics.

Abstract

For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ 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 $μ$ tends to infinity. If $D_r(μ)$ does not exists, we call $\underline D_r(μ)$ and $\overline D_r(μ)$, the lower and upper quantization dimensions of $μ$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, ν)$, where the mappings $f_i$ are bi-Lipschitz and the measure $ν$ 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.

Theorems & Definitions (27)

• Remark 1.1
• Remark 1.2
• Remark 1.5
• Remark 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Remark 3.2