Asymptotic quantization errors for in-homogeneous self-similar measures supported on self-similar sets

TL;DR

This paper investigates the asymptotic behavior of quantization errors for in-homogeneous self-similar measures on self-similar sets, establishing the quantization dimension and conditions for quantization coefficient finiteness.

Contribution

It proves the existence and exact value of the quantization dimension for such measures and provides conditions for the finiteness of the upper quantization coefficient.

Findings

Quantization dimension exists and is explicitly determined.

Lower quantization coefficient is always positive.

Upper quantization coefficient can be infinite, with conditions for finiteness provided.

Abstract

We study the quantization for a class of in-homogeneous self-similar measures $\mu$ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for $\mu$ of order $r\in(0,\infty)$ and determine its exact value $\xi_r$. Furthermore, we show that, the $\xi_r$-dimensional lower quantization coefficient for $\mu$ is always positive and the upper one can be infinite. We also give a sufficient condition to ensure the finiteness of the upper quantization coefficient.