The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

TL;DR

This paper investigates the quantization properties of in-homogeneous self-similar measures under certain conditions, establishing the existence and exact value of the quantization dimension and analyzing the quantization coefficients.

Contribution

It introduces a framework for quantization of in-homogeneous self-similar measures with an open set condition and determines the quantization dimension explicitly.

Findings

Existence of quantization dimension for the measure.

Explicit formula for the quantization dimension $\xi_r$.

Conditions for positivity and finiteness of quantization coefficients.

Abstract

Let $(g_i)_{i=1}^M$ be a family of contractive similitudes satisfying the open set condition. Let $\nu$ be a self-similar measure associated with $(g_i)_{i=1}^M$. We study the quantization problem for the in-homogeneous self-similar measure $\mu$ associated with a condensation system $((f_i)_{i=1}^N,(p_i)_{i=0}^N,\nu)$. Assuming a version of in-homogeneous open set condition for this system, we prove the existence of the quantization dimension for $\mu$ of order $r\in(0,\infty)$ and determine its exact value $\xi_r$. We give sufficient conditions for the $\xi_r$-dimensional upper and lower quantization coefficient to be positive or finite.