The quantization for Markov-type measures on a class of ratio-specified graph directed fractals

TL;DR

This paper investigates the asymptotic quantization error of Markov-type measures on ratio-specified graph directed fractals, establishing the existence and exact value of the quantization dimension and conditions for finiteness of quantization coefficients.

Contribution

It determines the quantization dimension for these measures and provides a necessary and sufficient condition for the finiteness of the upper quantization coefficient.

Findings

Quantization dimension exists and equals a value $s_{r}$

The $s_{r}$-dimensional lower quantization coefficient is positive

A condition for the finiteness of the $s_{r}$-dimensional upper quantization coefficient

Abstract

We study the asymptotic quantization error of order $r$ for Markov-type measures $\mu$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $\mu$ exists and determine its exact value $s_{r}$ in terms of spectral radius of a related matrix. We prove that the $s_{r}$-dimensional lower quantization coefficient of $\mu$ is always positive. Moreover, inspired by Mauldin-Williams's work on the Hausdorff measure of graph directed fractals, we establish a necessary and sufficient condition for the $s_{r}$-dimensional upper quantization coefficient of $\mu$ to be finite.