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

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.

References in corpus (1)

• The local quantization behavior of absolutely continuous probabilities

Cited by in corpus (2)

• Quantization dimensions of compactly supported probability measures via Rényi dimensions
• Quantization dimensions of negative order