Quantization dimension and stability for infinite self-similar measures with respect to geometric mean error

TL;DR

This paper extends the understanding of quantization dimension for infinite self-similar measures, showing it equals the Hausdorff dimension and analyzing its stability, which differs from finite systems.

Contribution

It establishes that the quantization dimension of infinite self-similar measures equals their Hausdorff dimension, extending finite case results to infinite systems.

Findings

Quantization dimension equals Hausdorff dimension for infinite measures.

Many finite-system tools fail in the infinite case.

Quantization dimension stability is analyzed for infinite systems.

Abstract

Let $\mu$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization dimension of the measure $\mu$ with respect to the geometric mean error. The quantization for infinite systems is different from the well-known finite case investigated by Graf and Luschgy. That is, many tools which are used in the finite setting, for example, existence of finite maximal antichains, fail in the infinite case. We prove that the quantization dimension of the measure $\mu$ is equal to its Hausdorff dimension which extends a well-known result of Graf and Luschgy for the finite case to an infinite setting. In the last section, we discuss the stability of quantization dimension for infinite systems.