A study on Quantization Dimension in complete metric spaces

TL;DR

This paper extends the theory of quantization dimension for invariant measures from Euclidean spaces to complete metric spaces, analyzing properties and continuity of the quantization dimension.

Contribution

It generalizes known results on quantization dimension of self-similar measures to the setting of complete metric spaces, including the study of its continuity.

Findings

Quantization dimension theory is extended to complete metric spaces.

Continuity of quantization dimension is established.

Results generalize Euclidean space findings to broader metric spaces.

Abstract

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a complete metric space. This generalizes the known results on quantization dimension of self-similar measures in the Euclidean space to a complete metric space. In the last part, continuity of quantization dimension is discussed.