Measurability of Multifractal Topological Entropy and Its Role in Multifractal Theory

TL;DR

This paper investigates the properties and measurability of multifractal topological entropy definitions, revealing their relationship with multifractal spectra and enhancing tools for multifractal analysis.

Contribution

It introduces and systematically analyzes the $(q, heta)$-Bowen and packing topological entropies, linking them to multifractal spectra and expanding theoretical understanding.

Findings

$(q, heta)$-packing entropy domain includes multifractal spectrum

Established measurability of multifractal topological entropies

Connected entropy definitions with multifractal analysis tools

Abstract

In this paper, we consider definitions including $(q, \vartheta)$-Bowen topological entropy and $(q, \vartheta)$-packing topological entropy. We systematically explore their properties and measurability and analyze the relationship between $(q, \vartheta)$-packing topological entropy and topological entropy on level sets. Furthermore, the study demonstrates that the domain of $(q, \vartheta)$-packing topological entropy encompasses the domain of the multifractal spectrum of local entropies, providing new perspectives and tools for multifractal analysis.