Local multifractal analysis in metric spaces

TL;DR

This paper investigates local multifractal properties of measures in doubling metric spaces, revealing the abundance of multifractal measures under mild conditions and introducing a local spectrum for finer analysis.

Contribution

It demonstrates the existence of many multifractal measures in metric spaces and develops a local spectrum for detailed local measure analysis.

Findings

Many multifractal measures exist under mild regularity conditions

Introduction of a local spectrum for finer local measure analysis

Enhanced understanding of local measure behavior in metric spaces

Abstract

We study the local dimensions and local multifractal properties of measures on doubling metric spaces. Our aim is twofold. On one hand, we show that there are plenty of multifractal type measures in all metric spaces which satisfy only mild regularity conditions. On the other hand, we consider a local spectrum that can be used to gain finer information on the local behaviour of measures than its global counterpart.