Measures and functions with prescribed homogeneous multifractal spectrum

TL;DR

This paper constructs measures and functions with specific homogeneous multifractal spectra, revealing that the spectrum's support must be an interval and demonstrating the spectrum's possible shapes using wavelet theory.

Contribution

It introduces a method to construct homogeneously multifractal measures and functions with prescribed spectra, and establishes a new constraint on the spectrum's support.

Findings

Spectrum support within [0,1] must be an interval

Constructed measures with spectra supported on [0,1] ∪ {2}

Used wavelet theory to build HM functions with prescribed spectra

Abstract

In this paper we construct measures supported in $[0,1]$ with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of $[0,1]$ has the same multifractal spectrum as the whole measure. The spectra $f$ that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of $[0,1]$ and satisfy $f(h)\leq h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure: the support of its spectrum within $[0,1]$ must be an interval. This result is a sort of Darboux theorem for multifractal spectra of measures. This result is optimal, since we construct a HM measure with spectrum supported by $[0,1] \cup {2}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.