Measures and functions with prescribed homogeneous multifractal spectrum

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 $f(h)\le h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure, that we call Darboux theorem for multifractal spectra of measures: the support of its spectrum within $[0,1]$ must be an interval. This result is optimal, because there exists a HM measure with spectrum supported by $[0,1]\cup \{2\}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.