# Measures and functions with prescribed homogeneous multifractal spectrum

### (Eötvös Loránd University)

**Event:** ERC Workshop on Geometric Measure Theory, Analysis in Metric Spaces and Real Analysis

**Date:** Oct 08, 2013,
**time:** 09:30

**Abstract.** This is a joint work with St\'ephane Seuret.

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 \( { [0,1] }\cup \{ 2 \}\).