# Measures and functions with prescribed homogeneous multifractal spectrum

## Zoltan Buczolich

### (Eötvös Loránd University)

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 ${ [0,1] }\cup \{ 2 \}$.