Measures and functions with prescribed homogeneous multifractal spectrum

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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)≤ h for all h∈ [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] ∪ 2. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.