Measures and functions with prescribed homogeneous multifractal spectrum
Zoltán Buzcolich
Eötvös University, Budapest, HungaryStéphane Seuret
Université Paris-Est Créteil, France
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Abstract
In this paper we construct measures supported in with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of has the same multifractal spectrum as the whole measure. The spectra that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of and satisfy for all . 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 must be an interval. This result is optimal, because there exists a HM measure with spectrum supported by . Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.
Cite this article
Zoltán Buzcolich, Stéphane Seuret, Measures and functions with prescribed homogeneous multifractal spectrum. J. Fractal Geom. 1 (2014), no. 3, pp. 295–333
DOI 10.4171/JFG/9