Homogeneous Multifractal Measures with Disjoint Spectrum and Monohölder Monotone Functions

Abstract

We proved in an earlier paper that the support of the multifractal spectrum of a homogeneously multifractal (HM) measure within $[0,1]$ must be an interval. In this paper we construct a homogeneously multifractal measure with spectrum supported by $[0,1] \cup \{ 2\}$. This shows that there can be a different behaviour for exponents exceeding one.

We also provide details of the construction of a strictly monotone increasing monohölder (and hence HM) function which has exact Hölder exponent one at each point. This function was also used in our paper about measures and functions with prescribed homogeneous multifractal spectrum.