Multifractal spectra of Moran measures without local dimension

Abstract

A measure without local dimension is a measure such that local dimension does not exist for any point in its support. In this paper, we construct such a class of Moran measures and study their lower and upper local dimensions. We show that the related "free energy" function ($L^q$-spectrum) does not exist. Nevertheless, we can obtain the full Hausdroff and packing dimension spectra for level sets defined by lower and upper local dimensions. They can be viewed as a generalized multifractal formalism.

References in corpus (1)

• Phase transitions for the multifractal analysis of self-similar measures

Cited by in corpus (2)

• A review on multifractal analysis of Hewitt-Stromberg measures
• On one map between singular expansions