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. 1 arXiv:1811.05075v1 [math.DS]

Some Fractal Properties of Sets Having the Moran Structure

“…Possible applications in the spectral theory of self-adjoint operators serve as an additional stimulus for a further investigation of singularly continuous measures [6]. For example, one can note the following researches of singular measures: singularity of Hewitt-Stromberg measures on Bedford-McMullen carpets [2], the mutual singularity of certain measures (see [6,8,24,46,47] and references therein), dimensions of measures [13,20,23].…”

“…Olsen [17] introduced a general form of multifractal formalism to interpret the statistical scaling properties of singular measures where the total mass or energy is spread over regions of phase space in an irregular way. The multifractal formalism aims at expressing the dimensions (the Hausdorff and packing dimensions) of the level sets in terms of the Legendre transform of some free energy function in analogy with the usual thermodynamic theory ( [1,23,47] and references therein).…”