On the mutual singularity of multifractal measures

Abstract

The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.

The mutual singularity of the relative multifractal measures

“…1. We note that our results, due to the use of the relative multifractal Hausdor and packing measures introduced in [6], appear as natural multifractal generalizations of some of the main results in [11,12,15,23].…”

“…The interesting case is, of course, the case where the measure ν is di erent from the normalized Lebesgue measure L n on an open and bounded set containing the support of µ. If ν is the normalized Lebesgue measure L n then our main results follow immediately from the (substantially more general) theorems in [11,12,15,23] (provided that certain conditions are satis ed). 3.…”

“…Possible applications in the spectral theory of self-adjoint operators serve as an additional stimulus for a further investigation of singularly continuous measures. The authors in [11,12,15,23] provided some examples of the mutual singularity of multifractal Hausdor and packing measures for graph directed self-similar measures in R n with totally disconnected support, cookie-cutter measures [15], for some homogeneous Moran measures [11,12], in the spacial case where ν is the Lebesgue measure L n . Also, in [8,9], M. Das proved that the relative multifractal Hausdor and packing measures are mutually singular for the self-similar measures satisfying the signi cantly weaker open set condition.…”

Remarks on the mutual singularity of multifractal measures

“…1. Our main results have been proved rigorously for graph directed self-similar measures in R n with totally disconnected support, cookie-cutter measures [10], for some homogeneous Moran measures [7,8] and for self-similar measures satisfying the significantly weaker open set condition [5,6]. Also, these results hold for Mandelbrot measures (see [3] for the definition).…”

“…to conclude that dim H (E(α)) = dim P (E(α)) = b * µ (α) = B * µ (α), where α = −B 0 µ (q). The authors in [7,8,10] provided some examples of the mutual singularity of multifractal Hausdorff and packing measures for graph directed self-similar measures in R n with totally disconnected support, cookie-cutter measures [10], for some homogeneous Moran measures [7,8] and for selfsimilar measures satisfying the significantly weaker open set condition [5,6]. The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular and provides a positive answer to Olsen's questions [10, Questions 7.1 and 7.2] in a more general setting.…”