Abstract
In the present work, we study the mutual singularity of multifractal Hausdorff and packing measures which provide a positive answer to Olsen’s questions in a more general framework. Our main results apply to a family of measures supported by the full 5-adic grid of [0, 1], namely the quasi-Bernoulli measures.
Highlights
Introduction and statements of resultsFor a long time, the interest of mathematicians in singularly continuous measures and probability distributions was fairly weak, which can be explained, on the one hand, by the absence of adequate analytic apparatus for specification and investigation of these measures, and, on the other hand, by a widespread opinion about the absence of applications of these measures.Due to the fractal explosion and a deep connection between the theory of fractals and singular measures, the situation has radically changed in the last years
Possible applications in the spectral theory of self-adjoint operators serve as an additional stimulus for a further investigation of singularly continuous measures
The purpose of this paper is to show that the multifractal Hausdorff and packing measures are mutually singular
Summary
The interest of mathematicians in singularly continuous measures and probability distributions was fairly weak, which can be explained, on the one hand, by the absence of adequate analytic apparatus for specification and investigation of these measures, and, on the other hand, by a widespread opinion about the absence of applications of these measures. The function q,t μ is not necessarily countably subadditive, the set function Hqμ,t is not necessarily monotone For these reasons, Olsen introduced the packing and Hausdorff measures denoted respectively by Pμq,t and Hμq,t are defined as following. One of the main importance of the multifractal measures Hμq,t and Pμq,t, and the corresponding dimension functions bμ, Bμ, and Λμ is due to the fact that the multifractal spectra functions fμ and Fμ are bounded above by the Legendre transforms of bμ and Bμ, respectively, i.e., fμ := dimH (E(α)) ≤ b∗μ(α) and Fμ := dimP (E(α)) ≤ Bμ∗(α) for all α ≥ 0 Such a minoration is related to the existence of an auxiliary measure which is supported by the set to be analyzed. The results of Theorems 1 and 2 hold if we replace the multifractal function Bμ by the function Λμ
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