On the Π0γ-completeness and Σ0γ-completeness of multifractal decomposition sets

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Abstract

The purpose of this paper twofold. Firstly, we establish Π0γ-completeness and Σ0γ-completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the Π0γ-completeness and Σ0γ-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if μ is a self-similar measure satisfying the strong separation condition and is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of μ defined by
{x ε ℝd | lim r ↘ 0 [log μ(B(x,r))/log r = α]} are Π03-complete provided they are non-empty.
Original languageEnglish
Pages (from-to)77-114
Number of pages38
JournalMathematika
Volume64
Issue number1
Early online date6 Feb 2018
DOIs
Publication statusPublished - 2018

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