On the Π0γ-completeness and Σ0γ-completeness of multifractal decomposition sets
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.
Citation
Olsen , L O R 2018 , ' On the Π 0 γ -completeness and Σ 0 γ -completeness of multifractal decomposition sets ' , Mathematika , vol. 64 , no. 1 , pp. 77-114 . https://doi.org/10.1112/S0025579317000365
Publication
Mathematika
Status
Peer reviewed
ISSN
0025-5793Type
Journal article
Collections
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