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
Rights
© 2018, University College London. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1112/S0025579317000365
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