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ON THE $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$-COMPLETENESS AND $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$-COMPLETENESS OF MULTIFRACTAL DECOMPOSITION SETS

Part of: Set theory Generalities Classical measure theory

Published online by Cambridge University Press:  06 February 2018

L. Olsen
L. Olsen*
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland email lo@st-and.ac.uk
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Abstract

The purpose of this paper twofold. Firstly, we establish $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$-completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$-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 $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$-completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if $\unicode[STIX]{x1D707}$ is a self-similar measure satisfying the strong separation condition and $\unicode[STIX]{x1D707}$ is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of $\unicode[STIX]{x1D707}$ defined by

$$\begin{eqnarray}\bigg\{x\in \mathbb{R}^{d}\,\bigg|\,\lim _{r{\searrow}0}\frac{\log \unicode[STIX]{x1D707}(B(x,r))}{\log r}=\unicode[STIX]{x1D6FC}\bigg\}\end{eqnarray}$$
are $\unicode[STIX]{x1D6F1}_{3}^{0}$-complete provided they are non-empty.

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 28A80: Fractals 54A05: Topological spaces and generalizations (closure spaces, etc.)
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 77 - 114
Copyright
Copyright © University College London 2018 

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