Regarding the set-theoretic complexity of the general fractal dimensions and measures maps
Bilel Selmi, Haythem Zyoudi
Abstract
Abstract Let ν be a Borel probability measure on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> {\mathbb{R}^{d}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {q,t\in\mathbb{R}} . This study takes a broad approach to the multifractal and fractal analysis problem and proposes an intrinsic definition of the general Hausdorff and packing measures by taking into account sums of the type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mi>i</m:mi> </m:munder> <m:mrow> <m:msup> <m:mi>h</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo></m:mo> <m:mi>h</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ν</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>B</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo></m:mo> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \sum_{i}h^{-1}(qh(\nu(B(x_{i},r_{i})))+tg(r_{i})) for some prescribed functions h and g . The aim of this paper is to study the descriptive set-theoretic complexity and measurability of these measures and related dimension maps.