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Mandelbrot Cascades and Related Topics

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Geometry and Analysis of Fractals

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 88))

Abstract

This article is an extended version of the talk given by the author at the conference “Advances in fractals and related topics”, in December 2012 at the Chinese Hong-Kong University. It gathers recent advances in Mandelbrot cascades theory and related topics, namely branching random walks, directed polymers on disordered trees, multifractal analysis, and dynamical systems.

The author thanks Dr Xiong Jin for his kind help in the figures elaboration.

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Authors and Affiliations

  1. LAGA (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, Paris-Sorbonne Cité, 99 avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Julien Barral

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  1. Julien Barral
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Correspondence to Julien Barral .

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Editors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Hongkong, Hong Kong SAR

    De-Jun Feng

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong SAR

    Ka-Sing Lau

Appendix: Hausdorff Measures and Dimension

Appendix: Hausdorff Measures and Dimension

Given \(g:\mathbb {R}_+\rightarrow \mathbb {R}_+\) a continuous non-decreasing function near 0 and such that \(g(0)=0\), and \(E\) a subset of \([0,1]\), the Hausdorff measure of \(E\) with respect to the gauge function \(g\) is defined as

$$ \mathcal {H}^g(E)=\lim _{\delta \rightarrow 0^+}\inf \Big \{\sum _{i\in \mathbb {N}}g(\text {diam}(U_i))\Big \}, $$

the infimum being taken over all the countable coverings \((U_i)_{i\in \mathbb {N}}\) of \(E\) by subsets of \(K\) of diameters less than or equal to \(\delta \).

If \(s\in \mathbb {R}_+^*\) and \(g(u)=u^s\), then \({\mathcal {H}}^g(E)\) is also denoted \({\mathcal {H}}^s(E)\) and called the \(s\)-dimensional Hausdorff measure of \(E\). Then, the Hausdorff dimension of \(E\) is defined as

$$ \dim E=\sup \{s> 0: {\mathcal {H}}^s(E)=\infty \}=\inf \{s> 0: {\mathcal {H}}^s(E)=0\}, $$

with the convention \(\sup \emptyset = 0\) and \(\inf \emptyset =\infty \).

For more information the reader is referred to [Fal03, Mat95].

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Barral, J. (2014). Mandelbrot Cascades and Related Topics. In: Feng, DJ., Lau, KS. (eds) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43920-3_1

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