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Inhomogeneous random coverings of topological Markov shifts

Published online by Cambridge University Press:  22 June 2017

STÉPHANE SEURET
STÉPHANE SEURET*
Affiliation:
Université Paris-Est, LAMA (UMR 8050) UPEMLV, UPEC, CNRS F-94010, Créteil, France. e-mail: seuret@u-pec.fr
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Abstract

Let $\mathscr{S}$ be an irreducible topological Markov shift, and let μ be a shift-invariant Gibbs measure on $\mathscr{S}$. Let (Xn)n ≥ 1 be a sequence of i.i.d. random variables with common law μ. In this paper, we focus on the size of the covering of $\mathscr{S}$ by the balls B(Xn, n−s). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim supn →+∞B(Xn, n−s) for every s ≥ 0, which depends on s and the multifractal features of μ. Our results include the inhomogeneous covering of $\mathbb{T}^d$ and Sierpinski carpets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 341 - 357
Copyright
Copyright © Cambridge Philosophical Society 2017 

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