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Inhomogeneous random coverings of topological Markov shifts
Published online by Cambridge University Press: 22 June 2017
- STÉPHANE SEURET*
- Affiliation:Université Paris-Est, LAMA (UMR 8050) UPEMLV, UPEC, CNRS F-94010, Créteil, France. e-mail: seuret@u-pec.fr
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 ballsB(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 everys ≥ 0, which depends ons 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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