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Inhomogeneous random coverings of topological Markov shifts


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, ns). 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, ns) 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.

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[1] Barral, J. and Seuret, S. Combining multifractal additive and multiplicative chaos. Commun. Math. Phys. 257 (2) (2005), 473497.
[2] Barral, J. and Seuret, S. Heterogeneous ubiquitous systems in and Hausdorff dimensions. Bull. Brazilian Math. Soc. 38 (3) (2007), 467515.
[3] Barral, J. and Seuret, S. Ubiquity and large intersections properties under digit frequencies constraints. Math. Proc. Camb. Phil. Soc. 145 (3) (2008), 527548.
[4] Barral, J., Ben Nasr, F. and Peyrière, J. Comparing multifractal formalisms: the neighbouring condition. Asian J. Math. 7 (2003), 149166.
[5] Barral, J. and Fan, A.H. Covering numbers of different points in Dvoretzky covering. Bull. Sci. Math. 129 (2005), 275317.
[6] Brown, G., Michon, G. and Peyrière, J. On the multifractal analysis of measures. J. Stat. Phys. 66 (1992), 775790.
[7] Collet, P. and Koukiou, F. Large deviations for multiplicative chaos. Commun. Math. Phys. 147 (1992), 329342.
[8] Collet, P., Lebowitz, J.L. and Porzio, A. The dimension spectrum of some dynamical systems. J. Stat. Phys. 47 (1987), 609644.
[9] Dodson, M., Melián, M., Pestana, D. and Velani, S. Patterson measure and Ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 3760.
[10] Dvoretzky, A. On covering the circle by randomly placed arcs. Pro. Nat. Acad. Sci. USA 42 (1956), 199203.
[11] Erdös, P. Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221254.
[12] Fan, A. H., Feng, D. J. and Wu, J. Recurrence, dimension and entropy. J. London Math. Soc. 64 (1) (2001), 229244.
[13] Fan, A. H., Schmeling, J. and Troubetzkoy, S. A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation. Proc. London Math. Soc. 107 (5) (2013), 11731219.
[14] Feng, D.J., Jarvenpäa, E., Jarvenpäa, M. and Suomala, V. Dimensions of random covering sets in Riemann manifolds. arXiv:1508.07881.
[15] Heurteaux, Y. Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), 309338.
[16] Jarvenpäa, E., Jarvenpäa, M., Koivusalo, H., Li, B. and Suomala, V. Hausdorff dimension of affine random covering sets in torus. Ann. Inst. Henri Poincaré Probab. Stat. 50 (4) (2014), 13711384.
[17] Jarvenpäa, E., Jarvenpäa, M., Koivusalo, B. Li, Suomala, V. and Xiao, Y. Hitting probabilities of random covering sets in torus and metric spaces. Preprint, arXiv: 1510.06630.
[18] Kahane, J.-P. Sur le recouvrement döun cercle par des arcs disposés au hasard C. R. Acad. Sci. Paris 248 (1956), 184186.
[19] Kahane, J.-P. Some random series of functions Camb. Stud. Adv. Math. 5 (Cambridge University Press, 1985).
[20] Li, B., Shieh, N.R. and Xiao, Y. Hitting probability and packing dimensions of the random covering sets. In: Applications of Fractals and Dynamical Systems in Science and Economics (Carfi, David, Lapidus, Michel L., Pearse, Erin P. J. and van Frankenhuijsen, Machiel, editors). Amer. math. soc. (2013).
[21] Ojala, T., Suomala, V. and Wu, M. Random cutout sets with spatially inhomogeneous intensities. Preprint 2015.
[22] Olsen, L. A multifractal formalism. Adv. Math. 116 (1995), 92195.
[23] Persson, T. A note on random coverings of Tori. Bull. London Math. Soc. 47 (1) (2015), 712.
[24] Ruelle, D. Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. Encyclopedia of Mathematics and its Applications 5 (Addison-Wesley Publishing Co., Reading, Mass., 1978).
[25] Shepp, L. Covering the circle with random arcs. Israel J. Math. 11 (1972), 328345.
[26] Tang, J. M. Random coverings of the circle with i.i.d. centers. Sci. China Math. 55 (6) (2015), 12571268.
[27] Tang, J. M. Hausdorff dimension of sets arising from Dvoretzky random covering. Acta. Mat. Sin. 57 (1) (2014).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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