Inhomogeneous random coverings of topological Markov shifts

Let y be an irreducible topological Markov shift, and let mu be a shift-invariant Gibbs measure on y. Let (X-n)(n >= 1) be a sequence of i.i.d. random variables with common law mu. In this paper, we focus on the size of the covering of y by the balls B(X-n,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 sup(n ->+infinity) B(X-n, n(-s)) for every s >= 0, which depends on s and the multifractal features of mu. Our results include the inhomogeneous covering of T-d and Sierpinski carpets.