Stephane Seuret: Inhomogeneous coverings of topological Markov shifts

Let \(S\) be an irreducible topological Markov shift, and let \(\mu\) be a shift-invariant Gibbs measure on \(S\). Let \((X_n)_{n\geq 1}\) be a sequence of i.i.d. random variables with common law \(\mu\). We are interested in the size of the set of points which are covered infinitely many times by the balls \(B(X_n, n^{-s})\). This generalizes the original Dvoretzky problem by considering random coverings of fractal sets by non homogeneously distributed balls. We compute the almost sure dimension of \(\limsup_{n\to +\infty} B(X_n, n^{-s})\) for every \(s\), which depends on \(s\) and the multifractal features of \(\mu\). Our results include the inhomogeneous covering of \(\zu^d\) and Sierpinski carpets.