Seminar

# Inhomogeneous coverings of topological Markov shifts

#### Fractal Geometry and Dynamics

#### 17 October 15:00 - 15:50

#### Stephane Seuret - Université Paris-Est Créteil Val-de-Marne

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.

Organizers

Kenneth Falconer

University of St Andrews

Maarit Järvenpää

University of Oulu

Antti Kupiainen

University of Helsinki

Francois Ledrappier

University of Notre Dame

Pertti Mattila

University of Helsinki