Multifractal of the Appolonian tiling

We analyse the multifractal properties of Gibbs measures and Markov type measures defined on the complementary set of the Appolonian tiling of the plane into circles or into spheres in higher dimension. Multifractal refers to a notion of size that involves the variations of the weight of the measures. We associate thermodynamical quantities such as free energy functions with partitions of exponentially decreasing diameters, and prove some regularity. Finally we introduce the correlation dimension which refers to a quantity that is the most accessible in numerical computations.