Multifractal of the Apollonian tiling (Q1279457)

Multifractal analysis is a useful technique to study the properties of singular measures. It gives some information about how densely the singularities of a measure are distributed. In this note, the author analyses the multifractal properties of both some Gibbs measures and Markov type measures. Multifractal refers to a notion of size that involves the variations of the weight of the measures. The author associates thermodynamical quantities such as free energy functions with partitions of exponentially decreasing diameters, and proves some regularity. Moreover, the author introduces the correlation dimension which refers to a quantity that is the most accessible in numerical computations.