Alex Rutar (University of St Andrews):
Title:
Geometric and Combinatorial Properties of Self-similar Measures
Abstract:
Multifractal analysis is generally concerned with understanding the fine scaling properties of measures. I will give a general introduction to multifractal analysis for a certain class of self-similar measures. In particular, while the multifractal analysis of such systems which have "no exact overlaps" is well-understood, the situation with "exact overlaps" is much more subtle. To address the latter case, I will present a matrix product representation of self-similar measures in the real line. This approach relates the validity of the multifractal formalism with connectivity properties of a certain finite or infinite graph. This provides new examples of overlapping self-similar measures satisfying the multifractal formalism for all $q\in\mathbb{R}$, and also gives insight into the failure of the multifractal formalism for self-similar measures.