THERMOGAMAS - Multifractal Analysis of Generalized Thue-Morse Measures (Philipp Gohlke)

Talk given during the THERMOGAMAS Workshop.

Abstract: The Thue-Morse measure is a spectral (diffraction) measure of the classical Thue-Morse sequence, and a prototypical example of a singular continuous measure. It can also be described as the equilibrium measure of a singular potential over the doubling map. This allows us to capture the local dimensions of the measure via tools from thermodynamic formalism. Some anomalies occur due to the singularity in the potential, including a super-polynomial decay around dyadic points.

These observations can be extended to a family of generalized Thue-Morse measures, each of them related to a 2-multiplicative sequence that is the fixed point of a substitution on a compact alphabet. We show that several thermodynamic quantities depend sensitively on the parameter and discuss the implications on the multifractal analysis of these measures. (based on joint work with M.Baake, M.Kesseböhmer, G.Lamprinakis, T.Schindler and J.Schmeling)