Scaling properties of the Thue–Morse measure

The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.