  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.
