Georgia Institute of Technology College of Sciences

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. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. In the generalized case, we consider Riesz products that can be regarded as diffraction measures of generalized Thue–Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising from the doubling map together with an observable exhibiting a logarithmic singularity. For this system, we develop a generalized thermodynamic formalism beyond the standard setting, which yields explicit formulas for Birkhoff and dimension spectra. A further novel aspect is the identification of a precise connection between these spectra and the L^q -spectrum of the underlying Riesz product. If time allows I will also give a link to the Fourier and quantization dimension. Moreover, this new formulation of the thermodynamic formalism allows to give precise large deviation results for unbounded observables with a qualitatively different result depending if the logarithmic singularity appears at a periodic or pre-periodic non periodic point.

The talk is based on joint work with Philipp Gohlke and Marc Kesseböhmer and work in progress with Matt Nicol.