Generalized Thue-Morse measures: spectral and fractal analysis

Abstract

We investigate a family of Riesz products and show that they 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. This new link allows us to determine, explicitly, the Fourier and quantization dimension, and to describe the spectral asymptotics of the associated Kre\u{i}n-Feller operator, providing new insights into the interplay between diffraction, fractal geometry, and spectral theory in the Thue-Morse context.