Generalized Thue-Morse measures: spectral and fractal analysis
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| Format: | Preprint |
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2025
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| _version_ | 1866911178170564608 |
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| author | Gohlke, Philipp Kesseböhmer, Marc Schindler, Tanja I. |
| author_facet | Gohlke, Philipp Kesseböhmer, Marc Schindler, Tanja I. |
| contents | 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 Kreĭn-Feller operator, providing new insights into the interplay between diffraction, fractal geometry, and spectral theory in the Thue-Morse context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Thue-Morse measures: spectral and fractal analysis Gohlke, Philipp Kesseböhmer, Marc Schindler, Tanja I. Dynamical Systems 37D35, 37B10, 52C23, 37A46, 62H30 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 Kreĭn-Feller operator, providing new insights into the interplay between diffraction, fractal geometry, and spectral theory in the Thue-Morse context. |
| title | Generalized Thue-Morse measures: spectral and fractal analysis |
| topic | Dynamical Systems 37D35, 37B10, 52C23, 37A46, 62H30 |
| url | https://arxiv.org/abs/2509.22109 |