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arXiv · 2607.29275

On the spectral analysis of dynamical M\"obius-Sarnak process and topological entropy of M\"obius fonction

el Houcein el Abdalaoui
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Abstract

By extending the Rokhlin-Sinai machinery relating to the entropy and countable Lebesgue component in the spectrum, we establish that the dynamical M\"obius-Sarnak process has a countable Lebesgue component. Inspired by recent work of M. Lin and the author, we extend the notion of spectral measure to all operators on Banach spaces. This generalization is further motivated by the Bellow-Losert extension of Wiener's notion of the spectral measure of sequences. Furthermore, we establish unconditionally that the topological entropy of the M\"obius flow is given by $\frac{6}{\pi^2}\log 3$. Among other consequences, we recover a recent result by el Abdalaoui-Nerurkar which asserts that for any quasi-generic measure for the M\"{o}bius function, the M\"{o}bius flow equipped with this measure has a countable Lebesgue component in its spectrum. It follows that the Sarnak M\"{o}bius orthogonality conjecture holds for any topological dynamical system with singular spectrum. We further show that all the potential spectral measures of he M\"{o}bius function are absolutely continuous with respect to Lebesgue measure.

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el Houcein el Abdalaoui. 2026-07-31. On the spectral analysis of dynamical M\"obius-Sarnak process and topological entropy of M\"obius fonction. https://arxiv.org/abs/2607.29275

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