0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

El Abdalaoui, El Houcein; Kasjan, Stanisław; Lemańczyk, Mariusz

doi:10.1090/proc/12683

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture
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by El Houcein El Abdalaoui, Stanisław Kasjan and Mariusz Lemańczyk PDF

Proc. Amer. Math. Soc. 144 (2016), 161-176 Request permission

Abstract:

We prove that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in \mathbb {N}$. Sarnak’s conjecture on orthogonality with the Möbius function is proved to hold for such dynamical systems. A step toward the proof of this result is the observation that in the class of uniquely ergodic models of a fixed coalescent automorphism, the conjecture is stable under continuous extensions. In particular, Sarnak’s conjecture holds for all dynamical systems given by regular Toeplitz sequences. A non-regular Toeplitz sequence which is not orthogonal to the Möbius function is constructed.

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