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


Authors: El Houcein El Abdalaoui, Stanisław Kasjan and Mariusz Lemańczyk
Journal: Proc. Amer. Math. Soc. 144 (2016), 161-176
MSC (2010): Primary 37B05; Secondary 47A40
DOI: https://doi.org/10.1090/proc/12683
Published electronically: June 23, 2015
MathSciNet review: 3415586
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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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El Houcein El Abdalaoui
Affiliation: Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Étienne du Rouvray, France
Email: elhoucein.elabdalaoui@univ-rouen.fr

Stanisław Kasjan
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin Street, 87-100 Toruń, Poland
Email: skasjan@mat.umk.pl

Mariusz Lemańczyk
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin Street, 87-100 Toruń, Poland
Email: mlem@mat.umk.pl

DOI: https://doi.org/10.1090/proc/12683
Received by editor(s): September 5, 2014
Received by editor(s) in revised form: November 25, 2014
Published electronically: June 23, 2015
Additional Notes: The third author’s research was supported by Narodowe Centrum Nauki grant DEC-2011/03/B/ST1/00407.
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society