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.References
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Additional Information
- 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
- MR Author ID: 333741
- 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
- MR Author ID: 112360
- Email: mlem@mat.umk.pl
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 161-176
- MSC (2010): Primary 37B05; Secondary 47A40
- DOI: https://doi.org/10.1090/proc/12683
- MathSciNet review: 3415586