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Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

Volume 229 / 2015

Tomasz Downarowicz, Stanisław Kasjan Studia Mathematica 229 (2015), 45-72 MSC: Primary 37B05; Secondary 37B10, 37A35, 11Y35. DOI: 10.4064/sm8314-12-2015 Published online: 3 December 2015


Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett–Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do all generalized Sturmian subshifts (not only the classical Sturmian subshift). We also give an example of an irregular Toeplitz subshift which meets our criterion. We give an example of a model of an odometer which is not even Toeplitz (it is weakly mixing), hence does not meet our criterion. However, for this example, we manage to produce a separate proof of Sarnak's conjecture. Next, we provide a class of Toeplitz sequences which fail Sarnak's conjecture (in a weak sense); all these examples have positive entropy. Finally, we examine the example of a Toeplitz sequence from [AKL] (which fails Sarnak's conjecture in the strong sense) and prove that it also has positive entropy (this proof has been announced in [AKL]).

This paper can be considered a sequel to [AKL], it also fills some gaps of [D].


  • Tomasz DownarowiczInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    Institute of Mathematics and Computer Science
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
  • Stanisław KasjanFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland

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