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

Tomasz Downarowicz ¹   ; Stanisław Kasjan ²

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland and Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland
² Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Studia Mathematica, Tome 229 (2015) no. 1, pp. 45-72

Résumé

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].

Détail

DOI : 10.4064/sm8314-12-2015

Keywords: although sarnaks conjecture holds compact group rotations irrational rotations odometers even known whether holds jewett krieger models rotations paper does long model topological extension via map establishes isomorphism equicontinuous model particular recover after akl regular toeplitz systems satisfy sarnaks conjecture another consequence generalized sturmian subshifts only classical sturmian subshift example irregular toeplitz subshift which meets criterion example model odometer which even toeplitz weakly mixing hence does meet criterion however example manage produce separate proof sarnaks conjecture provide class toeplitz sequences which fail sarnaks conjecture weak sense these examples have positive entropy finally examine example toeplitz sequence akl which fails sarnaks conjecture strong sense prove has positive entropy proof has announced akl paper considered sequel akl fills gaps nbsp