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

By Tomasz Downarowicz, Stanislaw Kasjan. 2015

49 citations recorded by the source corpus.

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. In particular, we reestablish (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 fits our criterion. We give an example of a model of an odometer which is not even Toeplitz (it is weakly mixing), hence does not fit 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 has positive entropy, as well (this proof has been announced in [AKL]). This paper can be considered a sequel to [AKL], it also fills some gaps of [D].

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01693dbf5b4c35098bae316fcbdb16a0a904c984

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In Algebra and Number Theory

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk
3. Sébastien Ferenczi
4. Terence Tao

In Discrete Mathematics and Combinatorics

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk
3. Sébastien Ferenczi

In Mathematical Physics

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk
3. el Houcein el Abdalaoui
4. Thierry de la Rue
5. Sébastien Ferenczi
6. Terence Tao

Citations in the corpus

• Sarnak’s Conjecture: What’s New (2017, background): “However, in [2] (and then [51]) there are examples of Toeplitz systems which are not Möbius orthogonal.”
• Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals (2015, unknown): “For those in which eigenfunctions are continuous, e.g. symbolic models like Sturmian models [5], Sarnak’s conjecture has also been proved to hold in view of a general criterion on lifting Sarnak’s conjecture by continuous uniquely ergodic extensions, [3], [9], [32].”
• The Chowla and the Sarnak conjectures from ergodic theory point of view (2014, background): “For further motivations and related results see [1, 11].”