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

Tomasz Downarowicz, and Stanisław Kasjan. "Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture." Studia Mathematica 229.1 (2015): 45-72. <http://eudml.org/doc/286497>.

@article{TomaszDownarowicz2015,
abstract = { 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]. },
author = {Tomasz Downarowicz, Stanisław Kasjan},
journal = {Studia Mathematica},
keywords = {odometer; Toeplitz flow; almost 1-1 extension; Möbius function; sarnak's conjecture; entropy},
language = {eng},
number = {1},
pages = {45-72},
title = {Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture},
url = {http://eudml.org/doc/286497},
volume = {229},
year = {2015},
}

TY - JOUR
AU - Tomasz Downarowicz
AU - Stanisław Kasjan
TI - Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture
JO - Studia Mathematica
PY - 2015
VL - 229
IS - 1
SP - 45
EP - 72
AB - 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].
LA - eng
KW - odometer; Toeplitz flow; almost 1-1 extension; Möbius function; sarnak's conjecture; entropy
UR - http://eudml.org/doc/286497
ER -