Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture
Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture
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DOI：
10.4064/sm8314-12-2015
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发表时间：
2015-02
期刊：
影响因子：
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通讯作者：
T. Downarowicz;S. Kasjan
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文献类型：
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作者：
T. Downarowicz;S. Kasjan
虽然Sarnak猜想适用于紧群旋转（无理旋转，里程计），但我们甚至不知道它是否适用于所有的Jewett-Krieger模型。在本文中，我们证明了它，只要模型是在同一个拓扑扩展。特别是，我们重新建立（后[AKL]），经常Toeplitz系统满足Sarnak猜想，并作为另一个结果，所以做所有广义Sturmian子移位（不仅是经典Sturmian子移位）。我们还给出了一个例子，一个不规则的Toeplitz子移位，符合我们的标准。我们给出了一个里程表模型的例子，它甚至不是Toeplitz（它是弱混合的），因此不符合我们的标准。然而，对于这个例子，我们设法产生一个单独的证明Sarnak猜想。接下来，我们提供了一类Toeplitz序列失败Sarnak猜想（在弱意义上），所有这些例子都有正熵。最后，我们研究的例子Toeplitz序列从[AKL]（失败Sarnak的猜想在强意义上），并证明它有积极的熵，以及（这一证明已宣布在[AKL]）。本文可以认为是[AKL]的续篇，也填补了[D]的一些空白.
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].