论文标题
简单的toeplitz subshifts：共生的组合特性和均匀性
Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles
论文作者
论文摘要
我们研究了Aperiodic简单的Toeplitz子缩影的组合特性，以及由它们定义的Jacobi运算符的光谱特性。更确切地说，我们为复杂性，重心复杂性以及足够大的单词长度，重复性而得出明确的公式。此外，我们提供了De Bruijn图的完整描述。我们表征了α-反应性，并基于Boshernitzan条件2011年的Liu和Qu的工作。这些组合结果也可以在[Arxiv：1801.08778]中找到。关于Jacobi操作员，我们表明他们在次移中几乎所有元素都有空的纯点光谱。这是2018年Grigorchuk，Lenz和Nagnibeda的结果。此外，这些频谱被证明是Lebesgue Measure Zero的一组。实际上，我们证明了每个局部恒定的SL（2，r） - 循环均均匀的说法。为此，我们使用所谓的领先序列条件进行子缩影，这源于与Grigorchuk，Lenz和Nagnibeda的合作，请参见[Arxiv：1906.01898]。这种方法使我们能够以统一的方式建立简单的Toeplitz subfrits和Sturmian subfrifts的共生均匀性。附录简要回顾了Jacobi操作员在简单的Toeplitz subshifts和Laplacians上的Schreier图形图上的连接。
We investigate combinatorial properties of aperiodic simple Toeplitz subshifts, as well as spectral properties of Jacobi operators defined by them. More precisely, we derive explicit formulas for complexity, palindrome complexity and, for sufficiently large word length, repetitivity. In addition we give a complete description of the de Bruijn graphs. We characterise alpha-repetitivity and, based on a work by Liu and Qu from 2011, the Boshernitzan condition. These combinatorial results can also be found in [arXiv:1801.08778]. Regarding the Jacobi operators, we show that they have empty pure point spectrum for almost all elements in the subshift. This generalises a result of Grigorchuk, Lenz and Nagnibeda from 2018. In addition the spectrum is shown to be a Cantor set of Lebesgue measure zero. In fact we prove the stronger statement that every locally constant SL(2,R)-cocycle is uniform. To do so, we use the so-called leading sequence condition for subshifts, which stems from a collaboration with Grigorchuk, Lenz and Nagnibeda, see [arXiv:1906.01898]. This approach allows us to establish uniformity of cocycles for simple Toeplitz subshifts and Sturmian subshifts in a unified way. An appendix briefly reviews the connection between Jacobi operators on simple Toeplitz subshifts and Laplacians on Schreier graphs of self-similar groups.
