Sturmian CMV矩阵的奇异连续谱
Purely Singular Continuous Spectrum for Sturmian CMV Matrices via Strengthened Gordon Lemmas
摘要：
戈登引理是指谱理论中的一类结果，证明了算子结构中的强局部重复排除了该算子本征值的存在。我们扩大最近的工作Ong和证明版本的戈登引理是有效的CMV矩阵，并不限制奇偶校验的规模上重复发生。在我们的方法中的关键成分是Damanik-Fillman-Lukic-Yessen公式，它涉及两类传递矩阵对于一个给定的CMV运营商。我们的结果可以应用于许多例子。我们应用我们的定理来完成分类的频谱类型的CMV矩阵的Sturmian Verblunsky系数，我们证明了这样的CMV矩阵有纯粹的奇异连续谱支持康托尔集为零勒贝格测度的所有（无理）频率和所有相位。我们还讨论了CMV矩阵的Verblunsky系数产生的一般编码的旋转应用。
英文摘要：
The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma which are valid for CMV matrices and which do not restrict the parity of scales upon which repetitions occur. The key ingredient in our approach is a formula of Damanik-Fillman-Lukic-Yessen which relates two classes of transfer matrices for a given CMV operator. There are many examples to which our result can be applied. We apply our theorem to complete the classification of the spectral type of CMV matrices with Sturmian Verblunsky coefficients; we prove that such CMV matrices have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure for all (irrational) frequencies and all phases. We also discuss applications to CMV matrices with Verblunsky coefficients generated by general codings of rotations.
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