Uniform Convergence of Schr?dinger Cocycles over Simple Toeplitz Subshift |
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Authors: | Qing-Hui Liu Yan-Hui Qu |
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Institution: | 1. Department of Computer Science and Engineering, Beijing Institute of Technology, Beijing, 100081, People??s Republic of China 2. Department of Mathematics, Tsinghua University, Beijing, 100084, People??s Republic of China
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Abstract: | For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95–123, 2006) proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. In this paper, we study simple Toeplitz subshifts. We give a criterion that simple Toeplitz
subshifts satisfy condition (B), and also give some sufficient conditions that they do not satisfy condition (B). However, we can still prove the uniformity of Schr?dinger cocycles over any simple Toeplitz subshift. As a consequence,
the related Schr?dinger operators have Cantor spectrum of Lebesgue measure 0. We also exhibit a fine structure for the spectrum,
and this helps us to prove purely singular continuous spectrum for a large class of simple Toeplitz potentials. |
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