Zero measure Cantor spectra for continuum one-dimensional quasicrystals |
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Authors: | Daniel Lenz Christian Seifert Peter Stollmann |
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Institution: | 1. Friedrich-Schiller-Universität Jena, Fakultät für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany;2. Technische Universität Hamburg-Harburg, Institut für Mathematik, 21073 Hamburg, Germany;3. Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany |
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Abstract: | We study Schrödinger operators on R with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts. |
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Keywords: | 35J10 47B80 47A10 47A35 |
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