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Uniformity Aspects of $\mathrm{SL}(2,\mathbb{R})$ Cocycles and Applications to Schr\'odinger Operators Defined Over Boshernitzan Subshifts
- Publication Year :
- 2022
-
Abstract
- We consider continuous $\mathrm{SL}(2,\mathbb{R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous $\mathrm{SL}(2,\mathbb{R})$ cocycles as $G_\delta$-sets. These results are then applied to Schr\"odinger operators with dynamically defined potentials. In the case where the base dynamics is given by a subshift satisfying the Boshernitzan condition, we show that for a generic continuous sampling function, the associated Schr\"odinger cocycles are uniform for all energies and, in the aperiodic case, the spectrum is a Cantor set of zero Lebesgue measure.<br />Comment: 24 pages
- Subjects :
- Mathematics - Dynamical Systems
Mathematical Physics
Mathematics - Spectral Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2207.12153
- Document Type :
- Working Paper