Uniformity aspects of ${\mathrm {SL}}(2,{\mathbb R})$ cocycles and applications to Schrödinger operators defined over Boshernitzan subshifts
Year: 2024
Type: article
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 Schrödinger 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 Schrödinger cocycles are uniform for all energies and, in the aperiodic case, the spectrum is a Cantor set of zero Lebesgue measure.
DOI: 10.1017/etds.2024.66
Authors: DAVID DAMANIK, DANIEL LENZ
Institutions: Rice University, Friedrich Schiller University Jena
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404549625