1 results
Uniformity aspects of
${\mathrm {SL}}(2,{\mathbb R})$ cocycles and applications to Schrödinger operators defined over Boshernitzan subshifts
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 20 November 2024, pp. 1-23
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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.
