Uniformity Aspects of SL(2,ℝ) Cocycles and Applications to Schrödinger Operators Defined Over Boshernitzan Subshifts

2022/07/25 by Damanik, David, Lenz, Daniel
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2207.12153

Abstract

We consider continuous SL(2,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 SL(2,ℝ) cocycles as G_δ-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.

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