Uniformity Aspects of $\mathrm{SL}(2,\mathbb{R})$ Cocycles and Applications to Schr\"odinger Operators Defined Over Boshernitzan Subshifts

TL;DR

This paper studies uniformity properties of $ ext{SL}(2,b R)$ cocycles over dynamical systems, applying results to Schrödinger operators over Boshernitzan subshifts, revealing generic uniformity and Cantor spectrum of zero measure.

Contribution

It characterizes uniform cocycles as $G_\delta$-sets and demonstrates generic uniformity and Cantor spectrum for Schrödinger operators over Boshernitzan subshifts.

Findings

Uniform cocycles form a $G_\delta$-set.

Generic continuous sampling functions yield uniform Schrödinger cocycles.

Spectrum is a Cantor set of zero Lebesgue measure in the aperiodic case.

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.