Zero measure Cantor spectra for continuum one-dimensional quasicrystals

TL;DR

This paper demonstrates that a broad class of continuum one-dimensional quasicrystal Schr"odinger operators have Cantor spectra of zero Lebesgue measure, linking spectral properties to dynamical systems of measures.

Contribution

It introduces a framework connecting measure-based potentials with dynamical systems, proving zero measure Cantor spectra for many aperiodic operator families.

Findings

Cantor spectra of zero Lebesgue measure are proven for various operator families.

Spectral properties are linked to zeros of Lyapunov exponents and transfer matrix non-uniformities.

The approach applies to operators generated by aperiodic subshifts.

Abstract

We study Schr\"odinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.