Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals

TL;DR

This paper proves that certain one-dimensional quasicrystals have a spectrum of zero Lebesgue measure, using ergodic theory and Lyapunov exponents, including new examples like Rudin-Shapiro, without relying on trace maps.

Contribution

It establishes the zero Lebesgue measure of the spectrum for a broad class of aperiodic subshifts using ergodic theorems, expanding previous results to new substitution systems.

Findings

Spectrum coincides with zeros of Lyapunov exponent under uniform existence.

All aperiodic subshifts with uniform positive weights have Cantor spectrum of zero measure.

Includes new examples such as Rudin-Shapiro substitution.

Abstract

The spectrum of one-dimensional discrete Schr\"odinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples as e.g. the Rudin-Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.