Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals

TL;DR

This paper investigates the spectral properties of one-dimensional random Jacobi operators linked to ergodic dynamical systems, demonstrating that under certain conditions, their spectrum is a measure-zero Cantor set, extending previous Schrödinger operator results.

Contribution

It generalizes earlier findings by characterizing the spectrum of Jacobi matrices for quasicrystals, showing it is a measure-zero Cantor set under Boshernitzan condition.

Findings

Spectrum is supported on a Cantor set with Lebesgue measure zero.

Characterization of spectrum via non-uniform transfer matrices and Lyapunov exponent.

Generalization of results from Schrödinger to Jacobi operators.

Abstract

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. In this context, we characterize the spectrum of these operators by non-uniformity of the transfer matrices and the set where the Lyapunov exponent vanishes. Adapting this result to subshifts satisfying the so-called Boshernitzan condition, it turns out that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schr\"odinger operators.