Singular Density of States Measure for Subshift and Quasi-Periodic Schr\"odinger Operators

TL;DR

This paper disproves Simon's subshift conjecture regarding the measure support of associated measures for aperiodic subshifts and shows that, under certain conditions, the density of states measure for Schr"odinger operators is singular, including in quasi-periodic cases.

Contribution

It disproves Simon's subshift conjecture and demonstrates the singularity of the density of states measure for various Schr"odinger operators under specific conditions.

Findings

Disproved Simon's subshift conjecture.

Proved the density of states measure is singular under certain assumptions.

Showed generic singularity of the measure for quasi-periodic operators.

Abstract

Simon's subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schr\"odinger operators. In addition we prove a weak version of the conjecture in the Schr\"odinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schr\"odinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.