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

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.