Strictly ergodic subshifts and associated operators

Abstract

Dedicated to Barry Simon on the occasion of his 60th birthday.Abstract. We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.

Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications

“…The main results in this area have been reviewed in [24,25,63]. In this section we will therefore focus on the recent progress and discuss why zero-measure spectrum is a consequence of Kotani theory when there is uniform convergence to the Lyapunov exponent.…”

Purely Singular Continuous Spectrum for CMV Operators Generated by Subshifts

“…In general, Schrödinger operators generated by these subshifts have received a lot of attention. [Sűt95], [Dam07] and [DEG12] are three extensive surveys of the many results in this area. Much of the impetus behind this research is the fact that these subshifts have been accepted as one-dimensional models of quasicrystals (quasicrystals are structures that are ordered and yet aperiodic; their discovery by Dan Shechtman won him the 2011 Nobel Prize in Chemistry), and so the corresponding Schrödinger operators provided insights into quasicrystalline properties.…”

Spectra of discrete Schrödinger operators with primitive invertible substitution potentials

“…Forthcoming joint work with with William Yessen [MY14] extends these results to Jacobi operators. A standard argument, which uses the minimality of the left shift on the dynamical hull generated by the substitution sequence (see Section 2.2 of this paper and [Dam07]) shows that the spectrum of H λ,α,ω is independent of initial point ω. Thus, we drop ω from the notation and denote the spectrum of the operator H λ,α by Σ λ,α .…”

“…See [DEG13] for a survey of results regarding Schrödinger operators (including higher dimensions) that model quasicrystals. Other related survey papers include [Bel92b], [BG95], [Dam07], and [Süt95].…”