Singular Continuous Phase for Schrödinger Operators Over Circle Maps with Breaks

Abstract

We consider Schrödinger operators over a class of circle maps including \(C^{2+\epsilon }\)-smooth circle maps with finitely many break points, where the derivative has a jump discontinuity. We show that in a region of the Lyapunov exponent—determined by the geometry of the dynamical partitions and \(\alpha \)—the spectrum of Schrödinger operators over every such map, is purely singular continuous, for every \(\alpha \)-Hölder-continuous potential V. As a corollary, we obtain that for every sufficiently smooth such map, with an invariant measure \(\mu \) and with rotation number in a set \(\mathcal {S}\), and \(\mu \)-almost all \(x\in {\mathbb {T}}^1\), the corresponding Schrödinger operator has a purely continuous spectrum, for every Hölder-continuous potential V. Set \(\mathcal {S}\) includes some Diophantine numbers of class \(D(\delta )\), for any \(\delta >1\).