Sasa Kocic
Singular continuous phase for Schroedinger operators over multicritical circle maps
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ABSTRACT. We consider a class of Schroedinger operators - referred to as Schroedinger operators over circle maps - that generalize one-frequency quasiperiodic Schroedinger operators, with a base dynamics given by an orientation-preserving homeomorphism of a circle T=R/Z, instead of a circle rotation. In particular, we consider Schroedinger operators over multicritical circle maps, i.e., circle diffeomorphisms with a finite
number of singular points where the derivative vanishes. We show that in a two-parameter region determined by the geometry of dynamical partitions and $lpha$ the spectrum of Schr dinger operators over every sufficiently smooth such map, is purely singular continuous, for every $lpha$-H lder-continuous potential V. For $lpha=1$, the region extends beyond the corresponding region for the almost Mathieu operator. As a corollary, we obtain that for every sufficiently smooth such map, with an invariant measure $\mu$ and with rotation number in a set S, and $\mu$-almost all x, the corresponding Schroedinger operator has a purely continuous spectrum, for every H lder-continuous potential V.