Sasa Kocic
Singular continuous phase for Schroedinger operators over circle diffeomorphisms with a singularity
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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 $T^1=R/Z$, instead of a circle rotation. In particular, we consider Schroedinger operators over circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity (circle maps with a break) or vanishes (critical circle maps). We show that in a two-parameter region --- determined by the geometry of dynamical partitions and $lpha$ --- the spectrum of Schroedinger operators over every sufficiently smooth such map, is purely singular continuous, for every $lpha$-H\"older-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 $S$ depending on the class of the considered maps, and $\mu$-almost all $x\in\Tt^1$, the corresponding Schroedinger operator has a purely continuous spectrum, for every H\"older-continuous potential $V$. For circle maps with a break, this set includes some Diophantine numbers with a Diophantine exponent $\delta$, for any $\delta>1$.