Sasa Kocic
Singular continuous phase for Schroedinger operators over circle maps with breaks
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ABSTRACT.  We consider Schroedinger operators over a class of circle maps including C^{2+epsilon}-smooth circle maps with finitely many break points. We show that in a region of the Lyapunov exponent --- determined by the geometry of the dynamical partitions and alpha --- the spectrum of Schroedinger operators over every such map, is purely singular continuous, for every alpha-Holder-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, and mu-almost all x in T^1, the corresponding Schroedinger operator has a purely continuous spectrum, for every Holder-continuous potential V. Set S includes some Diophantine numbers of class D(delta), for any delta>1.