Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum

Pages: 605 - 609, Issue special, November 2013

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Darren C. Ong - Department of Mathematics MS-136, Rice University, 6100 Main St, Houston, TX 77005, United States (email)

Abstract: As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.

Keywords:  Spectral theory, orthogonal polynomials on the unit circle, almost periodicity, quasiperiodic shifts, skew-shifts.
Mathematics Subject Classification:  Primary: 37E10, 47B99; Secondary: 39A24.

Received: September 2012;      Revised: December 2012;      Published: November 2013.