# American Institute of Mathematical Sciences

2013, 2013(special): 605-609. doi: 10.3934/proc.2013.2013.605

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

 1 Department of Mathematics MS-136, Rice University, 6100 Main St, Houston, TX 77005, United States

Received  September 2012 Revised  December 2012 Published  November 2013

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.
Citation: Darren C. Ong. Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Conference Publications, 2013, 2013 (special) : 605-609. doi: 10.3934/proc.2013.2013.605
##### References:
 [1] Michael Boshernitzan and David Damanik, Generic continuous spectrum for ergodic Schrödinger operators,, Communications in Mathematical Physics, 283 (2008), 647. [2] A. Khintchine, "Continued Fractions,", Dover, (1997). [3] Darren C. Ong, Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle,, Journal of Mathematical Analysis and Applications, 394(2) (2012), 633. [4] Zhenghe Zhang, Positive Lyapunov exponents for quasiperiodic Szëgo cocycles,, Nonlinearity, 25 (2012), 1771.

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##### References:
 [1] Michael Boshernitzan and David Damanik, Generic continuous spectrum for ergodic Schrödinger operators,, Communications in Mathematical Physics, 283 (2008), 647. [2] A. Khintchine, "Continued Fractions,", Dover, (1997). [3] Darren C. Ong, Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle,, Journal of Mathematical Analysis and Applications, 394(2) (2012), 633. [4] Zhenghe Zhang, Positive Lyapunov exponents for quasiperiodic Szëgo cocycles,, Nonlinearity, 25 (2012), 1771.
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