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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

Darren C. Ong 1,

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.
Keywords: almost periodicity, orthogonal polynomials on the unit circle, skew-shifts., Spectral theory, quasiperiodic shifts.
Mathematics Subject Classification: Primary: 37E10, 47B99; Secondary: 39A2.
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), 647-662.  Google Scholar

[2]

A. Khintchine, "Continued Fractions," Dover, 1997.  Google Scholar

[3]

Darren C. Ong, Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle, Journal of Mathematical Analysis and Applications, 394(2) (2012), 633-644.  Google Scholar

[4]

Zhenghe Zhang, Positive Lyapunov exponents for quasiperiodic Szëgo cocycles, Nonlinearity, 25 (2012), 1771-1797.  Google Scholar

show all references

References:
[1]

Michael Boshernitzan and David Damanik, Generic continuous spectrum for ergodic Schrödinger operators, Communications in Mathematical Physics, 283 (2008), 647-662.  Google Scholar

[2]

A. Khintchine, "Continued Fractions," Dover, 1997.  Google Scholar

[3]

Darren C. Ong, Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle, Journal of Mathematical Analysis and Applications, 394(2) (2012), 633-644.  Google Scholar

[4]

Zhenghe Zhang, Positive Lyapunov exponents for quasiperiodic Szëgo cocycles, Nonlinearity, 25 (2012), 1771-1797.  Google Scholar

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