American Institute of Mathematical Sciences

Advanced
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.   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.   Google Scholar

[4]

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

[4]

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

[1]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[2]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[3]

John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, Belinda Trotta. Dynamics of spacing shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4207-4232. doi: 10.3934/dcds.2013.33.4207
[4]

John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743
[5]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
[6]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[7]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[8]

Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323
[9]

Philipp Gohlke, Dan Rust, Timo Spindeler. Shifts of finite type and random substitutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5085-5103. doi: 10.3934/dcds.2019206
[10]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[11]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
[12]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
[13]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[14]

Kengo Matsumoto. On the Markov-Dyck shifts of vertex type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 403-422. doi: 10.3934/dcds.2016.36.403
[15]

Kevin McGoff, Ronnie Pavlov. Random $\mathbb{Z}^d$-shifts of finite type. Journal of Modern Dynamics, 2016, 10: 287-330. doi: 10.3934/jmd.2016.10.287
[16]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[17]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[18]

Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[19]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[20]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427

 Impact Factor: 

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences