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August  2016, 9(4): 1009-1023. doi: 10.3934/dcdss.2016039

Characterizations of uniform hyperbolicity and spectra of CMV matrices

David Damanik 1, , Jake Fillman 2, , Milivoje Lukic 3, and  William Yessen 1,

1. 

Department of Mathematics, Rice University, Houston, TX 77005, United States
2. 

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, United States
3. 

Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Received  September 2014 Revised  July 2015 Published  August 2016

We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.
Keywords: Linear cocycles, generalized eigenfunctions, CMV matrices, uniform hyperbolicity, orthogonal polynomials..
Mathematics Subject Classification: Primary: 37D20, 42C05; Secondary: 37A2.
Citation: David Damanik, Jake Fillman, Milivoje Lukic, William Yessen. Characterizations of uniform hyperbolicity and spectra of CMV matrices. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1009-1023. doi: 10.3934/dcdss.2016039
References:
[1]

Ju. M. Berezanskii, Expansions in Eigenfuncions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.  Google Scholar

[2]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. doi: 10.1007/s00209-009-0494-y.  Google Scholar

[3]

D. Damanik, J. Fillman, M. Lukic and W. Yessen, Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model, Int. Math. Res. Not., 2015 (2015), 7110-7129. doi: 10.1093/imrn/rnu158.  Google Scholar

[4]

D. Damanik, J. Fillman and D. C. Ong, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices, J. Math. Pures Appl., 105 (2016), 293-341. doi: 10.1016/j.matpur.2015.11.002.  Google Scholar

[5]

J. Geronimo and R. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations, 132 (1996), 140-178. doi: 10.1006/jdeq.1996.0175.  Google Scholar

[6]

F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139 (2006), 172-213. doi: 10.1016/j.jat.2005.08.002.  Google Scholar

[7]

R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Diff. Eq., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.  Google Scholar

[8]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.  Google Scholar

[9]

M. Lukic and D. Ong, Generalized Prüfer variables for perturbations of Jacobi and CMV matrices,, J. Math. Anal. Appl., ().   Google Scholar

[10]

P. Munger and D. Ong, The Hölder continuity of spectral measures of an extended CMV matrix, J. Math. Phys., 55 (2014), 093507, 10 pp. doi: 10.1063/1.4895762.  Google Scholar

[11]

D. Ong, Purely singular continuous spectrum for CMV operators generated by subshifts, J. Stat. Phys., 155 (2014), 763-776. doi: 10.1007/s10955-014-0974-2.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.  Google Scholar

[13]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems I., J. Diff. Eq., 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[14]

R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Diff. Eq., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[15]

J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.  Google Scholar

[16]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[17]

J.-C. Yoccoz, Some questions and remarks about SL$(2,\mathbbR)$ cocycles, Modern Dynamical Systems and Applications, 447-458, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[18]

Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity,, preprint, ().   Google Scholar

show all references

References:
[1]

Ju. M. Berezanskii, Expansions in Eigenfuncions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.  Google Scholar

[2]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. doi: 10.1007/s00209-009-0494-y.  Google Scholar

[3]

D. Damanik, J. Fillman, M. Lukic and W. Yessen, Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model, Int. Math. Res. Not., 2015 (2015), 7110-7129. doi: 10.1093/imrn/rnu158.  Google Scholar

[4]

D. Damanik, J. Fillman and D. C. Ong, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices, J. Math. Pures Appl., 105 (2016), 293-341. doi: 10.1016/j.matpur.2015.11.002.  Google Scholar

[5]

J. Geronimo and R. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations, 132 (1996), 140-178. doi: 10.1006/jdeq.1996.0175.  Google Scholar

[6]

F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139 (2006), 172-213. doi: 10.1016/j.jat.2005.08.002.  Google Scholar

[7]

R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Diff. Eq., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.  Google Scholar

[8]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.  Google Scholar

[9]

M. Lukic and D. Ong, Generalized Prüfer variables for perturbations of Jacobi and CMV matrices,, J. Math. Anal. Appl., ().   Google Scholar

[10]

P. Munger and D. Ong, The Hölder continuity of spectral measures of an extended CMV matrix, J. Math. Phys., 55 (2014), 093507, 10 pp. doi: 10.1063/1.4895762.  Google Scholar

[11]

D. Ong, Purely singular continuous spectrum for CMV operators generated by subshifts, J. Stat. Phys., 155 (2014), 763-776. doi: 10.1007/s10955-014-0974-2.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.  Google Scholar

[13]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems I., J. Diff. Eq., 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[14]

R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Diff. Eq., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[15]

J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.  Google Scholar

[16]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[17]

J.-C. Yoccoz, Some questions and remarks about SL$(2,\mathbbR)$ cocycles, Modern Dynamical Systems and Applications, 447-458, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[18]

Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity,, preprint, ().   Google Scholar

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