Yessen, William; Lukic, Milivoje; Fillman, Jake; Damanik, David Characterizations of uniform hyperbolicity and spectra of CMV matrices. (English) Zbl 1366.37077 Discrete Contin. Dyn. Syst., Ser. S 9, No. 4, 1009-1023 (2016). Summary: We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of \(\mathrm{GL}(2,\mathbb{C})\) 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. Cited in 21 Documents MSC: 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations Keywords:linear cocycle; uniform hyperbolicity; generalized eigenfunction; orthogonal polynomial PDFBibTeX XMLCite \textit{W. Yessen} et al., Discrete Contin. Dyn. Syst., Ser. S 9, No. 4, 1009--1023 (2016; Zbl 1366.37077) Full Text: DOI arXiv References: [1] Ju. M. Berezanskii, <em>Expansions in Eigenfuncions of Selfadjoint Operators</em>,, Amer. Math. Soc. (1968) [2] J. Bochi, Some characterizations of domination,, Math. Z., 263, 221 (2009) · Zbl 1181.37032 [3] D. Damanik, Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model,, Int. Math. Res. Not., 2015, 7110 (2015) · Zbl 1325.33007 [4] D. Damanik, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices,, J. Math. Pures Appl., 105, 293 (2016) · Zbl 1332.81066 [5] J. Geronimo, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle,, J. Differential Equations, 132, 140 (1996) · Zbl 0863.39003 [6] F. Gesztesy, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle,, J. Approx. Theory, 139, 172 (2006) · Zbl 1118.47023 [7] R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients,, J. Diff. Eq., 61, 54 (1986) · Zbl 0608.34056 [8] Y. Last, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators,, Invent. Math., 135, 329 (1999) · Zbl 0931.34066 [9] M. Lukic, Generalized Prüfer variables for perturbations of Jacobi and CMV matrices,, J. Math. Anal. Appl. · Zbl 1353.47062 [10] P. Munger, The Hölder continuity of spectral measures of an extended CMV matrix,, J. Math. Phys., 55 (2014) · Zbl 1297.81076 [11] D. Ong, Purely singular continuous spectrum for CMV operators generated by subshifts,, J. Stat. Phys., 155, 763 (2014) · Zbl 1325.47068 [12] M. Reed, <em>Methods of Modern Mathematical Physics, I: Functional Analysis</em>,, Academic Press (1972) · Zbl 0242.46001 [13] R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems I.,, J. Diff. Eq., 15, 429 (1974) · Zbl 0294.58008 [14] R. Sacker, A spectral theory for linear differential systems,, J. Diff. Eq., 27, 320 (1978) · Zbl 0372.34027 [15] J. Selgrade, Isolated invariant sets for flows on vector bundles,, Trans. Amer. Math. Soc., 203, 359 (1975) · Zbl 0265.58004 [16] B. Simon, <em>Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory</em>,, American Mathematical Society Colloquium Publications 54, 54 (2005) · Zbl 1082.42020 [17] J.-C. Yoccoz, Some questions and remarks about SL \((2,\mathbbR)\) cocycles,, Modern Dynamical Systems and Applications, 447 (2004) · Zbl 1148.37306 [18] Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity,, preprint This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.