Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, I. The essential support of the measure

Damanik, David; Munger, Paul; Yessen, William

doi:10.1016/j.jat.2013.04.001

Cited by 19 publications

(40 citation statements)

References 32 publications

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“…On the other hand, a point has a bounded forward orbit if and only if it belongs to a stable manifold of Λ λ (this is known and has been used since Casdagli's work [20]; an explicit proof was given in [39,Corollary 2.5], see also [78]). Since the spectrum is real and the (complexified) line of initial conditions maps R into the real subspace of the invariant surface, all intersection points must be real.…”

Section: Now Let Us Return To Our Case Where σmentioning

confidence: 99%

The Fibonacci Hamiltonian

2016

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We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and establish relations between its spectrum and spectral characteristics (namely, the optimal Hölder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) and the dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We also exhibit a connection between the spectral quantities and the thermodynamic pressure function. As a result, a detailed description of the spectral properties for all values of the coupling constant is obtained (in contrast to all previous quantitative results, D.

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Section: Now Let Us Return To Our Case Where σmentioning

confidence: 99%

The Fibonacci Hamiltonian

2016

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“…For some recent contributions on this topic we refer to [5,7,8,10,14,25,27,28,29,34,36] and references there in. Detailed accounts regarding the earlier research on these polynomials can be found, for example, in Szegő [35], Geronimus [18], Freud [17] and Van Assche [37].…”

Section: Introductionmentioning

confidence: 99%

Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

Bracciali

Ranga

Swaminathan

2016

Applied Numerical Mathematics

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When a nontrivial measure µ on the unit circle satisfies the symmetry dµ(e i(2π−θ) ) = −dµ(e iθ ) then the associated OPUC, say S n , are all real. In this case, in [12], Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zS n (z) + S * n (z)} and {zS n (z) − S * n (z)} satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1, 1]. The same authors, in [13], have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in [10] and then [8] the extension associated with the para-orthogonal polynomials zS n (z)− S * n (z) was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zS n (z) + S * n (z) for any nontrivial measure on the unit circle. Like in [10] and [8] chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.

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“…As a consequence, the said geometric restrictions can be lifted for all previous results (many of which are surveyed in [23], and some for other models appear in [13,14,31]; see also [12]).…”

Section: Introductionmentioning

confidence: 95%

“…Notice that we need the C from Lemma 2.17 to be bounded away from π 2 to effectively bound I(E 0 (λ), λ) from below as λ → ∞. [13,Section 2]). Now, if λ is of the form 4π 2 (a 2 − b 2 ) for a, b ∈ N with a > b, then with E = 4a 2 π 2 , it is evident from (23) that λ (E) = (1, 1, 1), which is a point on S 0 that is fixed under the action by f .…”

Section: 4mentioning

confidence: 99%

Mixed spectral regimes for square Fibonacci Hamiltonians

Fillman¹,

Takahashi²,

Yessen³

2016

J. Fractal Geom.

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For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states measure (i.e. one whose absolutely continuous and singular continuous components are both nonzero). Using the methods developed in this paper, we also show existence of parameter regimes for the square continuum Fibonacci Schrödinger operator yielding mixed interval-Cantor spectra. These examples provide the first explicit examples of an interesting phenomenon that has not hitherto been observed in aperiodic Hamiltonians. Moreover, while we focus only on the Fibonacci model, our techniques are equally applicable to models based on any two-letter primitive invertible substitution.

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Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, I. The essential support of the measure

Cited by 19 publications

References 32 publications

The Fibonacci Hamiltonian

The Fibonacci Hamiltonian

Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

Mixed spectral regimes for square Fibonacci Hamiltonians

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