Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

“…Liu and Wen [25] then extended the approach employed by Raymond. They study the case of general frequencies.…”

Dynamical bounds for Sturmian Schrödinger operators

“…If the sequence {x n (E)} n is bounded, then E lies in the spectrum of H β and if E is not in the spectrum, then {x n (E)} n is not bounded and grows super exponentially (see [10] and [9] for details). The main idea is to considerσ n for E ∈ C and not only in R. The set σ n is made of q n segments of R. Each band length is bounded from above and from below by decreasing to 0 (as n grows) constants depending of V , n and β [8]. By the Koebe distortion theorem, we can show that the height ofσ n is up to a constant the same as length.…”

“…La preuve qui suit celle de [7] utilise le résultat de [8] qui nécessite l'hypothèse V > 20. Le théorème suivant nous dit que cette situation ne peut s'étendre au cas D = +∞.…”

“…We assume that V > 20. Proofs require a previous result in [8] which needs that assumption. We denote the continued fraction expansion of β by…”

Strictly ergodic subshifts and associated operators

“…Strictly positive lower bounds for the Hausdorff dimension of the spectrum at all couplings λ follow from the Hausdorff continuity results of [29,83], to be discussed in Section 7. Several aspects of Raymond's work were used and extended in a number of papers [35,51,53,95,112]. In particular, Liu and Wen carried out a detailed analysis of the Hausdorff dimension of the spectrum in the general Sturmian case in the spirit of Raymond's approach; see [112].…”