Subcritical behavior for quasi-periodic Schrödinger cocycles with trigonometric potentials

A Lower Bound on the Lyapunov Exponent for the Generalized Harper’s Model

“…Remark. After our result was obtained, we learned that Marx, Shou, and Wellens obtained a general lower bound for the Lyapunov exponent improving Herman's results, also using the global theory [10]. However it is difficult to extract analytic quantitative results for a concrete potential from their estimates, although numerical results are possible, leading to estimates better than ours for certain parameters.…”

Large Coupling Asymptotics for the Lyapunov Exponent of Quasi-Periodic Schrödinger Operators with Analytic Potentials

“…Remark 2.3. As shown in [11] (see Remark 3.2, therein), the lower bound m(g) > 2 in Lemma 2.2 is in general optimal. Note that from (2.2), if m(g) > 2, σ(g) m(g) −1 yields…”

“…From a general point of view, a theorem by Ruelle [12] already guarantees that dominated splitting is an open property in the cocycle and that the LE is locally smooth about dominated splittings. In view of Theorem 1.1, we however need a more quantitative version of this result, which we proved in [11], see Proposition 3.1, therein:…”