Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Abstract

We show that if the base frequency is Diophantine, then the Lyapunov exponent of a C k quasi-periodic SL(2, R) cocycle is 1/2-Hölder continuous in the almost reducible regime, if k is large enough. As a consequence, we show that if the frequency is Diophantine, k is large enough, and the potential is C k small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is 1/2-Hölder continuous.

Hölder continuity of absolutely continuous spectral measure for multi-frequency Schrödinger operators

“…In this section, we give an iteration proposition proved in [7,21] as a generalization of the results in [15]. More precisely, let N = 2 r−r ′ |ln ǫ|, then we can distinguish two cases: • (Non-resonant case) if for any n ∈ Z d with 0 < |n| ≤ N , we have |2ξ− < n, α >| ≥ ǫ 1 10 ,…”

“…However, the results in [4,22] were restricted to one-frequency case, since a crucial technique in [4,22] is almost reducibility developed by Avila and Jitomirskaya in [3] based on quantitative Aubry duality and it seems nontrivial to generalize the method in [3] to multi-frequency case. While almost reducibility can also be got directly by classical KAM theory [7,10,11,15,21], and results in [7,10,11,21] do work in any dimension. This allows the possibility to study the regularity of distribution of absolutely continuous spectral measure in the multi-frequency case.…”

“…Recently, Wang and Zhang [26] obtained the weak Hölder continuity of Lyapunov exponent as a function of energies, for a class of C 2 quasi-periodic potentials and for any Diophantine frequency. More recently, Cai, Chavaudret, You and Zhou [7] proved sharp Hölder continuity of Lyapunov for quasi-periodic Schrödinger operator with small finitely differential potentials and Diophantine frequencies.…”

Large coupling asymptotics for the entropy of quasi-periodic operators

“…If m = 2, we have the following more precise iteration lemma, first appearing in [20], which is the refined version of [9,19].…”

Hölder Continuity of Spectral Measures for the Finitely Differentiable Quasi-Periodic Schrödinger Operators

“…What can we say about the regularity of the spectral measure of Schrödinger operators with finitely differentiable potential? Recall that Cai-Chavaudret-You-Zhou [10] have shown that if α ∈ DC d and V ∈ C k (T d , R) is small, then IDS of the Schrödinger operator is 1 2 -Hölder continuous. Recently, Zhao [24] also generalized [5] and [21] to the multi-frequency Schrödinger operators with V ∈ C ω (T d , R).…”