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

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