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Consider a class of quasi-periodically forced logistic maps T × [ 0 , 1 ] ↺: (θ , x) ↦ (θ + ω , c (θ) x (1 − x)) (T = R / Z) , where ω is an irrational frequency and α(θ) is a specific bimodal function. We prove that under weak Liouvillean condition on frequency, the strange non-chaotic attractor occurs with negative Lyapunov exponent. This extends the result in [Bjerklov, CMP, 2009]. [ABSTRACT FROM AUTHOR]

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schrödinger cocycles with the C2 cos-type potentials. In particular, the Lyapunov exponent is log-Hölder continuous at each Diophantine frequency. [ABSTRACT FROM AUTHOR]