Quasi-periodically Forced Logistic Map with Weak Liouvillean Frequency

Consider a class of quasi-periodically forced logistic maps

\(\mathbb{T}\times[0,1]\circlearrowleft:(\theta,x)\mapsto(\theta+\omega,c(\theta)x(1-x))\ \ \ (\mathbb{T}=\mathbb{R}/\mathbb{Z}),\) $$\mathbb{T}\times [0,1]\circlearrowleft :(\theta ,x)\mapsto (\theta +\omega ,c(\theta )x(1-x))(\mathbb{T}=\mathbb{R}/\mathbb{Z}),$$

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].