The Moebius function and continuous extensions of rotations

Let $f\colon \mathbb{T}\to \mathbb{R}$ be of class $C^{1+δ}$ for some $δ>0$ and let $c\in\mathbb{Z}$. We show that for a generic $α\in\mathbb{R}$, the extension $T_{c,f}\colon \mathbb{T}^2\to\mathbb{T}^2$ of the irrational rotation $Tx=x+α$, given by $T_{c,f}(x,u)=(x+α, u+cx+f(x))$ ($\bmod\ 1$) satisfies Sarnak's conjecture.

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