Central limit theorem for generalized Weierstrass functions

Let $f$ be a $C^{2+ε}$ expanding map of the circle and $v$ be a $C^{1+ε}$ real function of the circle. Consider the twisted cohomological equation $v(x) = α(f(x)) - Df(x) α(x)$ which has a unique bounded solution $α$. We prove that $α$ is either $C^{1+ε}$ or nowhere differentiable, and if $α$ is nowhere differentiable then the Newton quotients of $α$, after an appropriated normalization, converges in distribution to the normal distribution, with respect to the unique absolutely continuous invariant probability of $f$.

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