This work is based on the Ph.D. Thesis of Amanda de Lima. Let $f: \mathbb{S}^1 \mapsto \mathbb{S}^1$ be a $C^{2+\epsilon}$ expanding map of the circle and let $v:\mathbb{S}^1 \to \mathbb{R}$ be a $C^{1+\epsilon}$ function. Consider the twisted cohomological equation

$$
v(x) = \alpha(f(x)) - Df(x)\alpha(x),
$$

which has a unique bounded solution $\alpha$. We show that $\alpha$ is either $C^{1+\epsilon}$ or a continuous but nowhere differentiable function.

We show that if $\alpha$ is nowhere differentiable then

$$
\lim_{h\to 0} \mu \left\{x:\frac{\alpha(x+h)-\alpha(x)}{\sigma \ell h\sqrt{-\log |h|}} \le y\right\} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{y} e^{-\frac{t^2}{2}} dt.
$$

where $\sigma$ and $\ell$ are positive constantsand $\mu$ is the SBR measure of $f$. In particular $\alpha$ is not a Lipchitz function on any subset with positive Lebesgue measure.