Central limit theorem for generalized Weierstrass functions

Abstract

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

16 pages. To appear in Stochastics and Dynamics