Central limit theorem for generalized Weierstrass functions

TL;DR

This paper proves a central limit theorem for the Newton quotients of solutions to a cohomological equation related to expanding circle maps, revealing a dichotomy in regularity and a normal distribution convergence.

Contribution

It establishes a novel CLT for Newton quotients of solutions to a cohomological equation in dynamical systems, highlighting a regularity dichotomy.

Findings

Solutions are either very smooth or nowhere differentiable.

Newton quotients of nowhere differentiable solutions converge to a normal distribution.

The result connects regularity properties with probabilistic limit theorems.

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 $f$.