Subdiffusive behavior generated by irrational rotations

Abstract

The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums $y_n(x)=\sum_{k=0}^{n-1}\psi (x+k\alpha)$, where $\psi$ is a periodic function of bounded variation and $\alpha$ an irrational number. It is known that no diffusion process will be observed. Nevertheless, we find a picewise constant function $\psi$ and an increasing sequence of integer $(n_j)_j$ such that the limit distribution of the sequence $(y_{n_j}/\sqrt j)_j$ is Gaussian (with stricly positive variance). If $\alpha$ is of constant type, we show that the sequence $(n_j)_j$ may be taken to grow exponentially (this is close to optimal in some sense, and one has $||y_{n_j}||_{\mathrm L^2}\sim \max_{0\le k\le n_j}||y_k||_{\mathrm L^2}$ as $j\to\infty$). We give an heuristic link with the theory of expanding maps of the interval.