Subdiffusive behavior generated by irrational rotations

TL;DR

This paper investigates the asymptotic behavior of sums involving irrational rotations and demonstrates conditions under which subdiffusive behavior and Gaussian limit distributions emerge, challenging the typical understanding of deterministic diffusion.

Contribution

It introduces specific functions and sequences showing Gaussian limit distributions in irrational rotation sums, revealing new insights into deterministic diffusion phenomena.

Findings

Existence of sequences where normalized sums converge to Gaussian distributions.

For constant type irrationals, sequences can grow exponentially, approaching optimality.

Heuristic connections to expanding maps of the interval are proposed.

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.