A renormalization approach to irrational rotations

TL;DR

This paper presents a renormalization method to analyze properties of irrational rotations on the circle, providing sharp results on diffusion processes and applications in dynamical systems.

Contribution

It introduces a novel renormalization approach that unifies the study of irrational rotations and their associated diffusion phenomena.

Findings

Sharp results on diffusion of walks generated by irrational rotations

Unified framework for analyzing properties of irrational rotations

Applications to dynamical systems and related processes

Abstract

We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $\beta \mapsto \set{\alpha+\beta}$, $\alpha \in \R\setminus \Q$. In particular we obtain sharp results for the diffusion of the walk on $\Z$ generated by the location of points of the sequence $\{n\alpha +\beta\}$ on a binary partition of the unit interval. Finally we give some applications of our method.