A renormalization approach to irrational rotations

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 $$\beta \mapsto \{\alpha + \beta\}$$ , $$\alpha \in {\mathbb{R}}\!\setminus\! {\mathbb{Q}}$$ . In particular, we obtain sharp results for the diffusion of the walk on $${\mathbb{Z}}$$ generated by the location of points of the sequence {n α + β} on a binary partition of the unit interval. Finally, we give some applications of our method.