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

27 pages