A renormalization approach to irrational rotations

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Bonanno, Claudio
Isola, Stefano
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Abstract
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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.
Comment: 27 pages
Keywords
Mathematics - Dynamical Systems, Mathematics - Number Theory, 37E05, 37A25, 37A45
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