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Annali di Matematica Pura ed Applicata

, Volume 188, Issue 2, pp 247–267 | Cite as

A renormalization approach to irrational rotations

  • Claudio Bonanno
  • Stefano IsolaEmail author
Article

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.

Keywords

Irrational rotations Continued fractions Renormalization Diffusion 

Mathematics Subject Classification (2000)

37E05 37A25 37A45 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica ApplicataUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CamerinoCamerinoItaly

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