Journal of Modern Dynamics (JMD)

The Khinchin Theorem for interval-exchange transformations

Pages: 123 - 183, Issue 1, January 2011      doi:10.3934/jmd.2011.5.123

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Luca Marchese - Section de mathématiques, case postale 64, 2–4 Rue du Lièvre, 1211 Genève, Switzerland (email)

Abstract: We define a Diophantine condition for interval-exchange transformations. When the number of intervals is two, that is, for rotations on the circle, our condition coincides with the classical Khinchin condition. We prove for interval-exchange transformations the same dichotomy as in the Khinchin Theorem. We also develop several results relating the Rauzy-Veech algorithm with homogeneous approximations for interval-exchange transformations.

Keywords:  Interval-exchange transformations, Diophantine conditions, Rauzy– Veech induction, Teichmüller dynamics.
Mathematics Subject Classification:  Primary: 37A20, 37E05; Secondary: 32G15, 11K55.

Received: April 2010;      Revised: December 2010;      Available Online: April 2011.