The Khinchin Theorem for interval-exchange transformations

Luca Marchese

doi:10.3934/jmd.2011.5.123

Article Contents

2011, Volume 5, Issue 1: 123-183. Doi: 10.3934/jmd.2011.5.123

This issue Previous Article Integrability and Lyapunov exponents Next Article Tori with hyperbolic dynamics in 3-manifolds

The Khinchin Theorem for interval-exchange transformations

Luca Marchese^1,

1.
Section de mathématiques, case postale 64, 2–4 Rue du Lièvre, 1211 Genève

Received: April 2010

Revised: December 2010

Published: April 2011

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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.

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Mathematics Subject Classification: Primary: 37A20, 37E05; Secondary: 32G15, 11K55.

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