Khinchin theorem for interval exchange transformations

TL;DR

This paper extends Khinchin's theorem to interval exchange transformations by defining a diophantine condition, establishing a dichotomy, and relating it to the Rauzy-Veech algorithm, thus advancing understanding of Diophantine approximation in dynamical systems.

Contribution

It introduces a diophantine condition for i.e.t.s, generalizes Khinchin's theorem, and connects it with the Rauzy-Veech algorithm, providing new insights into their approximation properties.

Findings

Established a Khinchin-type dichotomy for i.e.t.s.

Defined a diophantine condition aligning with classical case for rotations.

Linked Rauzy-Veech algorithm with homogeneous approximations.

Abstract

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