Regular Interval Exchange Transformations over a Quadratic Field

TL;DR

This paper extends classical results on quadratic irrationals to interval exchange transformations over quadratic fields, demonstrating finiteness properties using a two-sided Rauzy induction.

Contribution

It generalizes Lagrange's theorem to interval exchange transformations over quadratic fields, employing a two-sided Rauzy induction approach.

Findings

Finite number of transformations up to homothety from quadratic field intervals

Extension of classical continued fraction results to interval exchange transformations

Use of two-sided Rauzy induction to analyze transformation dynamics

Abstract

We describe a generalization of a result of Boshernitzan and Carroll: an extension of Lagrange's Theorem on continued fraction expansion of quadratic irrationals to interval exchange transformations. In order to do this, we use a two-sided version of the Rauzy induction. In particular, we show that starting from an interval exchange transforma- tion whose lengths are defined over a quadratic field and applying the two-sided Rauzy induction, one can obtain only a finite number of new transformations up to homothety.