Dynamics of Non-Classical Interval Exchanges

TL;DR

This paper studies non-classical interval exchanges, a generalization of interval exchange maps related to measured foliations, demonstrating their unique ergodicity through a refinement process exhibiting uniform distortion.

Contribution

It introduces non-classical interval exchanges, analyzes their Rauzy induction, and proves their almost sure unique ergodicity using uniform distortion properties.

Findings

The refinement process exhibits uniform distortion.

Almost every non-classical interval exchange is uniquely ergodic.

Uniform distortion was independently established by Avila-Resende.

Abstract

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff's simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property called {\it uniform distortion}. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane's conjecture: almost every non-classical interval exchange is uniquely ergodic. Uniform distortion has been independently shown by Avila-Resende.