Ergodic Properties of Compositions of Interval Exchange Maps and Rotations

TL;DR

This paper investigates the ergodic behavior of compositions involving interval exchange transformations and rotations, demonstrating that most such compositions are uniquely ergodic.

Contribution

It proves that for any interval exchange transformation, almost all rotations produce a composition that is uniquely ergodic, extending understanding of ergodic properties in dynamical systems.

Findings

For any interval exchange transformation T, there exists a full measure set of  in [0,1) such that T composed with R_ is uniquely ergodic.

Most compositions of interval exchange maps with rotations are uniquely ergodic.

The result applies to a broad class of transformations, indicating typical ergodic behavior in these systems.

Abstract

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of \alpha in [0, 1) so that T composed with R_{\alpha} is uniquely ergodic, where R_{\alpha} is rotation by \alpha.