Continuous Interval Exchange Actions

TL;DR

This paper classifies all continuous one-parameter actions of interval exchange transformations, showing they essentially reduce to rotations on a torus, thus revealing their fundamental structure.

Contribution

It provides a classification of continuous homomorphisms from R to the group of interval exchange transformations, demonstrating they factor through torus rotations.

Findings

All one-parameter interval exchange actions are conjugate to torus rotations.

The classification relies on a suitable topological group structure on E.

The results unify the understanding of continuous interval exchange actions.

Abstract

Let E denote the group of all interval exchange transformations on [0,1). Given a suitable topological group structure on E, it is possible to classify all one-parameter interval exchange actions (continuous homomorphisms from R to E). In particular, up to conjugacy in E, any one-parameter interval exchange action factors through a rotational torus action.