Coboundaries of 3-IETs

TL;DR

This paper characterizes when differentiable functions are coboundaries for typical 3-interval exchange transformations, revealing conditions on boundary values and the existence of rare counterexamples.

Contribution

It provides a precise criterion for coboundaries in 3-IETs and explores the existence of exceptional cases using skew product analysis.

Findings

Differentiable functions with zero boundary values are coboundaries for typical 3-IETs.

Counterexamples exist with non-zero boundary values, but are rare.

The study employs properties of associated skew products to derive results.

Abstract

In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function with absolutely continuous derivative with bounded variation, whose integral and integral of its derivative is 0, is a coboundary for typical 3-IET if and only if the values at the endpoints of the domain are zero. We also show the existence of rare counterexamples for both cases of possible values at the endpoints of the interval. We obtain our result by studying the properties of associated skew products.