Coboundaries of 3-IETs

In this note, we investigate coboundaries of interval exchange transformations of three intervals, or 3-IETs.

More precisely, we study differentiable functions whose derivative is absolutely continuous, and whose integral is zero and whose derivative also has integral zero.

We show that these functions are coboundaries for a typical 3-IET if and only if their values at the endpoints of the domain are zero.

We also show the existence of exceptional counterexamples for both possible endpoint behaviors.

Our results are obtained by studying the properties of associated skew products.