Coboundaries of 3-IETs

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.