By using the properties of symmetric interval exchange transformations, we show how to control Birkhoff sums of functions satisfying certain "oddity" assumptions. In particular we show ergodicity of skew products over a typical symmetric interval exchange transformation given by some cocycles with symmetric logarithmic singularities as well as by a characteristic function of the interval (0,1/2).

This is joint work with Frank Trujillo and Corinna Ulcigrai.