Frank Trujillo, University of Zûrich
Real skew-product extensions of an ergodic transformation with respect to a mean zero cocycle are known to be recurrent and conservative. However, establishing ergodicity for a specific skew-product is generally very difficult.
For almost every interval exchange transformation (IET) with a symmetric permutation, we construct explicit examples of cocycles for which the associated skew-product is ergodic. More precisely, for almost every IET with a symmetric permutation, we provide sufficient conditions on a cocycle having logarithmic singularities for the real skew-product extension to be ergodic. As a corollary, we obtain the ergodicity of certain extensions of locally Hamiltonian flows in genus two having isomorphic saddles. This is a joint work with Przemysław Berk and Corinna Ulcigrai.