On the ergodicity of infinite antisymmetric extensions of symmetric IETs

Abstract

We consider skew product extensions over symmetric interval exchange transformations on the unit interval $[0,1)$ with respect to the cocycle $f={\chi }_{(0,1/2)}-{\chi }_{(1/2,1)}$. We prove that for almost every interval exchange transformation $T$ with symmetric combinatorial data, the skew product $T_f:[0,1)\times \mathbb{Z}\to [0,1)\times \mathbb{Z}$ given by $T_f(x,r)=(T(x),r+f(x))$ is ergodic with respect to the product of the Lebesgue and counting measures.