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 = χ(0,½) - χ(½,1). We prove that for almost every interval exchange transformation T with symmetric combinatorial data, the skew product Tf : [0,1) x Z → [0,1) given by Tf(x,r) = (T(x),r+f(x)) is ergodic with respect to the product of the Lebesgue and counting measures.