Ergodic properties of infinite extension of symmetric interval exchange transformations

Abstract

We prove that skew products with the co cycle given by the function f (x) = a(x-1/2) with a not equal 0 are ergodic for every ergodic symmetric IET in the base, thus giving the full characterization of ergodic extensions in this family. Moreover, we prove that under an additional natural assumption of unique ergodicity on the IET, we can replace f with any differentiable function with a non-zero sum of jumps. Finally, by considering weakly mixing IETs instead of just ergodic, we show that the skew products with co cycle given by f have infinite ergodic index.