Ergodic properties of infinite extension of symmetric interval exchange transformations

TL;DR

This paper characterizes when certain skew products over symmetric interval exchange transformations are ergodic, extending the understanding of their ergodic properties and showing conditions for infinite ergodic index.

Contribution

It provides a full characterization of ergodic extensions for skew products with linear cocycles over symmetric IETs, including cases with differentiable functions and weakly mixing IETs.

Findings

Skew products with linear cocycles are ergodic over ergodic symmetric IETs.

Under unique ergodicity, the result extends to differentiable functions with non-zero jump sums.

Skew products over weakly mixing IETs have infinite ergodic index.

Abstract

We prove that skew products with the cocycle given by the function $f(x)=a(x-1/2)$ with $a\neq 0$ 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 cocycle given by $f$ have infinite ergodic index.