Ergodicity of skew-products over typical IETs

TL;DR

This paper proves that a broad class of infinite measure-preserving skew-product systems, built over typical interval exchange transformations with piecewise constant functions, are ergodic, expanding understanding of their long-term statistical behavior.

Contribution

It establishes ergodicity for skew-products over typical IETs with piecewise constant functions, a significant extension in the study of infinite measure systems.

Findings

Systems are ergodic with respect to Lebesgue measure for typical parameters.

Ergodicity holds for a broad class of skew-products over IETs.

Results contribute to the understanding of statistical properties of infinite measure systems.

Abstract

We prove ergodicity of a class of infinite measure preserving systems, called skew-products. More precisely, we consider systems of the form \[ {T_f}:{[0, 1) \times \mathbb{R}}\to{[0, 1) \times \mathbb{R}},\quad {T_f(x, t)}:={(T(x), t+f(x))}, \] where $T$ is an interval exchange transformation and $f$ is a piece-wise constant function with a finite number of discontinuities. We show that such system is ergodic with respect to ${Leb}_{[0,1)\times \mathbb{R}}$ for a typical choice of parameters of $T$ and $f$.