Ergodicity of $\Z^2$ Extension of Irrational Rotation on the Circle

TL;DR

This paper investigates the ergodic properties of a $ ext{Z}^2$ skew product extension of irrational circle rotations, focusing on how the extension's structure influences its ergodic components.

Contribution

It provides a detailed analysis of the ergodic components of $ ext{Z}^2$ extensions of irrational rotations, a topic not extensively explored before.

Findings

Characterization of ergodic components for the extension

Conditions under which the extension is ergodic

Insights into the structure of invariant measures

Abstract

We consider skew product extension of irrational rotations on the circle by $\Z^2$ determined by an integer valued function as well as a fixed point on the circle. We study ergodic components of such extension.