Ergodicity in Riesz spaces

TL;DR

This paper clarifies the concept of ergodicity in Riesz spaces, extending classical ergodic theorems and exploring the relationship between mixing and ergodicity in this mathematical framework.

Contribution

It defines the notion of ergodicity in Riesz spaces and discusses its connection with mixing, filling a gap in the theoretical understanding of ergodic properties in this setting.

Findings

Clarification of ergodicity concept in Riesz spaces

Analysis of the relationship between mixing and ergodicity

Extension of classical ergodic theorems to Riesz spaces

Abstract

The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Riesz space setting.