A Rokhlin Lemma for Noninvertible Totally-Ordered Measure-Preserving Dynamical Systems

TL;DR

This paper extends the classical Rokhlin lemma to noninvertible, totally-ordered measure-preserving dynamical systems, demonstrating that a version of the lemma holds under a mild aperiodicity condition.

Contribution

It introduces a Rokhlin lemma adaptation for noninvertible systems with total order structures, expanding the applicability of the classical result.

Findings

Rokhlin lemma can be adapted to noninvertible systems with total order

Requires a slight extension of aperiodicity for the adaptation

Comparison with previous noninvertible Rokhlin lemmas

Abstract

Let $(X,\mathcal{F},\mu,T)$ be a not necessarily invertible non-atomic measure-preserving dynamical system where the $\sigma$-algebra $\mathcal{F}$ is generated by the intervals according to some total order. The main result is that the classical Rokhlin lemma may be adapted to such a situation assuming a slight extension of aperiodicity. This result is compared to previous noninvertible versions of the Rokhlin lemma.