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

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.