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

The paper proves a Rokhlin lemma for nigh aperiodic shuffleable systems by a self-contained argument that does not assume standardness; it builds n-chains using a pseudometric tied to the order-generated σ-algebra, applies Zorn’s lemma to obtain a maximal m-chain, and then constructs an n-chain covering ≥1−ε of the space (Theorem 9) . The candidate solution reduces “nigh aperiodic” to aperiodic (which is fine) but then applies Avila–Candela’s towers theorem, which requires a standard probability space; shuffleable systems need not be standard or complete, as the paper explicitly discusses and illustrates with pathologies (Definition 5; Example and discussion; completed shuffleable systems; Proposition 16) . Thus, the model’s Step 2 invokes a result outside its domain of validity and does not ensure the base set E lies in the original F(<).