Kitai’s Criterion for Composition Operators

The paper proves (and explicitly constructs) an invertible measurable system whose composition operator on L^p is topologically mixing yet fails Kitai’s Criterion, via a careful MRC/KRC analysis and a tailored space X; this is Theorem 3.5, with the construction and verification detailed in their proof and supported by their equivalence theorems for mixing and Kitai’s Criterion . By contrast, the candidate’s simpler atomic example on Z is not mixing: by the paper’s shift criterion (Theorem 5.1), mixing of the backward shift requires that both e_{j+k} and e_{j−k} tend to 0, which fails here because the negative weights stay at 1, so ‖e_{j−k}‖ ≡ 1 . The candidate’s direct mixing proof also overlooks the persistent “left-tail” contribution T^n u, so T^n x^(n) cannot approximate v in norm.