Discrete-time dynamics, step-skew products, and pipe-flows

The paper’s Theorem 5 (weak conditional convergence of a time-3 sampled perturbed pipe-flow to a given step-skew product) is essentially correct in construction and intent, but one step informally invokes “mixing” where the proof mechanism actually needs multiple mixing (or an explicitly Bernoulli/multiply-mixing driver) to justify near-independence across separated windows; this is easily fixed by strengthening the driver assumption and is hinted at elsewhere in the text via Lemma 3.1 (multiple mixing) and its use in the switching argument (eqs. (20)–(23)) . The candidate model reproduces the junction/pipe network faithfully and smartly picks a Poisson-configuration translation flow (which supplies exact independence across disjoint windows), but it bases the junction’s “which exit to open” decision on membership of the entire window sample, which is non-causal in the prescribed skew-product form ẏ = V(ω(t), y). The paper implements a causal aggregator via a weighted time-average excitation (eq. (21)) to realize the choice with the right law; the model omits this causalization detail. Thus, the paper needs a minor hypothesis sharpened, and the model needs a causal routing mechanism to be complete. Core constructions (junctions, pipes, gluing lemma, and the weak conditional convergence statement (28) in Theorem 5) align well .