Asymptotics of ODE’s flows everywhere or almost-everywhere in the torus: from rotation sets to homogenization of transport equations

The paper proves two core equivalences: (A) Db = {ζ} if and only if X(t,x)/t → ζ for Lebesgue-a.e. x under the existence of an a.e.-positive invariant density σ0 with div(σ0 b)=0 (Theorem 2.1), and (B) when div b=0, that same almost-everywhere asymptotics is equivalent to homogenization of the oscillatory transport equation to a constant drift ζ (Theorem 3.2) . The model’s Part A matches the paper’s logic (Birkhoff averages plus invariant densities) though framed via conditional expectations; the paper’s proof uses an auxiliary lemma on invariant functions/measures (Lemma 2.1) to reach the same conclusion . For Part B, the model obtains homogenization through a characteristic-based argument using the flow identity Φε(t,x)=εX(t/ε,x/ε), while the paper establishes it via two-scale convergence and an invariance argument that identifies the effective flux as U b = u ζ; both routes are standard and consistent, yielding the same limit and identification ζ = ∫Yd b (for div b=0) . Thus the results agree; the model’s proof is correct and largely aligns with the paper conceptually, but differs in technical presentation for the homogenization step.