PROPERTIES OF ACTION-MINIMIZING SETS AND WEAK KAM SOLUTIONS VIA MATHER’S AVERAGING FUNCTIONS

The uploaded paper proves the equivalence of five statements (Theorem 1.1) linking differentiability of β on ∂α(c), the exposedness of c for α, and disjointness of Mather/Aubry/Mañé sets for different cohomology classes. The candidate solution establishes the same equivalences using standard convex duality, ρ(Mc)=∂α(c), and semistatic time-averaging to produce common minimizing measures—exactly the tools the paper also uses (via Proposition 2.2, Proposition 3.1, and weak KAM facts). The paper’s proof of (i)⇔(ii) uses subdifferentials and Proposition 3.1, while the model’s proof uses an equivalent convex-analysis lemma; the (ii)⇔(v) direction is handled in the paper by an inclusion/equivalence between supports and Mañé sets, and by time-averaging along semistatic curves in the model. These are substantively the same strategy. See Theorem 1.1, Proposition 2.2, and Proposition 3.1 in the paper for the formal statements and proof structure .