Aubry-Mather Theory for Contact Hamiltonian Systems II

Theorem 1.3 is stated and proved in the paper using a Busemann-type construction (Lemma 2.1) to obtain backward/forward weak KAM solutions from semi-static curves and then establishing the covering, local, and global characterizations for ѱ and Ñ (see Theorem 1.3 and Section 2). This route is designed to handle the failure of non-expansiveness of the Lax–Oleinik semigroups under |∂H/∂u| ≤ λ. By contrast, the model’s proof outline assumes semigroup monotonicity/contractivity under (H3) and defines a backward weak KAM solution from a single basepoint on a positively semi-static orbit; it also mixes the positive/negative directions when proving the covering property. These steps bypass the paper’s key Busemann construction and rely on properties that the paper explicitly warns may fail. Hence the model’s argument is incomplete and directionally inconsistent, whereas the paper’s proof is coherent and correct (Theorem 1.3, Lemma 2.1, Proposition 2.2, Lemma 2.5, and the closedness argument). Citations: Theorem 1.3 and its definitions appear in 1.3–1.4 and are proved in Section 2; see Theorem 1.3 and discussion around Σ̃v±, Ĩv±, and the semigroup limits v± (lim T±t v∓) in the introduction and Section 2; the Busemann construction is Lemma 2.1 in Section 2; Prop. 2.2 gives Gv− ⊆ Ñ− and πÑ− = M; Lemma 2.5 yields the differentiability equality and calibration identities at Iv−; the closedness of Ñ is proved in Section 2.4.