Time-periodic solutions of contact Hamilton-Jacobi equations on the circle

The paper proves (and details) the existence of infinitely many nontrivial time‑periodic viscosity solutions on the circle under (H1),(H2),(H4) together with (A), via weak KAM/implicit action techniques and the min–plus representation of the backward semigroup T⁻_t (Theorem 1.1 and Lemmas 2.1–2.6) . By contrast, the model’s core premise—that the Cauchy semigroup is an L∞ strict contraction with factor e^{-δ t} under ∂H/∂u ≤ −δ—is false here: the paper recalls that, in this contact setting, the solution semigroups are e^{κ t}-expansive, not contractive (Proposition 3.4(3)), and it uses the inf–convolution representation T⁻_t φ(x)=inf_y h_{y,φ(y)}(x,t) crucially (Proposition 3.4(5)) . The additional transversality hypothesis (A) is essential in the construction (e.g., B(x)=∂H/∂p≠0 on Λ_{u+}), contrary to the model’s claim that it “plays no role” .