ON AMENABLE SEMIGROUPS OF RATIONAL FUNCTIONS

The candidate solution reproduces Pakovich’s Theorem 1.1 for polynomial semigroups and follows the same structural route: left reversibility excludes special families (monomials/Chebyshev) and forces containment in a centralizer C(P); equivalently, S embeds into a semidirect-type semigroup S_{Γ,R} with γ_R(Γ)=Γ, which yields amenability; and when all degrees are ≥2, all pairs have a common iterate. These are exactly the paper’s equivalences (1)–(5) and the “furthermore” clause (common iterates) for non-special polynomial semigroups not conjugate to Z or T, as stated and proved in Theorem 1.1 and Section 7 (together with Theorems 7.5, 7.6, 7.7 and 1.4) . The only over-strong step in the model is asserting C(P)=S_{Aut(P),P} outright; the paper phrases this more carefully via C(P)=S_{Γ,R} for some R∈C(P) of minimal degree and suitable Γ (Theorem 7.5), which suffices for the chain (5)⇒(4) and aligns with the rest of the argument . Overall, the logic and implications match the paper, with minor presentational differences.