On amenability and measure of maximal entropy for semigroups of rational maps: II.

The paper’s Theorem 1 constructs an S-invariant probability measure from a left-invariant mean associated to the Koopman anti-representation by embedding C(Ĉ) into L∞(S) via H(φ)(g)=∫Kg(φ)dσ and then applying the Riesz representation theorem; invariance follows from left-translation on L∞(S) . The model’s solution embeds C(Ĉ) using point evaluations Fφ(s)=φ(s(x0)) (also elements of Xρ by definition of ρ-amenability ) and applies the same mean-then-Riesz strategy. Both arguments rely on the Koopman operator KR(φ)=φ∘R and the same left-invariance mechanism; they differ only in the choice of the test functionals (Dirac vs. an arbitrary σ), so they are essentially the same proof.