CRITERIA FOR EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES OF HOLOMORPHIC SELF-MAPS IN THE UNIT DISC

The paper proves Theorem 1.2 case-by-case and hinges on the characterization that ψ_t commutes with φ for all t>0 iff the Koenigs function H of (ψ_t) satisfies H∘φ = H + c (Proposition 3.3). The candidate solution follows the same reduction and treats (a)–(f) exactly as in the paper: (b) via the centralizer structure in the hyperbolic/zero-step case; (c) using the canonical model and two incommensurable periods; (d) via Theorem 1.3; (e) via Proposition 8.1; and (f) via Theorem 1.4. These match the paper’s Section 9 proofs and supporting results, with only a minor notation conflation between the base space S and the image Ω = H(D). Overall, both are correct and essentially the same argument. See Theorem 1.2 and its proof, Proposition 3.3, Theorem 1.3, Proposition 8.1, and Theorem 1.4 in the paper .