Infinite Compositions and Complex Dynamics; Generalizing Schröder and Abel Functions

The paper’s Theorem 4.1 aims to construct an inverse Schröder function under assumptions (1)–(3) via an iterative correction vn and a Banach fixed point argument, but the key contraction step is not justified: the sup-norm estimate that introduces a factor q<1 from the rescaling w↦λw is asserted without conditions ensuring strict contraction, and the O(w)-asymptotic near w=0 is invoked without a complete derivation from earlier results that are stated for a different scaling regime. The model’s solution gives a cleaner plan (Denjoy–Wolff/Koenigs coordinate + logarithmic covering), but it silently assumes the existence of a simply connected, forward-invariant ‘tail’ A∞ on which a univalent Fatou–Abel coordinate exists; this is plausible under the hypotheses yet not proved in the write-up. Hence both proofs have important gaps.