KNEADING THEORY FOR ITERATION OF MONOTONOUS FUNCTIONS ON THE REAL LINE

The paper proves that equality of kneading sequences (with matching domains and orientations and under a separability hypothesis) characterizes combinatorial equivalence, and—under an additional gap-endpoint condition—extends to a topological conjugacy on cl(CG). This is Theorem 1.1, with the “only if” direction covered by Proposition 3.6 and the “if” direction by Theorem 3.12 plus Corollary 3.13; the extension to closures is proved in the commentary following Theorem 1.1 (item (2)) (see Theorem 1.1 and its proof, and the chain Proposition 3.6 → Proposition 3.7 → Proposition 3.10 → Theorem 3.12 → Corollary 3.13 → extension argument ). The candidate solution reconstructs the same equivalence: it defines φ(fg(c)) = f̃g(c̃), proves well-definedness using future separability and addresses/Kneading data, shows order preservation via the orientation homomorphism, verifies domain compatibility, establishes conjugacy on CG, and extends φ by order-continuous limits to cl(CG) using the gap-endpoint condition. These steps mirror the paper’s approach and rely on the same key hypotheses (separability—past and future—and matching orientations/domains). Minor technical points (restricting “for all s ∈ S” to admissible words and explicitly invoking past-separability where needed) are easily patched and align with the paper’s formal lemmata. Hence both are correct and essentially the same proof strategy.