ALMOST POLYNOMIAL-LIKE MAPS IN POLYNOMIAL DYNAMICS

The paper proves that under renormalizable wedges and eventual mapping of critical root points to repelling cycles, P|AP(W) is only guaranteed to be topologically conjugate to a lower-degree polynomial, via an almost polynomial-like (APL) surgery and APL straightening. It explicitly provides cases where AP(W) is not the filled set of any polynomial-like restriction of P (e.g., the cubic example in Figure 1), contradicting the model’s claim that one can always construct a holomorphic polynomial-like restriction with K(f)=AP(W) under these weaker assumptions. The model’s argument matches only the stronger classical setting of Theorem 1.2 (no critical or outward parabolic root points), not the paper’s main theorem.