On Hyperbolic Sets of Polynomials

The paper proves Theorem 1.1 (J∞ contains no hyperbolic sets) by an accessibility-and-angles argument together with a combinatorial lemma for the doubling map, culminating in a contradiction via preimage-counting in continua; the proof is complete and self-contained. The model’s proposed proof is not sound as written: it crucially invokes an unproven assumption that each renormalization F_n is persistently recurrent and that ω(F_n,0) is a minimal set (a nontrivial Lyubich result whose applicability here is neither justified nor shown to hold for all infinitely renormalizable maps), and it also contains a logical misstep when deducing ω(f,0) ⊂ ⋃_j f^j(J_n). Without these, the model’s argument does not go through.