RELATION BETWEEN HÉNON MAPS WITH BIHOLOMORPHIC ESCAPING SETS

The paper proves the main theorem rigorously: if I_H^+ and I_F^+ are biholomorphic, then there exist α, β, γ with α^{d+1} = β and β^{d−1} = γ^{d−1} = 1 such that β p_H(y) = α p_F(α y), a_H = γ a_F, and F ≡ L ◦ B ◦ H ◦ B with the stated linear maps B and L. This is explicitly stated and derived in Theorem 1.1 and the ensuing sections (lifting to the canonical cover, deck group analysis, the Q-polynomial relation β Q_H(ζ) = Q_F(α ζ), and the passage from Q to p via Böttcher expansions) . By contrast, the model’s argument contains critical inaccuracies: (i) it writes the key identity as β Q_H(ζ) = α^{d+1} Q_F(α ζ), which would force β = α^{2d+2} upon comparing leading terms, contradicting the paper’s α^{d+1} = β; the paper’s correct relation is β Q_H(ζ) = Q_F(α ζ) ; (ii) it asserts α^{d−1} = 1 (not implied by α^{d+1} = β and β^{d−1} = 1), and (iii) it claims a_H/a_F = β, whereas the paper only proves a_H^{d−1} = a_F^{d−1} and writes a_H = γ a_F with γ^{d−1} = 1, independent of β . The model also compresses the delicate step from Q-identities to p-identities using an unsubstantiated “triangular correspondence,” whereas the paper supplies a careful derivation via Böttcher expansions (Sections 4–5) . Hence, while the model’s final conclusion matches the paper’s theorem, its proof has substantive errors.