DYNAMICS OF TRANSCENDENTAL HÉNON MAPS III: INFINITE ENTROPY

The paper proves that any transcendental Hénon map F(z,w)=(f(z)−δw,z) has infinite topological entropy via a two-case analysis on the rescalings fn(z)=f(nz)/n: (i) quasi-normal family ⇒ F is Hénon-like of arbitrarily large degree on a fixed bidisk, hence entropy ≥ log d for arbitrarily large d (using Dujardin’s result), and (ii) not quasi-normal ⇒ by an Ahlfors–islands argument one obtains a forward-invariant compact set carrying a symbolic dynamics with ≥ k−2 symbols, giving arbitrarily large entropy. These steps are stated and proved in Theorems 3.2 and 3.9, with the key Lemma 3.8 and Proposition 3.4 in the quasi-normal case, and Lemma 3.10 and the entropy count in the non–quasi-normal case. The candidate solution reproduces the same conjugacy, the same two-case split, the Hénon-like reduction, and the islands/graph-lifting construction, differing only in notation (R vs. n) and a minor counting constant (m vs. k−2); substantively, it matches the paper’s proof. See Theorem 1.1 and the outlined strategy in the introduction, and the detailed arguments in Section 3 of the paper.