On the (Dis)connection Between Growth and Primitive Periodic Points

The paper proves that for any prescribed sequences of periods and rates there exists an order-zero holomorphic map F: C^2 → C^2 with at least m_n isolated p_n-primitive periodic points in B(0, 2^n+1), and with log M_F(r) = O((log r)^2), using a dispatcher function via Hörmander’s ∂̄-technique and a symmetrized Cornalba–Shiffman construction; isolation is checked via a nondegeneracy of d(F^{p_n})−Id at the constructed points (Observation 2) and growth is controlled uniformly (Observation 3) . The candidate’s proposal F(z,w)=(z,w^2) achieves ν_p(F,R)=∞ for all p and R≥2 by producing positive-dimensional families (vertical lines) of periodic points, which are not isolated; this evades the intended (and essential) isolated-point requirement emphasized by the authors in the abstract and enforced in the proof, so it does not solve the same problem the paper addresses .