The Parabolic and Near-Parabolic Renormalization for a Class of Polynomial Maps and Its Applications

The candidate solution is a faithful specialization of Zhang’s result to the one-parameter family P_α(z)=e^{2πiα}z(1+z)^m, relying on exactly the parabolic/near‑parabolic renormalization scheme and the Buff–Chéritat–style area argument developed in the paper. The paper proves existence of non‑renormalizable polynomials with a Cremer fixed point and positive‑area Julia set for degrees ≥22 (Theorem 1.1), and Section 3 works directly with P_α. Minor clarifications: the renormalization framework is proved for m≥22 (so deg P_α=m+1≥23), and the near‑parabolic renormalization requires small complex α with angle bounds and certain high‑type conditions. These do not contradict the candidate’s existence claim for m≥22 but should be noted.