2021/03/01 by Yimin Wang, Wang, Yimin
Mathematics · #37F10 #37F20 #37F25 #37F30 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2103.00732
openalex publication_date 2021/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f0 be a polynomial of degree d1+d2 with a periodic critical point 0 of multiplicity d1-1 and a Julia critical point of multiplicity d2. We show that if f0 is primitive, free of neutral periodic points and non-renormalizable at the Julia critical point, then the straightening map χf0:\mathcal C(λf0) → \mathcal Cd1 is a bijection. More precisely, fm0 has a polynomial-like restriction which is hybrid equivalent to some polynomial in \mathcal Cd1 for each map f ∈ \mathcal C(λf0), where m0 is the period of 0 under f0. On the other hand, f0 can be tuned with any polynomial g∈ \mathcal Cd1. As a consequence, we conclude that the straightening map χf0 is a homeomorphism from \mathcal C(λf0) onto the Mandelbrot set when d1=2. This together with the main result in [SW] solve the problem for primitive tuning for cubic polynomials with connected Julia sets thoroughly.