Factorization of the quadratic Misiurewicz-Thurston polynomials

2025/06/21 by N Mihalache, Mihalache, Nicolae, François Vigneron +1
Mathematics · #37F10 #37F20 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2506.17662

openalex publication_date 2025/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note provides the complete factorization of the Misiurewicz-Thurston polynomial q_ℓ,n=p_ℓ+n(z) - p_ℓ(z) over ℂ, which plays a central role in the study of the Mandelbrot set, where p₀(z) = 0, p_n+1(z) = p_n(z)² + z. The roots can be classified into two categories. First, there are hyperbolic points hyp(k) for any divisor k of n, which are parameters whose critical orbits are of exact period k. Those are roots of q_ℓ,n with multiplicity \lfloor (ℓ -1)/(k) \rfloor + 2. Next are the points mis(j,k) for 2≤ j≤ ℓ whose critical orbits are pre-periodic of exact period k with an exact pre-period j. Those are simple roots of q_ℓ,n.

Citations

• Centers of hyperbolic components in the Mandelbrot set, up to period 32, rounded to nearest 10^-26
• Misiurewicz Points for Polynomial Maps and Transversality

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