Rigidity and non local connectivity of Julia sets of some quadratic polynomials

2007/10/24 by Genadi Levin, Levin, Genadi
Mathematics · #30D05 #37F45 #37F50 #37G15 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.CV #math.DS #msc:30D05 #msc:37F45 #msc:37F50 #msc:37G15

paper · pdf · doi:10.48550/arxiv.0710.4406

Final version of the Addendum, minor changes. To appear

openalex publication_date 2007/10/24 · arxiv created 2013/05/29 · arxiv updated 2015/03/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

For an infinitely renormalizable quadratic map f_c: z↦ z²+c with the sequence of renormalization periods k_m and rotation numbers t_m=p_m/q_m, we prove that if \limsup k_m^-1log |p_m|>0, then the Mandelbrot set is locally connected at c. We prove also that if \limsup |t_m+1|^1/q_m<1 and q_m→ ∞, then the Julia set of f_c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. Abstract of the Addendum: We improve one of the main results of the above paper.

Related

• Local connectivity of the Julia set of real polynomials
• Geometry of quadratic polynomials: moduli, rigidity and local connectivity
• Local connectivity of Julia sets: expository lectures
• Local connectivity of Julia sets for unicritical polynomials
• The Renormalization Method and Quadratic-Like Maps