Local connectivity of the Mandelbrot set at some satellite parameters of bounded type

2018/08/30 by Dzmitry Dudko, Dudko, Dzmitry, Mikhail Lyubich +1 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1808.10425

openalex publication_date 2018/08/30 · openalex created_date 2023/03/16 · openalex updated_date 2026/07/28

Abstract

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

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