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arXiv · 1808.10425

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

Dzmitry Dudko·Mikhail Lyubich
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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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Dzmitry Dudko, Mikhail Lyubich. 2018-08-30. Local connectivity of the Mandelbrot set at some satellite parameters of bounded type. https://arxiv.org/abs/1808.10425

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