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

TL;DR

This paper investigates the local connectivity and geometric properties of the Mandelbrot set and Julia sets near specific parameters, revealing new examples of local connectivity and self-similarity in complex dynamics.

Contribution

It demonstrates local connectivity of the Mandelbrot set at certain satellite parameters and shows Julia sets are locally connected with positive area, introducing new cases of these properties.

Findings

Mandelbrot set is locally connected at certain satellite parameters

Julia sets are locally connected and have positive area

Mandelbrot set exhibits self-similarity near Siegel parameters

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.