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

Dzmitry Dudko, Mikhail Lyubich

Published 2018-08-30Version 1

We consider the unstable manifold of a pacman renormalization operator constructed in [DLS]. It comprises rescaled limits of quadratic polynomials. Every such limit admits a maximal extension to a sigma-proper branched covering of the complex plane. Using methods and ideas from transcendental dynamics, we show that certain maps on the unstable manifold are hybrid equivalent to quadratic polynomials. This allows us to construct a stable lamination in the space of pacmen. Combined with hyperbolicity of pacman renormalization, we obtain various scaling results near the main cardioid of the Mandelbrot set. As a consequence, we show that the Mandelbrot set is locally connected at certain infinitely renormalizable parameters of bounded satellite type (which provide first examples of this kind). Moreover, the corresponding Julia sets are locally connected as well. We then show, adapting the method of [AL2], that these Julia sets have positive area. (Note that the examples constructed in [BC] are of unbounded satellite type, while the examples from [AL2] are of bounded primitive type).