Local Connectivity of the Mandelbrot Set at Some Satellite Parameters of Bounded Type

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 periodic type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed by the authors jointly with N. Selinger in [DLS] as a global transcendental family. It is the first occasion when external rays and puzzles of limiting transcendental maps are applied to study the Polynomial dynamics.