Pacman Renormalization and local connectivity of the Mandelbrot set at some satellite parameters of bounded type.
We consider the unstable manifold of a pacman renormalization operator constructed in a joint work with Mikhail Lyubich and Nikita Selinger. Maps on the unstable manifold are 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 prove 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 and have positive area.
Joint work with Mikhail Lyubich.