Pacman Renormalization and Scaling of Satellite Mandelbrot Copies near Siegel Points

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. We prove that each of these periodic points is hyperbolic with one-dimensional unstable manifold. We follow up with a description of the global structure of this unstable manifold, viewed as a one-parameter transcendental family, which yields various scaling laws for satellite Mandelbrot sets near Siegel parameters.
Based upon joint work with Dima Dudko and Nikita Selinger.