Pacman renormalization and selfsimilarity of the Mandelbrot set near Siegel parameters
Abstract
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 quadraticlike and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with onedimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.01206
 Bibcode:
 2017arXiv170301206D
 Keywords:

 Mathematics  Dynamical Systems
 EPrint:
 wrt version 1: Appendix B is revised  the proof of Theorem B.8 is simplified