Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters

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. We show that Siegel renormalization periodic po...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2020-07, Vol.33 (3), p.653-733
Main Authors: Dudko, Dzmitry, Lyubich, Mikhail, Selinger, Nikita
Format: Article
Language:English
Subjects:
Dynamical Systems
Mathematics
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Summary: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. 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 one-dimensional unstable manifold, resolving a long-standing problem. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.
ISSN:0894-0347
1088-6834
DOI:10.1090/jams/942