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

Dudko, Dzmitry; Lyubich, Mikhail; Selinger, Nikita

doi:10.1090/jams/942

Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters
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by Dzmitry Dudko, Mikhail Lyubich and Nikita Selinger

J. Amer. Math. Soc. 33 (2020), 653-733

DOI: https://doi.org/10.1090/jams/942

Published electronically: June 16, 2020

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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 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.

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