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

TL;DR

This paper develops a unifying renormalization framework for the Mandelbrot set near Siegel parameters, revealing self-similarity and scaling laws through hyperbolic periodic points in Pacman Renormalization Theory.

Contribution

It introduces Pacman Renormalization Theory, linking quadratic-like and Siegel renormalizations, and extends McMullen's Siegel periodic points to this framework, proving their hyperbolicity.

Findings

Periodic points are hyperbolic with one-dimensional unstable manifolds.

Scaling laws for satellite components near Siegel parameters are established.

The framework unifies previous surgery and renormalization approaches.

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. 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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