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.

中文翻译：

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Siegel 参数附近 Mandelbrot 集的 Pacman 重整化和自相似性

在 1980 年代，Branner 和 Douady 发现了一项与 Mandelbrot 集的各个肢体相关的手术。我们将此手术置于“吃豆子重整化理论”的框架中，该理论结合了二次类重整化和 Siegel 重整化的特征。我们表明 Siegel 重整化周期点（由 McMullen 在 1990 年代构建）可以提升为 pacman 重整化周期点。然后我们证明这些周期点是具有一维不稳定流形的双曲线。因此，我们获得了 Mandelbrot 卫星组件中心在相应 Siegel 参数附近的缩放定律。