Siegel圆盘的柱面重整化

Siegel圆与重整化不动点

Siegel disks and renormalization fixed points

本文构造了Siegel圆盘映射柱重整化的双曲不动点。

Golden mean Siegel disk universality and renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like ma…

Renormalization and Siegel disks for complex Hénon maps

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.

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 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 t…

Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Universality and Non-Rigidity

It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the g…

Siegel圆盘的柱面重整化和一个构造性的可测黎曼映射定理

Cylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem

研究了具有非理性中立不动点的二次多项式的Siegel圆盘的边界是自相似的，该不动点具有黄金平均旋转数.这种自相似性的几何对于一大类全纯映射是普适的。在文献中已经提出了这种普遍性的重整化解释。然而，这种解释的一个成分，重整化的双曲性，尚未得到证明。目前的工作考虑了一个圆柱重整化-一种新的类型的重整化的全纯映射与西格尔盘，这是更适合于双曲性证明。全纯映射的柱重整化的一个关键元素是$\field{C}$的一个子集到双无限柱$\field{C} / \field{Z}$的动力学商的共形同构。这种共形同构的构造是一个隐式过程，可以使用可测黎曼映射定理来执行。给出了可测黎曼映射定理的一个构造性证明，并得到了所需共形同构的数值逼近的严格界。这种一致化共形坐标的控制对于严格的计算机辅助研究柱体重整化是至关重要的。