ON DYNAMICS OF CUBIC SIEGEL POLYNOMIALS
ON DYNAMICS OF CUBIC SIEGEL POLYNOMIALS
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发表时间：
1998
期刊：
影响因子：
9.4
通讯作者：
Renormalizable Cubics
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文献类型：
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作者：
Renormalizable Cubics
设f是复平面上d ≥ 2次的多项式，考虑下列命题：（Ad）“如果f有一个固定的有界型旋转数的Siegel圆，则θ是一个通过f的某个临界点的拟圆。“（B d）“如果f有一个固定的Siegel圆盘，使得φ是一个通过f的某个临界点的拟圆，则φ的旋转数是有界型的。“陈述（A 2）是Douady，Ghys，赫尔曼和Shishikura的定理，（B d）是开的，甚至对d = 2也是如此，本文的主要目的是证明（A 3）：定理14.7。设P是一个三次多项式，它有一个固定的旋转数为θ的Siegel圆盘.设θ为有界型。然后，我们证明了三次Siegel多项式的动力学性质的几个结果，其中沿着，我们证明了三次Siegel多项式的动力学性质。实际上，我们研究了三次参数空间中的一维切片Pcm 3（θ），该空间由具有给定旋转数θ的固定Siegel圆盘的所有三次函数组成.许多结果适用于一般的Brjuno型θ。西格尔圆盘是全纯动力系统中准周期运动的例子。设p是有理映射f：C → C的无理中立不动点。这意味着f（p）= p和乘数f ′（p）= λ的形式为e 2πiθ，其中旋转数0 < θ < 1是无理数。称p是可线性化的，如果在p附近存在一个全纯坐标变换，使f与刚性转动z → λz共轭。这种线性化可能的最大区域是一个单连通区域，称为f的以p为中心的西格尔圆盘。换句话说，存在一个共形同构h：（D，0）<$−→（λ z，p）使得h（λz）= f（h（z））对所有z ∈ D，且λ z不包含在任何具有此性质的更大区域中。虽然Siegel圆盘是f的Fatou集的一个分支，但Siegel圆盘的边界是Julia集的一个子集。每一个被刺穿的西格尔圆盘{p}都由动态定义的...
Let f be a polynomial of degree d ≥ 2 in the complex plane and consider the following statements: (A d) " If f has a fixed Siegel disk ∆ of bounded type rotation number, then ∂∆ is a quasicircle passing through some critical point of f. " (B d) " If f has a fixed Siegel disk ∆ such that ∂∆ is a quasicircle passing through some critical point of f , then the rotation number of ∆ is bounded type. " Statement (A 2) is a theorem of Douady, Ghys, Herman, and Shishikura, (B d) is open, even for d = 2, and the main object of this work is to prove (A 3): Theorem 14.7. Let P be a cubic polynomial which has a fixed Siegel disk ∆ of rotation number θ. Let θ be of bounded type. Then the boundary of ∆ is a quasicircle which contains one or both critical points of P. Along the way, we prove several results about the dynamics of cubic Siegel polynomials. In fact, we study the one-dimensional slice P cm 3 (θ) in the cubic parameter space which consists of all cubics with a fixed Siegel disk of a given rotation number θ. Many of the results apply to general θ of Brjuno type. Siegel disks are examples of quasiperiodic motion in holomorphic dynamical systems. Let p be an irrationally indifferent fixed point of a rational map f : C → C. This means that f (p) = p and the multiplier f ′ (p) = λ is of the form e 2πiθ , where the rotation number 0 < θ < 1 is irrational. p is called linearizable if there exists a holomorphic change of coordinates near p which conjugates f to the rigid rotation z → λz. The largest domain on which this linearization is possible is a simply-connected domain ∆ which is called the Siegel disk of f centered at p. In other words, there exists a conformal isomorphism h : (D, 0) ≃ −→ (∆, p) such that h(λz) = f (h(z)) for all z ∈ D, and ∆ is not contained in any larger domain with this property. While the Siegel disk ∆ is a component of the Fatou set of f , the boundary of ∆ is a subset of the Julia set. Every punctured Siegel disk ∆ {p} is foliated by dynamically-defined …