On Dynamics of Cubic Siegel Polynomials

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 …