On PZ type Siegel disks of the sine family

Abstract

We prove that for typical rotation numbers 0 < θ < 1, the boundary of the Siegel disk of f θ (z) = e 2πiθ sin(z) centered at the origin is a Jordan curve which passes through exactly two critical points π/2 and −π/2. Statement of the main resultLet 0 < θ < 1 be an irrational number and [a 1 , . . . , a n , . . .] be its continued fraction. We say that θ is of bounded type if sup{a n } < ∞. The set of bounded type irrational numbers has zero Lebesgue measure. The following theorem was proved in [10]. THEOREM 1.1. Let 0 < θ < 1 be a bounded type irrational number. Then the Siegel disk of e 2πiθ sin(z) centered at the origin is a quasi-disk with boundary containing exactly two critical points π/2 and −π/2. Theorem 1.1 was proved by performing quasi-conformal (qc) surgery on an appropriate premodel. In [7] the authors there proposed a surgery technique to construct Jordan Siegel disks with typical rotation numbers. This surgery technique was pioneered by Haissinsky, who used it to transform an attracting basin into a parabolic basin, and is now called transqc surgery. Petersen asked if Theorem 1.1 can be extended to the case of typical rotation numbers through trans-qc surgery. The main purpose of this paper is to prove the following theorem.MAIN THEOREM. For typical irrational numbers 0 < θ < 1, the Siegel disk of e 2πiθ sin(z) centered at the origin is a Jordan domain with boundary containing exactly two critical points π/2 and −π/2. Compared with qc surgery, the main difficulty in performing trans-qc surgery is to verify the integrability of certain degenerate Beltrami differentials. This often requires some delicate area estimates. In [7] the authors used Petersen puzzles developed in [6] to obtain the desired estimate for the Douady-Ghys Blaschke model. Such puzzle construction does not exist in constructing Siegel disks of entire functions. In fact it is not known if the

Polynomial Siegel disks are typically Jordan domains

“…It is easy to see that Z n is the outer half of an open neighborhood of T. See Figure 2 Proof. The argument is completely the same as the one used in the proof of Proposition 8.2 in [36]. The reader may refer to [36] for the details.…”

“…As we mentioned before the main difficulty in performing a trans-qc surgery is to verify the integrability of certain degenerate Beltrami differential. The key idea used in the proof of Lemma 2.2 is a method developed in [36] which allows us to obtain a uniform area estimate. Lemma 2.3 (Tukia, [31]).…”

“…The remaining part of this section is devoted to the proof of Lemma 4.16. The idea of the proof is adapted from [36]. Before we present the proof, let us introduce some notations and terminologies first.…”

“…This is the place where we use transqc surgery. The tool developed in [36] will play a role here. It allows us to make a uniform area estimate for the Beltrami differentials of a special class of premodels.…”

On Some Advances in Dynamics of One Variable Complex Functions

“…In [64], a result on Siegel disk is rigorously shown as: For 0 < θ < 1, the boundary of the Siegel disk of f θ (z) = e 2πiθ sin(z) centered at the origin is a Jordan curve which goes through exactly 2 critical points π/2 and −π/2. Here θ refers as of Petersen-Zakari (PZ) type and [a 1 , a 2 , .…”