Abstract We prove that for any positive integer n, there is a polynomial P ( z ) with degree n such that the entire function G ( z ) = P ( sin ⁡ ( z ) ) has a bounded type Siegel disk bounded by a quasi-circle in the plane which passes through at least one critical point of G. In addition, let 0 θ 1 be an irrational number of bounded type. We prove that for any integer k ≥ 0 , the boundary of the Siegel disk of G ( z ) = ∫ 0 sin ⁡ ( z ) e 2 π i θ ( 1 − s 2 ) k d s centered at the origin is a quasi-circle passing through exactly two critical points π / 2 and − π / 2 with multiplicity k + 1 , respectively.

中文翻译：

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正弦族的有界型 Siegel 圆盘

摘要 我们证明，对于任何正整数 n，存在度数为 n 的多项式 P ( z ) 使得整个函数 G ( z ) = P ( sin ⁡ ( z ) ) 具有由拟平面中通过 G 的至少一个临界点的圆。另外，设 0 θ 1 为有界类型的无理数。我们证明对于任何整数 k ≥ 0 ，G ( z ) = ∫ 0 sin ⁡ ( z ) e 2 π i θ ( 1 − s 2 ) kds 的边界是一个以原点为中心的拟圆分别通过重数为 k + 1 的两个临界点 π / 2 和 − π / 2 。