On the dynamics of \(e^{2 \pi i \theta} \sin (z)\)
The paper concerns the dynamics of the so-called Siegel entire function \(f_{\theta}(z) = \exp(2\pi i \theta )\sin z\), where \(0 < \theta < 1\) is a bounded-type irrational number. Here, an irrational number \(0< \theta<1\) is said to be of bounded type if \(\sup\{a_n\} < \infty\), where \([a_1. \cdots, a_n, \cdots]\) is its continued fraction. Then \(f_{\theta}\) has a Siegel disk centered at the origin which has rotation number \(\theta\). The author proves the following interesting result: the boundary of the Siegel disk of the entire function \(f_{\theta}(z) = \exp(2\pi i \theta)\sin z\) is a quasi-circle which passes through exactly two critical points \(\pi/2\) and \(-\pi/2\).
scientific article; zbMATH DE number 5015554