On the non-escaping set of \(e^{2\pi i\theta} \sin (z)\)
Let \(\theta\) be an irrational number of bounded type and let \(f_{\theta}(z)=e^{2\pi i\theta}\sin z\). The author proved in a previous paper that the Siegel disk of \(f_{\theta}\) around 0 is bounded by a quasicircle passing through the critical points \(\pi/2\) and \(-\pi/2\). The main result of this paper is that the set of all \(z\) in the Julia set of \(f_{\theta}\) which do not tend to \(\infty\) under iteration has measure zero. In contrast, by a result of \textit{C. McMullen} [Trans. Am. Math Soc. 300, 329-342 (1987; Zbl 0618.30027)], the set of all \(z\) which tend to \(\infty\) under iteration has positive measure. Besides the methods of his previous paper, the author also uses ideas of \textit{C. L. Petersen} [Acta Math. 177, No. 2, 163-224 (1996; Zbl 0884.30020)] who proved that the Julia set of \(e^{2\pi i\theta}z+z^2\) has measure zero for \(\theta\) of bounded type.
scientific article; zbMATH DE number 5320460