Area of non-escaping parameters of the sine family
Complex sine functions belong to the rather ``nice'' examples in dynamics of transcendental entire functions. They embed naturally into the space \(z\mapsto a\cdot e^{i z}+b\cdot e^{- i z}\) with complex \(a\) and \(b\) (see also the results on the ``cosine'' family \(z\mapsto a\cdot e^{z}+b\cdot e^{-z}\) in [\textit{D. Schleicher}, Duke Math. J. 136, No. 2, 343--356 (2007; Zbl 1114.37024)]). These functions have exactly two singular values but no asymptotic values which causes, for certain dynamically significant parameters, that the topological dynamics of the Julia sets and escaping sets are somewhat ``opposite'' to the behaviour in the exponential case.NEWLINENEWLINENEWLINEIn the article under review the authors consider the one-parameter family \(f_{\lambda}: z \mapsto\lambda\cdot \sin z\) with \(\lambda\in\mathbb C\backslash\{0\}\). It is not difficult to check that the set of singular values consists of the two simple critical values \(\pm\lambda\), and none of them is an asymptotic value. Furthermore, \(f_{\lambda}^n (\lambda) \to \infty\) for \(n\to\infty\) if and only if \(f_{\lambda}^n (-\lambda) \to \infty\), hence it is sensible to call NEWLINE\[NEWLINE\Lambda:=\{\lambda\in\mathbb C\{0\}:f_\lambda^n (\lambda)\nrightarrow\infty\}NEWLINE\]NEWLINE the set of non-escaping parameters. (The authors show that \(\Lambda\) is symmetric w.r.t. both axes but the reviewer was not able to see a clarification of the set of singular values of \(f_\lambda\), hence the dynamical meaning of \(\Lambda\) remains somewhat unclear.)NEWLINEThe authors prove that the intersection of \(\Lambda\) with any vertical strip of finite width has finite area.NEWLINENEWLINE The proof is fairly technical and various estimates of measure and density of forward images of suitably chosen rectangles are performed. The main tools seem to be \textit{C. McMullen}'s [Trans. Am. Math. Soc. 300, 329--342 (1987; Zbl 0618.30027)] criterion for nested intersections to have positive Lebesgue measure plus certain lemmas from \textit{H. Schubert}'s article [Proc. Am. Math. Soc. 136, No. 4, 1251--1259 (2008; Zbl 1136.37027)]. It would have been nice if the authors had sketched their proof ideas in the introduction. Additional motivation for considering exactly this problem in exactly this setting would also be valuable for the readers. It also seems that except the two references stated above, the other bibliographical listings are rather for historical and introductive purpose.
scientific article; zbMATH DE number 6062418