Some transcendental entire functions with irrationally indifferent fixed points

Abstract

Let $S$ be the set of all transcendental entire functions of the form $P(z) \exp (Q(z))$, where $P$ and $Q$ are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in $S$ with irrationally indifferent fixed points as follows: (1) We construct functions in $S$ with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for $f \in S$. (2) We construct functions in $S$ with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases. (3) For any integer $d \geq 2$ and some $c \in {\mathbf C} \setminus \{0\}$, we show that the function of the form $e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in {\mathbf R} \setminus {\mathbf Q})$ has a Siegel point at the origin if and only if $\theta$ is a Brjuno number. This is an extension of Geyer's result in [11]. (4) For the function of the form $(e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in {\mathbf R} \setminus {\mathbf Q}, \alpha \in {\mathbf C} \setminus \{0\})$, we show that if $\alpha$ and $\theta$ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose $\alpha$ and $\theta$ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].