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 ∈ 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 ≥ 2 and some c ∈ C \ {0}, we show that the function of the form e^2πiθz(1 + cz)^d−1e^z (θ ∈ R\Q) has a Siegel point at the origin if and only if θ is a Brjuno number. This is an extension of Geyer's result in [11].

(4) For the function of the form (e^2πiθz + αz²)e^z (θ ∈ R\Q, α ∈ C\{0}), we show that if α and θ 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 α and θ 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].