Entire functions of slow growth whose Julia set coincides with the plane

We construct a transcendental entire function f with J(f) = C such that f has arbitrarily slow growth; that is, log |f(z)| ≤ φ(|z|) log |z| for |z| > r0, where φ is an arbitrary prescribed function tending to infinity. For an entire function f we denote the Julia set by J(f). By definition, it is the complement of the maximal open set F (f), the set of normality, where the iterates f form a normal family. While for polynomials the Julia set always has empty interior, for transcendental functions it may coincide with the whole complex plane C. The first example with this property was given by Baker [1] and later Misiurewicz [16] showed that this is the case for the exponential function. There are several methods of constructing such examples (besides [1] and [16] we refer to [3, p. 74], [4, p. 155, p. 172], [7, p. 167-168] [8, p. 225], [9, p. 625], [10, p. 610], [12]) but none of them seems to be applicable to entire functions of arbitrarily slow growth, the main problem being to exclude the possibility of a wandering component of the set of normality where the iterates tend to infinity. That such a wandering component may indeed occur for functions of arbitrarily slow growth was shown by Baker [2] and Hinkkanen [11]. Notice that for entire functions of order less than 1/2 there is always a sequence of critical values tending to infinity (see [13, p. 1788]). This makes usual arguments for the proof of the absence of wandering domains hard to apply. Theorem 1 Let t 7→ φ(t) : [0,∞) → [1,∞) be an arbitrary increasing function tending to ∞ as t → ∞. Then there exists an entire function f ∗Supported by EPSRC grant GR/L 35546 at Imperial College and by NSF grant DMS9800084