On Julia limiting directions of meromorphic functions

Abstract

Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2π) is called a Julia limiting direction of f if there is an unbounded sequence {z_n} in the Julia set J(f) satisfying lim_n→∞ arg z_n = θ (mod 2π). We denote by L(f) the set of all Julia limiting directions of f. Our main result is that, for any non-empty compact set E ⊆ [0, 2π) and ρ ∈ [0, ∞], there are an entire function f of infinite lower order and a transcendental meromorphic function g of order ρ such that L(f) = L(g) = E. In addition, we have also constructed some transcendental entire functions whose lower order is ρ ∈ (1/2, ∞) and whose L(f) coincides with a certain kind of compact set. To prove our results, we have established a criterion for a direction θ to be a Julia limiting direction of a function by utilizing the growth rate of the function in the direction θ. The criterion may be of independent interest.