On Julia limiting directions of meromorphic functions
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ON JULIA LIMITING DIRECTIONS OF MEROMORPHIC FUNCTIONS
BY
Jun Wang School of Mathematical Sciences, Fudan University Shanghai 200433, P. R. China e-mail: [email protected] AND
Xiao Yao∗ School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 and Shanghai Center of Mathematical Sciences, Fudan University Shanghai 200433, P. R. China e-mail: [email protected]
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 {zn } in the Julia set J(f ) satisfying limn→∞ arg zn = 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.
∗ Corresponding Author.
Received November 24, 2017 and in revised form July 26, 2019
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J. WANG AND X. YAO
Isr. J. Math.
1. Introduction and main results Let f be a transcendental meromorphic function in the complex plane C. A point z is in the Fatou set F (f ) of f if and only if there is a neighborhood of z on which the iterates f n (n = 1, 2, . . .) of f are well defined and {f n } forms a normal family. The complement of F (f ) in C is the Julia set J (f ) of f . Standard references and recent research progress in transcendental iteration theory can be found in [5, 7, 8, 19]. Set E :={f : f is transcendental entire}, M :={f : f is transcendental meromorphic and has at least one pole}. For each f ∈ E, Baker [2] proved that J (f ) cannot be contained in any finite union of straight lines. However, it is not true for some functions in M owing to the fact that J (tan z) = R. From a viewpoint of angular distribution, Qiao [16] introduced the limiting direction of a Julia set. A value θ ∈ [0, 2π) is said to be a limiting direction of the Julia set of f if there is an unbounded sequence {zn } ⊆ J (f ) such that lim arg zn = θ
n→∞
(mod 2π).
We denote by L(f ) the set of all limiting directions of the Julia set of f . For brevity, we call a limiting direction of the Julia set of f a Julia limiting direction of f in this paper. Further, we identify [0, 2π) with the circle S1 := {z ∈ C : |z| = 1} and intervals in [0, 2π) with arcs on the circle for convenience of understanding. The set L(f ) is a non-empty closed set in [0, 2π) and will reveal the properties of the large-scale geometry of the Julia set. Since any rational function, as well as any polynomial, can be treated as a map between two Riemann spheres, it makes no sense for u
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