Radial Distribution of Julia Sets of Entire Solutions to Complex Difference Equations
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Radial Distribution of Julia Sets of Entire Solutions to Complex Difference Equations Jinchao Chen, Yezhou Li and Chengfa Wu Abstract. In this paper, entire solutions f of a class of nonlinear difference equations are studied. By considering the order and deficiency of the coefficients in the equations, we investigate the properties of the radial distribution of the Julia set of f , and estimate the lower bound of the measure of the set defined by the common limiting directions of Julia sets of shifts of f . Mathematics Subject Classification. 30D35, 34M10, 37F10. Keywords. Radial distribution, Julia set, nonlinear difference equation, deficiency.
1. Introduction and Main Results In this paper, we investigate the common limiting directions of Julia sets of shifts of f , which is an entire solution of the difference equation An (z)Pn (f (z + c1 ), . . . , f (z + cm )) + · · · + A1 (z)P1 (f (z + c1 ), . . . , f (z + cm )) = A0 (z).
(1.1)
In this equation, Ai , i = 0, . . . , n, are entire functions, ck , k = 1, . . . , m, are distinct complex numbers, and Pj , j = 1, . . . , n, are distinct polynomials in m variables with degree less than d, that is, Pj (f (z + c1 ), . . . , f (z + cm )) =
λ=(k1 ,...,km )∈Λj
aλ
m
[f (z + ci )]ki ,
i=1
finite multi-indices where aλ are nonzero complex numbers, Λj consists of m of the form λ = (k1 , . . . , km ), ki ∈ N, and maxλ∈Λj { i=1 ki } < d. The following example shows the existence for the entire solution of Eq. (1.1). This work is supported by National Natural Science Foundation of China (Grant No. 11571049 and No. 11101048) and China Scholarship Council. J. Chen, Y. Li and C. Wu contributed equally to this work.
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Example 1.1. The difference equation 1 1 f (z) + f (z + 1) + · · · + N f (z + N ) = (N + 1)ez e e has an entire solution f (z) = ez , where N is a positive integer. To accurately describe and study our problems, we use classical Nevanlinna theory of value distribution in this paper. We use some standard notations such as the proximity function m(r, f ), Nevanlinna counting function N (r, f ), Nevanlinna characteristic function T (r, f ), maximum modulus function M (r, f ), and some basic results (see [12,15,24]). The order and lower order of f are, respectively, defined by ρ(f ) = lim sup r→∞
log+ T (r, f ) log+ T (r, f ) , μ(f ) = lim inf , r→∞ log r log r
where log+ x = max{0, log x}, x > 0. The deficiency of the value a is defined by δ(a, f ), and if δ(a, f ) > 0, we say that a is a deficient value of f . Some basic results on complex dynamics of transcendental meromorphic functions are also needed here [20,26]. It is known that for a transcendental meromorphic function f , its Julia set J(f ) is nonempty and closed, and its Fatou set F (f ) is open. Moreover, if f is a transcendental entire function, then J(f ) is unbounded. Baker [2] proved that J(f ) cannot be covered by a finite set of straight lines. A ray ending at the origin arg z = θ, θ ∈ [0, 2π] is called a limiting direction of the Julia se
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