Note on wandering domains in the dynamics of solutions of certain difference equations
- Guowei Zhang^{1}Email author
https://doi.org/10.1186/s13662-015-0359-0
© Zhang; licensee Springer 2015
Received: 24 October 2014
Accepted: 6 January 2015
Published: 30 January 2015
Abstract
In this note we study the value distribution of solutions of certain difference equations analogous to differential equations, the finite order solutions of which do not have wandering domains. Meanwhile, the nonexistence of wandering domains of solutions with finite order of these difference equations is proved. Thus the nonexistence of wandering domains of solutions of these difference and differential equations is similar in some extent.
Keywords
MSC
1 Introduction and main results
Let f be a nonlinear meromorphic function, the Fatou set \(F(f)\) is the set of points \(z\in\mathbb{C}\) such that iterates of f, \((f^{n})_{n\in\mathbb{N}}\), form a normal family in some neighborhood of z. The complement of \(F(f)\) is called the Julia set \(J(f)\) of f. The Fatou set is open and completely invariant. If U is a component of \(F(f)\), then \(f^{n}(U)\) lies in some component \(U_{n}\) of \(F(f)\). If \(U_{n}\neq U_{m}\) for all \(n\neq m\), then U is called a wandering domain of f. Otherwise U is called pre-periodic and \(U_{n}=U\) for some \(n\in\mathbb{N}\), then U is called periodic. An introduction to iteration theory can be found in [1].
Sullivan [2] proved that rational functions do not have wandering domains. However, transcendental meromorphic functions may have wandering domains (for example, see [2–6]), while many classes of meromorphic functions do not have wandering domains (for example, see [3, 7–12]). In [13], the nonexistence of wandering domains is proved by Wang for a meromorphic function f of finite order satisfying some first order nonlinear differential equations, see the following two theorems.
Theorem A
Theorem B
We also assume that the readers are familiar with basic Nevanlinna’s value distribution theory and its standard notations such as \(m(r,f)\), \(N(r,f)\), \(T(r,f)\). \(S(r, f)\) denotes any term satisfying \(S(r, f)=o(T(r, f))\) as \(r\rightarrow\infty\) outside some exceptional set of finite measure; see [14, 15] as references for Nevanlinna theory. Also, we use the notations \(\sigma(f)\), \(\lambda(f)\) to denote the order of f, exponent of convergence of zeros of f, respectively, as usual. Halburd and Korhonen [16, 17], Chiang and Feng [18] established a version of Nevanlinna theory based on difference operators independently. After that many difference equations analogous to differential equations have been studied.
In this note, we study the value distribution and dynamical properties of the solutions of difference equations which are analogous to differential equations (1) and (2). We use \(f_{c}\) to denote the shift \(f(z+c)\) of \(f(z)\), where c is a nonzero constant.
We obtain the following results with regard to equation (3).
Theorem 1
Let \(f(z)\) be a finite order meromorphic solution of (3), then \(\max\{t,n\}\geq p\geq n-t\).
Theorem 2
If equation (3) admits a finite order meromorphic solution f, then f is rational.
The following example shows that there are rational solutions satisfying equation (3).
Example
Suppose that \(n=t\), \(c=1\), then \(f=a+\frac{1}{z}\) satisfying equation (3), where \(P(z,f)= (\frac{az^{2}+(a+1)z}{az+1} )^{n}f^{n}\).
By Sullivan’s no existence of wandering domains for rational function, we obtain the following dynamical property for the finite order solutions of (3), which is similar to the dynamical property of solutions of equation (1).
Corollary 1
The finite order meromorphic solutions of (3) do not have wandering domains.
Theorem 3
In the following, we shall show the properties of the solutions of the following difference equation (7), which is a special case of (6).
Theorem 4
Some ideas of this theorem are from [19]. Recall the following theorem about the nonexistence of wandering domains for a class of entire functions, which is due to Baker [4].
Theorem C
Combining Theorem 4 and Theorem C, obviously, we have the corollary below.
2 Preliminary lemmas
The following lemma introduced by Laine and Yang [20] is an analogue of findings of Mohonko and Mohonko [21] on differential equations.
Lemma 1
([20])
Lemma 2
([18])
The following result is due to Valiron and Mohonko, one can find the proof in Laine’s book [22, p.29].
Lemma 3
3 Proof of theorems
Proof of Theorem 1
Proof of Theorem 2
Proof of Theorem 3
Proof of Theorem 4
Since \(T(z)\) is a polynomial, observing the both sides of (19), we know that \(\frac{T^{n}(z)}{q(z)T(z+c)^{m}}\) must be a constant. Without loss of generality, we set \(T^{n}(z)=q(z)T(z+c)^{m}\). If \(T(z+c)\) has a zero that is not a zero of \(T(z)\), we get a contradiction immediately. Hence every zero of \(T(z+c)\) must be a zero of \(T(z)\), but maybe with different multiplicity. In other words, every distinct zero of \(T(z)\) must be a zero of \(T(z-c)\). Since \(c\neq0\) and \(n>m\), \(m|n\), by continuing inductively, \(T(z)\) has infinitely many zeros, this is a contradiction. Hence \(T(z)\) cannot have any zeros, in which case \(T(z)\) and \(q(z)\) are constants. By (18), f is of the form \(f=e^{\alpha(z)}\), where \(\alpha(z)\) is a nonconstant polynomial. □
Declarations
Acknowledgements
The author is very grateful to the editor and the referees for their insightful and constructive comments and suggestions, which have led to an improved version of this paper. This work was supported by the Tianyuan Fund for Mathematics of NSFC (No. 11426035).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Bergweiler, W: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N.S.) 29, 151-188 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Sullivan, D: Quasiconformal homeomorphisms and dynamics I: solution of the Fatou-Julia problem on wandering domains. Ann. Math. 122, 401-418 (1985) View ArticleMATHGoogle Scholar
- Baker, IN: An entire function which has wandering domains. J. Aust. Math. Soc. A 22, 173-176 (1976) View ArticleMATHGoogle Scholar
- Baker, IN: Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. (3) 49, 563-576 (1984) View ArticleMATHGoogle Scholar
- Baker, IN: Some entire functions with multiply-connected wandering domains. Ergod. Theory Dyn. Syst. 5, 163-169 (1985) View ArticleMATHGoogle Scholar
- Eremenko, A, Lyubich, M: Examples of entire functions with pathological dynamics. J. Lond. Math. Soc. 36, 454-468 (1987) MathSciNetGoogle Scholar
- Baker, IN, Kotus, J, Lü, YN: Iterates of meromorphic functions IV: critically finite functions. Results Math. 22, 651-656 (1992) View ArticleMATHMathSciNetGoogle Scholar
- Baker, IN, Singh, A: Wandering domains in the iteration of compositions of entire functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 20, 149-153 (1995) MATHMathSciNetGoogle Scholar
- Bergweiler, W: Newton’s method and a class of meromorphic functions without wandering domains. Ergod. Theory Dyn. Syst. 13, 231-247 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Bergweiler, W, Haruta, M, Kriete, H, Meier, HG, Terglane, N: On the limit functions of iterates in wandering domains. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 18, 369-375 (1993) MATHMathSciNetGoogle Scholar
- Bergweiler, W, Terglane, N: Weakly repelling fixpoints and the connectivity of wandering domains. Trans. Am. Math. Soc. 348, 1-12 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Bergweiler, W, Wang, Y: On the dynamics of composite entire functions. Ark. Mat. 36, 31-39 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Wang, YF: Wandering domains in the dynamics of certain meromorphic functions. Bull. Aust. Math. Soc. 59, 99-104 (1999) View ArticleMATHGoogle Scholar
- Hayman, W: Meromorphic Functions. Clarendon, Oxford (1964) MATHGoogle Scholar
- Yi, HX, Yang, CC: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995) (in Chinese) Google Scholar
- Halburd, RG, Korhonen, R: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477-487 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Halburd, RG, Korhonen, R: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 94, 463-478 (2006) MathSciNetGoogle Scholar
- Chiang, YM, Feng, SJ: On the Nevanlinna characteristic of \(f(z+\eta)\) and difference equations in the complex plane. Ramanujan J. 16, 105-129 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Wen, ZT, Heittokangas, J, Laine, I: Exponential polynomials as solutions of certain nonlinear difference equations. Acta Math. Sin. 28, 1295-1306 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Laine, I, Yang, CC: Clunie theorems for difference and q-difference polynomials. J. Lond. Math. Soc. 76(3), 556-566 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Mohonko, AZ, Mohonko, VD: Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations. Sib. Mat. Zh. 15, 1305-1322 (1974) MathSciNetGoogle Scholar
- Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993) View ArticleGoogle Scholar