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Composite entire functions with no unbounded Fatou components. (English) Zbl 1132.30017
Let \(f\) be a transcendental entire function. It is known that the Julia set \(J(f)\) is unbounded, so the Fatou set \(F(f)\) cannot contain a neighborhood of \(\infty\). In response to a question of I. N. Baker [J. Aust. Math. Soc., Ser. A 30, 483–495 (1981; Zbl 0474.30023)], Y. Wang [Isr. J. Math. 121, 55–60 (2001; Zbl 1054.37028)] showed that if \(f\) has order less than \(1/2\) but positive lower order, every component of \(F(f)\) is bounded.
The paper under review considers a class of functions \(\Lambda\) for which for \(\varepsilon> 0\), \(\log L(r, f)> (1-\varepsilon)\log M(r, f)\) for all \(r\) outside a set of logarithmic density zero, and sets \(F= \bigcup_{k\geq 1} F_k\) where \(F_k\) is the set of transcendental entire functions for which \(\log\log M(r, f)\geq(\log r)^{1/k}\). If \(h= f_{N^0} f_{N-1}\circ\cdots\circ f\), where \(f_i\) are in \(F\cap\Lambda\) for \(i= 1,\dots N\), then it is shown that \(h\) has no unbounded Fatou component.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Fabry gaps
Full Text: DOI
[1] Anderson, J.M.; Hinkkanen, A., Unbounded domains of normality, Proc. amer. math. soc., 126, 3243-3252, (1998) · Zbl 0902.30021
[2] Baker, I.N., Zusammensetzungen ganzer funktionen, Math. Z., 69, 121-163, (1958) · Zbl 0178.07502
[3] Baker, I.N., The iteration of polynomials and transcendental entire functions, J. aust. math. soc. ser. A, 30, 483-495, (1981) · Zbl 0474.30023
[4] Bergweiler, W., Iteration of meromorphic functions, Bull. amer. math. soc., 29, 151-188, (1993) · Zbl 0791.30018
[5] Cao, C.; Wang, Y., Boundedness of Fatou components of holomorphic maps, J. dynam. differential equations, 16, 377-384, (2004) · Zbl 1081.37027
[6] Fuchs, W.H.J., Proof of a conjecture of G. Pólya concerning gap series, Illinois J. math., 7, 661-667, (1963) · Zbl 0113.28702
[7] Hayman, W.K., Angular value distribution of power series with gaps, Proc. London math. soc. (3), 24, 590-624, (1972) · Zbl 0239.30035
[8] Hua, X.H.; Yang, C.C., Dynamics of transcendental functions, (1998), Gordon and Breach Science Pub.
[9] Morosawa, S.; Nishimura, Y.; Taniguchi, M.; Ueda, T., Holomorphic dynamics, (2000), Cambridge Univ. Press · Zbl 0979.37001
[10] A.P. Singh, M. Taniguchi, Escaping sets of composite entire functions, preprint
[11] Sons, L.R., An analogue of a theorem of W.H.J. Fuchs on gap series, Proc. London math. soc. (3), 21, 525-539, (1970) · Zbl 0206.08801
[12] Stallard, G.M., The iteration of entire functions of small growth, Math. proc. Cambridge philos. soc., 114, 43-55, (1993) · Zbl 0782.30023
[13] Wang, Y., Bounded domains of the Fatou set of an entire function, Israel J. math., 121, 55-60, (2001) · Zbl 1054.37028
[14] Wang, Y., On the Fatou set of an entire function with gaps, Tohoku math. J., 53, 163-170, (2001) · Zbl 0991.30015
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