Dynamics of permutable transcendental entire functions. (English) Zbl 1131.37051
Summary: Let \(f\) and \(g\) be two permutable transcendental entire functions. Assume that \(f\) has the form \(f(z) = p(z) + p_1(z)e^{q_1(z)}+
p_2(z)e^{q_2(z)}\). We shall investigate the dynamical properties of \(f\) and \(g\) and show that they have the same Julia sets
and Fatou sets, i.e. \(J(f) = J(g)\). This relates to an open question due to I. N. Baker [Proc. Lond. Math. Soc. (3) 49, 563–576 (1984; Zbl 0523.30017)].
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Citations:
Zbl 0523.30017References:
| [1] | I.N. Baker, Wandering domains in the iteration of entire functions , Proc. London Math. Soc. 49 (1984), 563-576. · Zbl 0523.30017 · doi:10.1112/plms/s3-49.3.563 |
| [2] | W. Bergweiler, Iteration of meromorphic functions , Bull. Amer. Math. Soc. 29 (1993), 151-188. · Zbl 0791.30018 · doi:10.1090/S0273-0979-1993-00432-4 |
| [3] | ——–, On the Julia sets of analytic self-maps of the punctured plane , Analysis 15 (1995), 252-256. · Zbl 0829.30014 |
| [4] | W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions , Math. Proc. Cambridge Philos. Soc. 126 (1999), 565-574. · Zbl 0939.30019 · doi:10.1017/S0305004198003387 |
| [5] | C.T. Chuang and C.C. Yang, Fix-points and factorization of meromorphic functions , World Scientific, Singapore, 1990. · Zbl 0743.30027 |
| [6] | A.E. Eremenko, On the iterates of entire functions , Dynam. Sys. Ergodic Theory, Banach Center Publ., Vol. 23, Polish Scientific Publ., Warsaw, 1989, pp. 339-345. · Zbl 0692.30021 |
| [7] | A.E. Eremenko and M.Yu. Lyubich, Dynamical properties of some classes of entire functions , Ann. Inst. Fourier 42 (1992), 989-1020. · Zbl 0735.58031 · doi:10.5802/aif.1318 |
| [8] | P. Fatou, Sur les équations fonctionelles , Bull. Soc. Math. France 47 (1919), 161-271; 48 (1920), 33-94, 208-314. · JFM 47.0921.02 |
| [9] | ——–, Sur l’ itération des fonctions transcendentes entières , Acta Math. 47 (1926), 337-370. · JFM 52.0309.01 · doi:10.1007/BF02559517 |
| [10] | F. Gross and C.F. Osgood, On fixed points of composite entire functions , J. London Math. Soc. 28 (1983), 57-61. · Zbl 0487.30015 · doi:10.1112/jlms/s2-28.1.57 |
| [11] | W.K. Hayman, Meromorphic functions , Clarendon Press, Oxford, 1964. · Zbl 0115.06203 |
| [12] | X.H. Hua and C.C. Yang, Dynamics of transcendental functions , Gordon and Breach Science Publ., New York, 1998. · Zbl 0934.30021 |
| [13] | G. Julia, Mémoire sur l’ itération des fractions rationnelles , J. Math. Pures Appl. 1 (1918), 47-245. · JFM 46.0520.06 |
| [14] | T. Kobayashi, Permutability and unique factorizability of certain entire functions , Kodai Math. J. 3 (1980), 8-25. · Zbl 0433.30023 · doi:10.2996/kmj/1138036114 |
| [15] | L.W. Liao and C.C. Yang, Julia sets of two permutable entire functions , J. Math. Soc. Jap. 1 (2004), 169-176. · Zbl 1049.37036 · doi:10.2969/jmsj/1191418700 |
| [16] | ——–, Some further results on the Julia sets of two permutable entire functions , Rocky Mountain J. Math. 35 (5) (2005), 1657-1674. · Zbl 1099.30013 · doi:10.1216/rmjm/1181069655 |
| [17] | S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic dynamics , Cambridge Univ. Press, Cambridge, 2000. · Zbl 0979.37001 |
| [18] | T.W. Ng, Permutable entire functions and their Julia sets , Math. Proc. Cambridge Philos. Soc. 131 (2001), 129-138. · Zbl 0979.30015 · doi:10.1017/S0305004101005084 |
| [19] | K.K. Poon and C.C. Yang, Dynamical behavior of two permutable entire functions , Ann. Polon. Math. 168 (1998), 159-163. · Zbl 0896.30021 |
| [20] | F.Y. Ren and W.S. Li, An affirmative answer to a problem of Baker , J. Fudan Univ. Natur. Sci. 36 (1997), 231-233. · Zbl 0921.30020 |
| [21] | W. Schmidt, On the periodic stable domains of permutable rational functions , Complex Variables Theory Appl. 17 (1992), 149-152. · Zbl 0751.30029 · doi:10.1080/17476939208814507 |
| [22] | X.L. Wang and C.C. Yang, On the Fatou components of two permutable transcendental entire functions , J. Math. Anal. Appl. 278 (2003), 512-526. · Zbl 1024.30013 · doi:10.1016/S0022-247X(02)00664-9 |
| [23] | J.H. Zheng and Z.Z. Zhou, Permutability of entire functions satisfying certain differential equations , Tôhoku Math. J. 40 (1988), 323-330. · Zbl 0638.30024 · doi:10.2748/tmj/1178227977 |
| [24] | L.M. Zhu, D.G. Yang and X.L. Wang, On the Julia sets of permutable transcendental entire functions , J. Southeast Univ. 18 (2002), 270-273. · Zbl 1056.37063 |
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