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Residual Julia set for functions with the Ahlfors’ property. (English) Zbl 1322.37020
A meromorphic function in a domain \(D\) is said to have the \(k\)-island property if for any \(k\) disjoint Jordan domains \(B_1,\dots,B_k\) and any domain \(U\) intersecting \(\partial D\) there exists \(j\in \{1,\dots,k\}\) and a domain \(V\subset U\cap D\) such that \(f:V\to B_j\) is univalent. By the Ahlfors five islands theorem, functions meromorphic in the plane satisfy this property with \(k=5\). A component of the Julia set of a meromorphic function is called buried if it does not intersect the boundary of a Fatou component.
The authors give conditions ensuring that for functions with the \(k\)-islands property, the Julia set has buried components, or even that singleton buried components are dense in the Julia set.
MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:
[1] Abikoff W., Ann. Math. Stud. 66 pp 1– (1971)
[2] DOI: 10.1007/BF02392264 · Zbl 0258.30016
[3] DOI: 10.1007/BF02420945 · Zbl 0012.17204
[4] DOI: 10.1007/BF01181396 · Zbl 0168.04002
[5] DOI: 10.1007/BF01110294 · Zbl 0172.09502
[6] Baker I.N., J. Anal. 8 pp 121– (2000)
[7] DOI: 10.1080/17476930008815263 · Zbl 1020.30027
[8] Baker I.N., Ergodic Theory Dyn. Syst. 21 pp 647– (2001)
[9] DOI: 10.1112/jlms/s2-42.2.267 · Zbl 0726.30022
[10] Baker I.N., Ergodic Theory Dyn. Syst. 11 pp 241– (1991)
[11] Baker I.N., Ergodic Theory Dyn. Syst. 11 pp 603– (1991)
[12] DOI: 10.1007/BF03323112 · Zbl 0774.30024
[13] Beardon A.F., Ann. Acad. Sci. Fenn. Ser. A.I. Math. 16 pp 173– (1991) · Zbl 0757.30034
[14] DOI: 10.1090/S0273-0979-1993-00432-4 · Zbl 0791.30018
[15] DOI: 10.1090/S1088-4173-00-00057-6 · Zbl 0954.30012
[16] DOI: 10.1080/17476939608814947 · Zbl 0865.30040
[17] A. Bolsch, Iteration of meromorphic functions with countably many singularities, dissertation, Technical Universität, Berlin, 1997. · Zbl 0901.30022
[18] DOI: 10.1112/S0024609399005950
[19] DOI: 10.1080/17476939708814991 · Zbl 0877.30011
[20] Domínguez P., Ann. Acad. Sci. Fenn. Ser. A.I. Math. 23 pp 225– (1998)
[21] DOI: 10.1080/10236190903203879 · Zbl 1192.37063
[22] DOI: 10.1017/CBO9780511735233.008
[23] DOI: 10.1080/17476930903394671 · Zbl 1206.37024
[24] A.L. Epstein, Towers of finite type complex analytic maps, Ph.D. thesis, City University of New York, 1993.
[25] A.L. Epstein, Dynamics of finite type complex analytic maps. I: Global structure theory, Manuscript.
[26] DOI: 10.1007/BF02559517 · JFM 52.0309.01
[27] M.E. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Imperial College, London, 1994.
[28] DOI: 10.1007/978-1-4613-9602-4_3
[29] DOI: 10.1017/S0143385797069848 · Zbl 0867.30021
[30] S.Morosawa, Y.Nishimura, M.Taniguchi, and T.Ueda, Holomorphic Dynamics, Cambridge University Press, UK, 2000. · Zbl 0979.37001
[31] M.H.A.Newman, Elements of the Topology of Plane Set of Points, Cambridge University Press, UK, 1954.
[32] DOI: 10.1112/blms/bdr112 · Zbl 1291.37067
[33] DOI: 10.1017/S0143385712000259 · Zbl 1277.37076
[34] R. Oudkerk, Iteration of Ahlfors and Picard functions which overflow their domain, Manuscript. · Zbl 1025.37028
[35] Qiao J.Y., Chin. Sci. Bull. 39 (7) pp 529– (1994)
[36] Qiao J.Y., Sci. China Ser. A. 38 pp 1409– (1995)
[37] DOI: 10.1007/BF02931834 · Zbl 0910.30028
[38] DOI: 10.1090/S0002-9939-08-09650-0 · Zbl 1214.37036
[39] DOI: 10.1007/s11854-012-0023-5 · Zbl 1285.37013
[40] Stallard G.M., Ann. Acad. Sci. Fenn. Ser. A.I. Math. 18 pp 273– (1993)
[41] DOI: 10.2307/1971308 · Zbl 0589.30022
[42] DOI: 10.1017/S0305004106009388 · Zbl 1099.30014
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