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Totally disconnected Julia set for different classes of meromorphic functions. (English) Zbl 1333.37023
Let \(X\) be a Riemann surface. A map \(f\) is of “finite type” if \(f:D\rightarrow X\) is analytic where \(D\subset X\) is an open set, \(f\) is not locally constant, \(f\) has only finitely singular values and every isolated singularity of \(f\) is essential. This notion was introduced by A. Epstein [Towers of finite type complex analytic maps. New York: City University of New York, Graduate Center (PhD Thesis) (1993)].
In the short paper under review, the authors study the dynamics of finite type maps. They describe the global structure of the Julia set of \(f\). More precisely, they prove that if \(f\) is a generic finite type maps having an attracting fixed point which basin contains all singular values of \(f\), then the Julia set \(J_f\) of \(f\) is totally disconnected.
Their proof follows closely the one of I. N. Baker et al. [Ergodic Theory Dyn. Syst. 21, No. 3, 647–672 (2001; Zbl 0990.37033)] in the case of meromorphic maps: they first prove that the attracting basin is totally invariant. In a second time, they prove that the complement of an open neighborhood of the attracting fixed point is a finite union of disks. Finally, they conclude by a classical argument. One of the key ingredients is the classification of Fatou components for finite type maps established by Epstein.
They also provide some examples.

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
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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References:
[1] I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146 – 153. · Zbl 0168.04002
[2] I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241 – 248. · Zbl 0711.30024
[3] I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. II. Examples of wandering domains, J. London Math. Soc. (2) 42 (1990), no. 2, 267 – 278. · Zbl 0726.30022
[4] I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 603 – 618. · Zbl 0774.30023
[5] I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22 (1992), no. 3-4, 651 – 656. · Zbl 0774.30024
[6] I. N. Baker and P. Domínguez, Some connectedness properties of Julia sets, Complex Variables Theory Appl. 41 (2000), no. 4, 371 – 389. · Zbl 1020.30027
[7] I. N. Baker, P. Domínguez, and M. E. Herring, Dynamics of functions meromorphic outside a small set, Ergodic Theory Dynam. Systems 21 (2001), no. 3, 647 – 672. · Zbl 0990.37033
[8] Andreas Bolsch, Repulsive periodic points of meromorphic functions, Complex Variables Theory Appl. 31 (1996), no. 1, 75 – 79. · Zbl 0865.30040
[9] A. Bolsch, Iteration of meromorphic functions with countably many singularities. Dissertation, Berlin (1997). · Zbl 0901.30022
[10] A. Bolsch, Periodic Fatou components of meromorphic functions, Bull. London Math. Soc. 31 (1999), no. 5, 543 – 555. · Zbl 0932.30022
[11] A. È. Erëmenko and M. Yu. Lyubich, Iterations of entire functions, Dokl. Akad. Nauk SSSR 279 (1984), no. 1, 25 – 27 (Russian).
[12] A. È. Erëmenko and M. Yu. Lyubich, The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, 1 – 70 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 563 – 634. · Zbl 0712.58036
[13] A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989 – 1020 (English, with English and French summaries). · Zbl 0735.58031
[14] Adam Lawrence Epstein, Towers of finite type complex analytic maps, ProQuest LLC, Ann Arbor, MI, 1993. Thesis (Ph.D.) – City University of New York.
[15] A.L Epstein, Dynamics of finite type complex analytic maps. I. Global structure theory, Manuscript.
[16] M.E. Herring, An extension of the Julia-Fatou theory of iteration. PhD thesis (1994), Imperial College, London.
[17] P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337 – 370 (French). · JFM 52.0309.01
[18] Linda Keen and Janina Kotus, Dynamics of the family \?tan\?, Conform. Geom. Dyn. 1 (1997), 28 – 57 (electronic). · Zbl 0884.30019
[19] Lasse Rempe, Philip J. Rippon, and Gwyneth M. Stallard, Are Devaney hairs fast escaping?, J. Difference Equ. Appl. 16 (2010), no. 5-6, 739 – 762. · Zbl 1201.30027
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