Totally disconnected Julia set for different classes of meromorphic functions

Domínguez, P.; Hernández, A.; Sienra, G.

doi:10.1090/S1088-4173-2014-00258-6

Totally disconnected Julia set for different classes of meromorphic functions
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by P. Domínguez, A. Hernández and G. Sienra

Conform. Geom. Dyn. 18 (2014), 1-7

DOI: https://doi.org/10.1090/S1088-4173-2014-00258-6

Published electronically: January 24, 2014

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Abstract:

We study a class of functions $\textbf {A}$ given by Epstein in Towers of finite type complex analytic maps, ProQuest LLC, Ann Arbor, MI, 1993, called finite-type maps. We extend a result related with the Julia set given by Baker, Domínguez in Some connectedness properties of Julia sets, Complex Variable Theory Appl. 41 (2000), 371–389, and Baker, Domínguez, and Herring in Dynamics of functions meromorphic outside a small set, Ergodic Theory Dynam. Systems 21 (2001), 647–672, to functions in class $\textbf {A}$.

References

I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146–153. MR 107015, DOI 10.1007/BF01181396
I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241–248. MR 1116639, DOI 10.1017/S014338570000612X
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. MR 1083445, DOI 10.1112/jlms/s2-42.2.267
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. MR 1145612, DOI 10.1017/S0143385700006386
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. MR 1189754, DOI 10.1007/BF03323112
I. N. Baker and P. Domínguez, Some connectedness properties of Julia sets, Complex Variables Theory Appl. 41 (2000), no. 4, 371–389. MR 1785150, DOI 10.1080/17476930008815263
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. MR 1836425, DOI 10.1017/S0143385701001328
Andreas Bolsch, Repulsive periodic points of meromorphic functions, Complex Variables Theory Appl. 31 (1996), no. 1, 75–79. MR 1423240, DOI 10.1080/17476939608814947
A. Bolsch, Iteration of meromorphic functions with countably many singularities. Dissertation, Berlin (1997).
A. Bolsch, Periodic Fatou components of meromorphic functions, Bull. London Math. Soc. 31 (1999), no. 5, 543–555. MR 1703869, DOI 10.1112/S0024609399005950
A. È. Erëmenko and M. Yu. Lyubich, Iterations of entire functions, Dokl. Akad. Nauk SSSR 279 (1984), no. 1, 25–27 (Russian). MR 769199
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. MR 1015124
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). MR 1196102, DOI 10.5802/aif.1318
Adam Lawrence Epstein, Towers of finite type complex analytic maps, ProQuest LLC, Ann Arbor, MI, 1993. Thesis (Ph.D.)–City University of New York. MR 2690048
A.L Epstein, Dynamics of finite type complex analytic maps. I. Global structure theory, Manuscript.
M.E. Herring, An extension of the Julia-Fatou theory of iteration. PhD thesis (1994), Imperial College, London.
P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337–370 (French). MR 1555220, DOI 10.1007/BF02559517
Linda Keen and Janina Kotus, Dynamics of the family $\lambda \tan z$, Conform. Geom. Dyn. 1 (1997), 28–57. MR 1463839, DOI 10.1090/S1088-4173-97-00017-9
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. MR 2675603, DOI 10.1080/10236190903282824