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Normal families of bicomplex holomorphic functions. (English) Zbl 1230.30032

Summary: We introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice \(\mathbf j = 0\).

MSC:

30G35 Functions of hypercomplex variables and generalized variables
30D45 Normal functions of one complex variable, normal families
15A66 Clifford algebras, spinors
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
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[1] Montel P., Ann. Ecole Norm. Sup. 24 pp 233– · JFM 38.0440.02 · doi:10.24033/asens.580
[2] DOI: 10.1007/978-1-4612-0907-2 · doi:10.1007/978-1-4612-0907-2
[3] DOI: 10.1142/1904 · doi:10.1142/1904
[4] DOI: 10.1090/S0273-0979-98-00755-1 · Zbl 1037.30021 · doi:10.1090/S0273-0979-98-00755-1
[5] DOI: 10.1080/17476938208814009 · Zbl 0503.30039 · doi:10.1080/17476938208814009
[6] Rochon D., Anal. Univ. Oradea 11 pp 71–
[7] DOI: 10.2307/2687027 · Zbl 02311713 · doi:10.2307/2687027
[8] Price G. B., An Introduction to Multicomplex Spaces and Functions (1991) · Zbl 0729.30040
[9] Shabat B. V., Introduction to Complex Analysis Part II: Functions of Several Variables (1992) · Zbl 0799.32001
[10] Avanissian V., Cellule d’harmonicité et Prolongement Analytique Complexe (1985)
[11] DOI: 10.1007/978-1-4612-6313-5 · doi:10.1007/978-1-4612-6313-5
[12] Rudin W., Real and Complex Analysis (1976) · Zbl 0954.26001
[13] Scheidemann V., Introduction to Complex Analysis in Several Variables (2005) · Zbl 1085.32001
[14] DOI: 10.1007/978-1-4612-4364-9 · doi:10.1007/978-1-4612-4364-9
[15] DOI: 10.1007/978-1-4612-4422-6 · doi:10.1007/978-1-4612-4422-6
[16] DOI: 10.1142/S0218348X0000041X · Zbl 0969.37021 · doi:10.1142/S0218348X0000041X
[17] DOI: 10.1142/S0218348X03002075 · Zbl 1041.37023 · doi:10.1142/S0218348X03002075
[18] Martineau É., Int. J. Bifurcat. Chaos 15 pp 1–
[19] Chuang C. T., Fix-Points and Factorization of Meromorphic Functions (1990) · Zbl 0743.30027
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