Fatou and Julia like sets
Abstract
UDC 517.5
For a familiy of holomorphic functions on an arbitrary domain, we introduce Fatou and Julia like sets, and establish some of their interesting properties.
References
A. F. Beardon, Iteration of rational functions, Grad. Texts Math., 132, Springer Verlag, New York (1991), https://doi.org/10.1007/978-1-4612-4422-6 DOI: https://doi.org/10.1007/978-1-4612-4422-6
W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29, no. 2, 151 – 188 (1993), https://doi.org/10.1090/S0273-0979-1993-00432-4 DOI: https://doi.org/10.1090/S0273-0979-1993-00432-4
C. Caratheodory, Theory of functions of a complex variable, vol. II, Chelsea Publ. Co., New York (1954).
L. Carleson, T. W. Gamelin, Complex dynamics, Springer, New York (1993), https://doi.org/10.1007/978-1-4612-4364-9 DOI: https://doi.org/10.1007/978-1-4612-4364-9
M. Cristea, Open discrete mapping having local $ACL^n$ inverses, Complex Var. and Elliptic Equat., 55, № 1-3, 61 – 90 (2010), https://doi.org/10.1080/17476930902998985 DOI: https://doi.org/10.1080/17476930902998985
W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford (1964).
A. Hinkkanen, G. J. Martin, The dynamics of semigroups of rational functions, I, Proc. London Math. Soc., 73, № 2, 358 – 384 (1996), https://doi.org/10.1112/plms/s3-73.2.358 DOI: https://doi.org/10.1112/plms/s3-73.2.358
D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, E. A. Sevost’yanov, Toward the theory of Orlicz – Sobolev classes, St.Petersburg Math. J., 25, № 6, 929 – 963 (2014), https://doi.org/10.1090/s1061-0022-2014-01324-6 DOI: https://doi.org/10.1090/S1061-0022-2014-01324-6
O. Martio, S. Rickman, J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1., 465, 1 – 13 (1970). DOI: https://doi.org/10.5186/aasfm.1969.448
R. Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc., 84, № 1, 35 – 43 (1982), https://doi.org/10.2307/2043804 DOI: https://doi.org/10.1090/S0002-9939-1982-0633273-X
V. I. Ryazanov, R. R. Salimov, E. A. Sevostyanov, On convergence analysis of space homeomorphisms, Sib. Adv. Math., 23, № 4, 263 – 293 (2013), https://doi.org/10.3103/s105513441304004 DOI: https://doi.org/10.3103/S1055134413040044
J. L. Schiff, Normal families, Springer (1993), https://doi.org/10.1007/978-1-4612-0907-2 DOI: https://doi.org/10.1007/978-1-4612-0907-2
W. Schwick, Repelling periodic points in the Julia sets, Bull. London Math. Soc., 29, № 3Б 314 – 316 (1997), https://doi.org/10.1112/S0024609396007035 DOI: https://doi.org/10.1112/S0024609396007035
N. Steinmetz, Rational iteration, Walter de Gruyter, Berlin (1993), https://doi.org/10.1515/9783110889314 DOI: https://doi.org/10.1515/9783110889314
L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc., 35, № 3, 215 – 230 (1998), https://doi.org/10.1090/S0273-0979-98-00755-1 DOI: https://doi.org/10.1090/S0273-0979-98-00755-1
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