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On topological properties of Fatou sets and Julia sets of transcendental entire functions. (English. Japanese original) Zbl 1411.37049

Sugaku Expo. 30, No. 2, 235-273 (2017); translation from Sūgaku 65, No. 3, 269-298 (2013).
The author surveys dynamical properties of transcendental entire functions of a single complex variable, with a particular focus on topological properties of Fatou sets and Julia sets.
A preliminary section starts with the usual basics and introduces fundamental notions in complex dynamics, such as normal families, critical points, Fatou set, Julia set, periodic and wandering component, attractive basins.
The next two sections are devoted to a discussion of connectedness and local connectedness of Julia sets. Further investigations into topological properties of Fatou and Julia sets can be found in the subsequent sections. Both classical and more recent results are discussed.
The last sections are devoted to escaping and fast escaping sets, and include several very recent developments, and some closing thoughts. A note added in proof at the end mentions some results that have appeared whilst the paper was finalized. Several interesting historical observations can be found in the paper and in the footnotes.
The contents listed above, together with numerous conjectures collected in the paper and a comprehensive list of references, should make this a valuable reference for researchers interested in learning more about dynamics of entire functions.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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
30D20 Entire functions of one complex variable (general theory)
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