Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class
Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class
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DOI：
10.1090/ecgd/373
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发表时间：
2022-10
期刊：
影响因子：
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通讯作者：
M. Kisaka;Hiroto Naba
中科院分区：
文献类型：
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作者：
M. Kisaka;Hiroto Naba
S S类是具有有限多个奇异值的所有整函数的集合。设S q⊂S S_q\子集S是恰好具有q，q个不同奇异值的所有超越整函数的集合。S_q中的f-∈S q f的Fatou-Shishikura不等式给出了它的Cremer圈的个数和它的直接吸引盆、抛物面盆和Siegel盘的圈的个数之和的上界Q_q。本文证明了f∈S q f在下列意义下是最可能的：对于满足该不等式的任意循环数的组合，S q中的某个T∈S q T实现了它.在我们的构造中，T T是结构有限的超越整函数。
The Speiser class S S is the set of all entire functions with finitely many singular values. Let S q ⊂ S S_q\subset S be the set of all transcendental entire functions with exactly q q distinct singular values. The Fatou-Shishikura inequality for f ∈ S q f\in S_q gives an upper bound q q of the sum of the numbers of its Cremer cycles and its cycles of immediate attractive basins, parabolic basins, and Siegel disks. In this paper, we show that the inequality for f ∈ S q f\in S_q is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some T ∈ S q T\in S_q realizes it. In our construction, T T is a structurally finite transcendental entire function.