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• W4304194100 abstract "The Speiser class <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of all entire functions with finitely many singular values. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript q Baseline subset-of upper S> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>S_qsubset S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of all transcendental entire functions with exactly <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct singular values. The Fatou-Shishikura inequality for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f element-of upper S Subscript q> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>fin S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives an upper bound <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f element-of upper S Subscript q> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>fin S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T element-of upper S Subscript q> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>Tin S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> realizes it. In our construction, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding=application/x-tex>T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a structurally finite transcendental entire function." @default.
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• W4304194100 date "2022-10-11" @default.
• W4304194100 modified "2023-09-25" @default.
• W4304194100 title "Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class" @default.
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• W4304194100 doi "https://doi.org/10.1090/ecgd/373" @default.
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