Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class

<p>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_q\subset 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">f\in 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">f\in 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">T\in 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.</p>