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

Abstract

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.