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

Abstract:

The Speiser class $S$ is the set of all entire functions with finitely many singular values. Let $S_q\subset S$ be the set of all transcendental entire functions with exactly $q$ distinct singular values. The Fatou-Shishikura inequality for $f\in S_q$ gives an upper bound $q$ 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 $f\in S_q$ is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some $T\in S_q$ realizes it. In our construction, $T$ is a structurally finite transcendental entire function.