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

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- by Masashi Kisaka and Hiroto Naba
- Conform. Geom. Dyn.
**26**(2022), 165-181 - DOI: https://doi.org/10.1090/ecgd/373
- Published electronically: October 11, 2022
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## 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.## References

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## Bibliographic Information

**Masashi Kisaka**- Affiliation: Department of Mathematical Sciences, Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
- MR Author ID: 335457
- Email: kisaka@math.h.kyoto-u.ac.jp
**Hiroto Naba**- Affiliation: Department of Mathematical Sciences, Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
- Email: naba.hiroto.78r@st.kyoto-u.ac.jp
- Received by editor(s): November 17, 2021
- Received by editor(s) in revised form: June 4, 2022
- Published electronically: October 11, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**26**(2022), 165-181 - MSC (2020): Primary 37F10; Secondary 30D05, 37F31
- DOI: https://doi.org/10.1090/ecgd/373
- MathSciNet review: 4494582