Escaping points of commuting meromorphic functions with finitely many poles
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Abstract:
Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply connected fast escaping wandering domains. By combining this with a recent result of Tsantaris, we obtain the strongest statement (to date) regarding the Julia sets of commuting meromorphic functions. In order to highlight the difference to the entire case, we show that transcendental meromorphic functions with finitely many poles have orbits that alternate between approaching a pole and escaping to infinity at strikingly fast rates.References
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Additional Information
- Gustavo R. Ferreira
- Affiliation: The Open University, Milton Keynes, MK7 6AA, United Kingdom
- ORCID: 0000-0002-7330-0018
- Email: gustavo.rodrigues-ferreira@open.ac.uk
- Received by editor(s): April 28, 2020
- Received by editor(s) in revised form: February 3, 2021
- Published electronically: November 4, 2021
- Communicated by: Filippo Bracci
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 589-603
- MSC (2020): Primary 37F10; Secondary 30D05
- DOI: https://doi.org/10.1090/proc/15591
- MathSciNet review: 4356170