Totally ramified rational maps

Cui, Weiwei; Hu, Jun

doi:10.1090/ecgd/377

Totally ramified rational maps
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by Weiwei Cui and Jun Hu

Conform. Geom. Dyn. 26 (2022), 208-234

DOI: https://doi.org/10.1090/ecgd/377

Published electronically: December 6, 2022

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Abstract:

Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree $d\leq 6$ must be regularly ramified; (2) for any integer $d>6$, there exists a totally ramified rational map of degree $d$ which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree $10$. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree $7$ or $8$, up to pre- and post-composition by Möbius transformations.

References

Itai Benjamini, Sergei Merenkov, and Oded Schramm, A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature, Proc. Amer. Math. Soc. 132 (2004), no. 3, 641–647. MR 2019938, DOI 10.1090/S0002-9939-03-07147-8
Weiwei Cui, Entire functions arising from trees, Sci. China Math. 64 (2021), no. 10, 2231–2248. MR 4316137, DOI 10.1007/s11425-019-1644-0
Reinhard Diestel, Graph theory, 5th ed., Graduate Texts in Mathematics, vol. 173, Springer, Berlin, 2017. MR 3644391, DOI 10.1007/978-3-662-53622-3
Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745, DOI 10.1007/978-1-4684-9930-8
Ernesto Girondo and Gabino González-Diez, Introduction to compact Riemann surfaces and dessins d’enfants, London Mathematical Society Student Texts, vol. 79, Cambridge University Press, Cambridge, 2012. MR 2895884
Anatoly A. Goldberg and Iossif V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; With an appendix by Alexandre Eremenko and James K. Langley. MR 2435270, DOI 10.1090/mmono/236
Jun Hu, Francisco G. Jimenez, and Oleg Muzician, Rational maps with half symmetries, Julia sets, and multibrot sets in parameter planes, Conformal dynamics and hyperbolic geometry, Contemp. Math., vol. 573, Amer. Math. Soc., Providence, RI, 2012, pp. 119–146. MR 2964076, DOI 10.1090/conm/573/11393
Jun Hu, Oleg Muzician, and Yingqing Xiao, Dynamics of regularly ramified rational maps: I. Julia sets of maps in one-parameter families, Discrete Contin. Dyn. Syst. 38 (2018), no. 7, 3189–3221. MR 3809081, DOI 10.3934/dcds.2018139
Jun Hu and Yingqing Xiao, No Herman rings for regularly ramified rational maps, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1587–1596. MR 3910423, DOI 10.1090/proc/14347
Gareth A. Jones and Jürgen Wolfart, Dessins d’enfants on Riemann surfaces, Springer Monographs in Mathematics, Springer, Cham, 2016. MR 3467692, DOI 10.1007/978-3-319-24711-3
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280, DOI 10.1007/978-3-642-85590-0
S. Stoïlow, Leçons sur les principes topologiques de la théorie des fonctions analytiques. Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann, Gauthier-Villars, Paris, 1956 (French). MR 0082545