Abstract
We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain \({\mathcal O}(d^2)\) conservative maps of fixed degree d defined over \({\mathbb Q}\); this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on their \({\mathbb Q}\)-dynamics.
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R. Benedetto, P. Ingram, R. Jones, A. Levy, Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps. Duke Math. J. 163(13), 2325–2356 (2014)
E. Brezin, R. Byrne, J. Levy, K. Pilgrim, K. Plummer, A census of rational maps. Conform. Geom. Dyn. 4, 35–74 (2000)
K. Cordwell, S. Gilbertson, N. Nuechterlein, K.M. Pilgrim, S. Pinella, On the classification of critically fixed rational maps. Conform. Geom. Dyn. 19, 51–94 (2015)
M. Eskin, Stable reduction of three-point covers. Ph.D. thesis, Ulm University, 2015. https://doi.org/10.18725/OPARU-3280
F. Liu, B. Osserman, The irreducibility of certain pure-cycle Hurwitz spaces. Am. J. Math. 130(6), 1687–1708 (2008)
D. Lukas, M. Manes, D. Yap, A census of quadratic post-critically finite rational functions defined over \(\mathbb {Q}\). LMS J. Comput. Math. 17(Suppl. A), 314–329 (2014)
B. Osserman, Rational functions with given ramification in characteristic p. Compos. Math. 142(2), 433–450 (2006)
B. Osserman, Linear series and the existence of branched covers. Compos. Math. 144(1), 89–106 (2008)
F. Pakovich, Conservative polynomials and yet another action of \(\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)\) on plane trees. J. Théor. Nombres Bordeaux 20(1), 205–218 (2008)
K.M. Pilgrim, An algebraic formulation of Thurston’s characterization of rational functions. Ann. Fac. Sci. Toulouse Math. (6) 21(5), 1033–1068 (2012)
S.M. Ruiz, An algebraic identity leading to Wilson’s theorem. Math. Gaz. 80(489), 579–582 (1996)
L. Schneps (ed.), The Grothendieck Theory of Dessins D’enfants. London Mathematical Society Lecture Note Series, no. 200 (Cambridge University Press, Cambridge, 1994)
J. Sijsling, J. Voight, On Computing Belyi Maps, Numéro consacré au trimestre “Méthodes arithmétiques et applications”, automne 2013. Publ. Math. Besançon Algèbre Théorie Nr., vol. 2014/1 (Presses Univ. Franche-Comté, Besançon, 2014), pp. 73–131
J.H. Silverman, The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241 (Springer, New York, 2007)
J.H. Silverman, Moduli Spaces and Arithmetic Dynamics. CRM Monograph Series, vol. 30 (American Mathematical Society, Providence, 2012)
D. Tischler, Critical points and values of complex polynomials. J. Complexity 5(4), 438–456 (1989)
H. Völklein, Groups as Galois Groups. Cambridge Studies in Advanced Mathematics, vol. 53 (Cambridge University Press, Cambridge, 1996)
D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 306(10–11), 1072–1075 (2006)
A. Zvonkin, Belyi functions: examples, properties, and applications. http://www.labri.fr/perso/zvonkin/Research/belyi.pdf
Acknowledgements
This project began at the Women in Numbers Europe 2 conference at the Lorentz Center. We thank the Lorentz Center for providing excellent working conditions, and we thank the Association for Women in Mathematics for supporting WIN-E2 and other research collaboration conferences for women through their NSF ADVANCE grant. We also thank the referee for numerous helpful comments, all of which greatly improved the paper.
MM partially supported by NSF-HRD 1500481 (AWM ADVANCE grant) and by the Simons Foundation grant #359721.
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Anderson, J., Bouw, I.I., Ejder, O., Girgin, N., Karemaker, V., Manes, M. (2018). Dynamical Belyi Maps. In: Bouw, I., Ozman, E., Johnson-Leung, J., Newton, R. (eds) Women in Numbers Europe II. Association for Women in Mathematics Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-74998-3_5
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DOI: https://doi.org/10.1007/978-3-319-74998-3_5
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