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A census of quadratic post-critically finite rational functions defined over $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Q}$

Published online by Cambridge University Press:  01 July 2014

David Lukas
Affiliation:
University of Hawaii, Honolulu, HI 96822,USA email [email protected]
Michelle Manes
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822,USA email [email protected]
Diane Yap
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822,USA email [email protected]

Abstract

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We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

Type
Research Article
Copyright
© The Author(s) 2014 

References

Benedetto, R., Ingram, P., Jones, R. and Levy, A., ‘Critical orbits and attracting cycles in $p$-adic dynamics’, Preprint, 2012, arXiv:1201.1605v1.Google Scholar
Brezin, E., Byrne, R., Levy, J., Pilgrim, K. and Plummer, K., ‘A census of rational maps’, Conform. Geom. Dyn. 4 (2000) 3574.CrossRefGoogle Scholar
Faber, X., ‘Rational functions with a unique critical point’, Int. Math. Res. Not. IMRN (2014) no. 3, 681699.CrossRefGoogle Scholar
Granlund, T. et al. , GNU Multiple Precision Arithmetic Library (Version 5.1.3), Free Software Foundation, Inc., 2013, http://www.gmplib.org/.Google Scholar
Jones, R. and Manes, M., ‘Galois theory of quadratic rational functions’, Preprint, 2011,arXiv:1101.4339v3.Google Scholar
Manes, M., ‘ℚ-rational cycles for degree-2 rational maps having an automorphism’, Proc. Lond. Math. Soc. (3) 96 (2008) no. 3, 669696.CrossRefGoogle Scholar
Manes, M., ‘Moduli spaces for families of rational maps on ℙ1’, J. Number Theory 129 (2009) no. 7, 16231663.CrossRefGoogle Scholar
Manes, M. and Yasufuku, Y., ‘Explicit descriptions of quadratic maps on ℙ1 defined over a field K’, Acta Arith. 148 (2011) no. 3, 257267.CrossRefGoogle Scholar
Milnor, J., ‘Geometry and dynamics of quadratic rational maps’, Experiment. Math. 2 (1993) no. 1, 3783.CrossRefGoogle Scholar
Nelson, P., Downs, J. and Poznyakoff, S., GNU gdbm (Version 1.10), Free Software Foundation, Inc., 2011, http://www.gnu.org/software/gdbm/.Google Scholar
Olson, L. D., ‘Points of finite order on elliptic curves with complex multiplication’, Manuscripta Math. 14 (1974) 195205.CrossRefGoogle Scholar
Poonen, B., ‘The classification of rational preperiodic points of quadratic polynomials over ℚ: a refined conjecture’, Math. Z. 228 (1998) no. 1, 1129.CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of dynamical systems, Graduate Texts in Mathematics 241 (Springer, 2007).CrossRefGoogle Scholar
Silverman, J. H., Moduli spaces and arithmetic dynamics, CRM Monograph Series 30 (American Mathematical Society, Providence, RI, 2012).CrossRefGoogle Scholar
Stein, W. A. et al. , Sage Mathematics Software (Version 5.12). The Sage Development Team, 2013, http://www.sagemath.org.Google Scholar