Geodesic
Cubic Polynomials with Periodic Cycles of a Specified Multiplier
Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 318-344

Voir la notice de l'article provenant de la source Cambridge University Press

We consider cubic polynomials $f\left( z \right)\,=\,{{z}^{3}}\,+\,az\,+\,b$ defined over $\mathbb{C}\left( \lambda\right)$ , with a marked point of period $N$ and multiplier $\lambda$ . In the case $N\,=\,1$ , there are infinitely many such objects, and in the case $N\,\ge \,3$ , only finitely many (subject to a mild assumption). The case $N\,=\,2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field ${{\cup }_{n\ge 1}}\,\mathbb{C}\left( {{\lambda }^{1/n}} \right)$ .
DOI : 10.4153/CJM-2011-093-8
Mots-clés : 37P35, cubic polynomials, periodic points, holomorphic dynamics 
Ingram, Patrick. Cubic Polynomials with Periodic Cycles of a Specified Multiplier. Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 318-344. doi: 10.4153/CJM-2011-093-8
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[1] [1] Berker, S., Epstein, A. L., and Pilgrim, K. M., Remarks on the period three cycles of quadratic rational maps. Nonlinearity 16(2003), no. 1, 93–100. Google Scholar | DOI

[2] [2] Bonifant, A., Kiwi, J. and Milnor, J. Cubic polynomial maps with periodic critical point. II. Escape regions. Conform. Geom. Dyn. 14(2010), 68–112. Google Scholar | DOI

[3] [3] Fastenberg, L. A., Computing the Mordell-Weil ranks of cyclic covers of elliptic surfaces. Proc. Amer. Math. Soc. 129(2001), no. 7, 1877–1883. Google Scholar | DOI

[4] [4] Hindry, M. and Silverman, J. H., The canonical height and integral points on elliptic curves. Invent. Math. 93(1988), no. 2, 419–450. Google Scholar | DOI

[5] [5] Milnor, J., Remarks on iterated cubic maps. Experiment. Math. 1(1992), no. 1, 5–24. Google Scholar

[6] [6] Milnor, J., Dynamics in one complex variable. Third ed. Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. Google Scholar

[7] [7] Morton, P., On certain algebraic curves related to polynomial maps. Compositio Math. 103(1996), no. 3, 319–350. Google Scholar

[8] [8] Shaferevich, I. R., Basic Algebraic Geometry 1. Varieties in projective space. Second ed. Spring-Verlag, Berlin, 1994. Google Scholar

[9] [9] Shioda, T., On the Mordell-Weil lattices. Comment. Math. Univ. St. Paul. 39(1990), no. 2, 211–240. Google Scholar

[10] [10] Silverman, J. H., The Arithmetic of dynamical systems. Graduate texts in mathematics, 241, Springer, New York, 2007. Google Scholar

[11] [11] Vojta, P., Mordell's conjecture over function fields. Invent. Math. 98(1989), 115–138. http://dx.doi.org/10.1007/BF01388847 Google Scholar

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