New York Journal of Mathematics
Volume 20 (2014) 507-605


John R. Doyle, Xander Faber, and David Krumm

Preperiodic points for quadratic polynomials over quadratic fields

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Published: May 30, 2014
Keywords: Arithmetic dynamics, quadratic polynomials, preperiodic points, Uniform Boundedness Conjecture, quadratic points
Subject: 37P35, 14G05

To each quadratic number field K and each quadratic polynomial f with K-coefficients, one can associate a finite directed graph G(f,K) whose vertices are the K-rational preperiodic points for f, and whose edges reflect the action of f on these points. This paper has two main goals. (1) For an abstract directed graph G, classify the pairs (K,f) such that the isomorphism class of G is realized by G(f,K). We succeed completely for many graphs G by applying a variety of dynamical and Diophantine techniques. (2) Give a complete description of the set of isomorphism classes of graphs that can be realized by some G(f,K). A conjecture of Morton and Silverman implies that this set is finite. Based on our theoretical considerations and a wealth of empirical evidence derived from an algorithm that is developed in this paper, we speculate on a complete list of isomorphism classes of graphs that arise from quadratic polynomials over quadratic fields.


The second author was partially supported by an NSF postdoctoral research fellowship.

Author information

John R. Doyle:
Department of Mathematics, University of Georgia, Athens, GA 30602

Xander Faber:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822

David Krumm:
Department of Mathematics, Claremont McKenna College, Claremont, CA 91711