 

John R. Doyle, Xander Faber, and David Krumm
Preperiodic points for quadratic polynomials over quadratic fields view print


Published: 
May 30, 2014

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


Abstract
To each quadratic number field K and each quadratic polynomial f with Kcoefficients, one can associate a finite directed graph G(f,K) whose vertices are the Krational 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.


Acknowledgements
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
jdoyle@math.uga.edu
Xander Faber:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822
xander@math.hawaii.edu
David Krumm:
Department of Mathematics, Claremont McKenna College, Claremont, CA 91711
dkrumm@cmc.edu

