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Let X be a projective curve in P1 × P1 and φ be an endomorphism of degree ≥ 2 of P1 × P1, given by two rational functions by φ(z,w) = (f(z), g(w)) (i.e., φ = f × g), where all are defined over Q. In this paper, we prove a characterization of the existence of an infinite intersection of X(Q) with the set of φ-preperiodic points in P1×P1, by means of a binding relationship between the two sets of preperiodic points of the two rational functions f and g, in their respective P1-components. In turn, taking limits under the characterization of the Julia set of a rational function as the derived set of its preperiodic points, we obtain the same relationship between the respective Julia sets J (f) and J (g) as well. We then find various sufficient conditions on the pair (X,φ) and often on φ alone, for the finiteness of the set of φ-preperiodic points of X(Q). The finiteness criteria depend on the rational functions f and g, and often but not always, on the curve. We consider in turn various properties of the Julia sets of f and g, as well as their interactions, in order to develop such criteria. They include: topological properties, symmetry groups as well as potential theoretic properties of Julia sets. 1. Arakelov theory and the preperiodic points on a curve in P × P 1.1. Some important remarks. 1.2. Introduction. An endomorphism φ of a projective space, defined over a number field, determines an infinite set of algebraic preperiodic points. Let us denote by Prepφ(X) the intersection that a projective curve X in this space may have with this set over Q. We may ask, is Prepφ(X) Zariski dense? Under what conditions can this set be infinite? Is the curve X itself necessarily preperiodic under the morphism φ, which then would automatically provide for such an infinite intersection set? These questions are closely related to various conjectures and problems which historically guided the evolution of algebraic dynamics. In particular, the Bogomolov conjecture [5] (settled by Ullmo and Zhang) and its generalization, as well as the original dynamical Manin-Mumford conjecture of S. Zhang, which has since been refined in [35] and expanded by him, into what is a list of conjectures, during a recent conference. Received by the editors April 17, 2010 and, in revised form, August 10, 2010, September 27, 2010 and January 11, 2011. 2010 Mathematics Subject Classification. Primary 14G40; Secondary 37F10, 37P05, 37P30. This work consists of the author’s dissertation at Columbia University (1997), previously unpublished. The author takes this opportunity to extend his thanks to advisors L. Szpiro and S. Zhang for their help and support during that endeavor. 1Algebraic Dynamics Conference 2010, New York c ©2012 American Mathematical Society Reverts to public domain 28 years from publication

On the preperiodic points of an endomorphism of ℙ¹×ℙ¹ which lie on a curve