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

Abstract. Let X be a projective curve in ℙ ¹ × ℙ ¹ and φ{symbol} be an endomorphism of degree ≥ 2 of ℙ ¹ × ℙ ¹, given by two rational functions by φ{symbol}(z,w) = (f(z), g(w)) (i.e., φ{symbol} = 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{symbol}-preperiodic points in ℙ ¹×ℙ ¹, by means of a binding relationship between the two sets of preperiodic points of the two rational functions f and g, in their respective ℙ ¹-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{symbol}) and often on φ{symbol} alone, for the finiteness of the set of φ{symbol}-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.