ON THE PREPERIODIC POINTS OF AN ENDOMORPHISM OF P1 x P1 WHICH LIE ON A CURVE

Let X be a projective curve in P-1 x P-1 and phi be an endomorphism of degree >= 2 of P-1 x P-1, given by two rational functions by phi( z, w) = (f(z), g(w)) (i.e., phi = f x g), where all are defined over (Q) over bar. In this paper, we prove a characterization of the existence of an infinite intersection of X((Q) over bar) with the set of phi-preperiodic points in P-1 x 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, phi) and often on phi alone, for the finiteness of the set of phi-preperiodic points of X((Q) over bar). 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.