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

Abstract

Let X X be a projective curve in P 1 × P 1 \mathbb {P}^{1} \times \mathbb {P}^{1} and φ \varphi be an endomorphism of degree ≥ 2 \geq 2 of P 1 × P 1 \mathbb {P}^{1} \times \mathbb {P}^{1} , given by two rational functions by φ ( z , w ) = ( f ( z ) , g ( w ) ) \varphi (z,w)=(f(z),g(w)) (i.e., φ = f × g \varphi =f \times g ), where all are defined over Q ¯ \overline {\mathbb {Q}} . In this paper, we prove a characterization of the existence of an infinite intersection of X ( Q ¯ ) X(\overline {\mathbb {Q}}) with the set of φ \varphi -preperiodic points in P 1 × P 1 \mathbb {P}^{1} \times \mathbb {P}^{1} , by means of a binding relationship between the two sets of preperiodic points of the two rational functions f f and g g , in their respective P 1 \mathbb {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 ) \mathcal {J}(f) and J ( g ) \mathcal {J}(g) as well. We then find various sufficient conditions on the pair ( X , φ ) (X,\varphi ) and often on φ \varphi alone, for the finiteness of the set of φ \varphi -preperiodic points of X ( Q ¯ ) X(\overline {\mathbb {Q}}) . The finiteness criteria depend on the rational functions f f and g g , and often but not always, on the curve. We consider in turn various properties of the Julia sets of f f and g 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.