On the preperiodic points of an endomorphism of $\mathbb {P}^{1} \times \mathbb {P}^{1}$ which lie on a curve

Abstract:

Let $X$ be a projective curve in $\mathbb {P}^{1} \times \mathbb {P}^{1}$ and $\varphi$ be an endomorphism of degree $\geq 2$ of $\mathbb {P}^{1} \times \mathbb {P}^{1}$, given by two rational functions by $\varphi (z,w)=(f(z),g(w))$ (i.e., $\varphi =f \times g$), where all are defined over $\overline {\mathbb {Q}}$. In this paper, we prove a characterization of the existence of an infinite intersection of $X(\overline {\mathbb {Q}})$ with the set of $\varphi$-preperiodic points in $\mathbb {P}^{1} \times \mathbb {P}^{1}$, by means of a binding relationship between the two sets of preperiodic points of the two rational functions $f$ and $g$, in their respective $\mathbb {P}^{1}$-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 $\mathcal {J}(f)$ and $\mathcal {J}(g)$ as well. We then find various sufficient conditions on the pair $(X,\varphi )$ and often on $\varphi$ alone, for the finiteness of the set of $\varphi$-preperiodic points of $X(\overline {\mathbb {Q}})$. 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.