Degree Growth of Iterates of Curves and Likely Intersections

Abstract: We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}¹ \times \mathbb{P}<sup>1$ under a product polynomial endomorphism $\varphi=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f<sup>a,g<sup>b)$ for any $a,b\geq 1$, then the bidegree of $\varphi<sup>n(C)$ is asymptotic to $(°(g)<sup>n,°(f)<sup>n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f<sup>a,g<sup>b)$-preperiodic and $C&#39;$ is not totally invariant for $\varphi$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( \varphi^{n_i}(C)\cap C&#39; \right) $$ is Zariski dense in $C&#39;$.