Degree Growth of Iterates of Curves and Likely Intersections

Abstract

We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}^1 \times \mathbb{P}^1$ under a product polynomial endomorphism $φ=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\geq 1$, then the bidegree of $φ^n(C)$ is asymptotic to $(°(g)^n,°(f)^n)$. 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 $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $φ$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( φ^{n_i}(C)\cap C' \right) $$ is Zariski dense in $C'$.