Degree Growth of Iterates of Curves and Likely Intersections

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 $\phi =(f,g)$, 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 $a,b\ge 1$, then the bidegree of ${\phi }^n(C)$ is asymptotic to $(\mathrm{°}(g{)}^n,\mathrm{°}(f{)}^n)$. 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 $C^{\mathrm{\prime }}$ is not totally invariant for $\phi$, then for any infinite sequence $n_i$ of positive integers, the union of the intersections $$\bigcup _{i\ge 1}({\phi }^{n_i}(C)\cap C^{\mathrm{\prime }})$$ is Zariski dense in $C^{\mathrm{\prime }}$.