Simultaneous Periods for Families of Rational Maps Modulo Primes

Let $K$ be a number field, and $\varphi_{1},\ldots,\varphi_{g}\in K(t)$ be finitely many rational maps, each of degree at least $2$. We first show that for generic finite sets $\mathcal{A}_{1},\ldots,\mathcal{A}_{g}$ consisting entirely of points that are not $\varphi_{i}$-periodic, there exists a set of primes $\mathfrak p$ of $K$ of positive density such that for each $\mathcal{A}_{i}$ and every $\alpha\in\mathcal{A}_i$, $\alpha$ is not $\varphi_i$-periodic modulo $\mathfrak p$. The notion of genericity used here is defined in terms of the associated arboreal fields and is sharper than those previously used in the literature. Leveraging our proof in the generic case, we then show that the same conclusion holds for most \textit{expected} cases of non-generic sets $\mathcal{A}_{i}$. Finally, we apply our result to confirm the dynamical Mordell--Lang conjecture for coordinate-wise actions of a class of maps that includes rational maps that are generic in this sense.