The size of semigroup orbits modulo primes

Abstract: Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{p}$ of $K$, let $m_{\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\bmod\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\operatorname{\mathsf{N}}_{K/\mathbb{Q}}\mathfrak{p}$ is a set of density zero. For $V=\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.