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.
Categories: math.NT math.AG math.DS