Let $varphi: {mathbb P}^1 longrightarrow {mathbb P}^1$ be a rational map of degree greater than one defined over a number field $k$. For each prime ${mathfrak p}$ of good reduction for $varphi$, we let $varphi_{mathfrak p}$ denote the reduction of $v
arphi$ modulo ${mathfrak p}$. A random map heuristic suggests that for large ${mathfrak p}$, the proportion of periodic points of $varphi_{mathfrak p}$ in ${mathbb P}^1({mathfrak o}_k/{mathfrak p})$ should be small. We show that this is indeed the case for many rational functions $varphi$.
Let $f_1,...,f_gin {mathbb C}(z)$ be rational functions, let $Phi=(f_1,...,f_g)$ denote their coordinatewise action on $({mathbb P}^1)^g$, let $Vsubset ({mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)in ({mathbb P}^1)^g({mathbb C})$
be a nonpreperiodic point for $Phi$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $Phi^n(P) in V({mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n leq N$ such that $Phi^n(P) in V({mathbb C})$ is less than $log^k N$, where $log^k$ denotes the $k$-th iterate of the $log$ function. This can be interpreted as an analog of the gap principle of Davenport-Roth and Mumford.
