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.
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