A gap principle for dynamics

Abstract

Let $f_1,...,f_g\in {\mathbb C}(z)$ be rational functions, let $\Phi=(f_1,...,f_g)$ denote their coordinatewise action on $({\mathbb P}^1)^g$, let $V\subset ({\mathbb P}^1)^g$ 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.