A gap principle for dynamics

TL;DR

This paper establishes a sparsity result for the intersection of orbits of rational functions with subvarieties, showing that such intersections are extremely rare unless the subvariety contains a positive-dimensional periodic subvariety.

Contribution

It introduces a new gap principle for dynamics, demonstrating that the set of times when an orbit hits a subvariety is very sparse unless the subvariety contains a positive-dimensional periodic component.

Findings

The set of n with ^n(P) in V is very sparse.

For large N, the count of such n is less than a logarithmic iterate of N.

The result generalizes classical gap principles to a dynamical setting.

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.