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 $n\le N$ 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.

21 pages

References in corpus (4)

• The dynamical Mordell-Lang problem for etale maps
• Linear relations between polynomial orbits
• Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture
• Periodic points, linearizing maps, and the dynamical Mordell-Lang problem

Cited by in corpus (5)

• Integral Points and Relative Sizes of Coordinates of Orbits in P^N
• On intersections of polynomial semigroups orbits with plane lines
• Analytic relations on a dynamical orbit
• Bounding periods of subvarieties of (P^1)^n
• A Gap Principle for Subvarieties with Finitely Many Periodic Points