Rational dynamical systems, $S$-units, and $D$-finite power series

Abstract

Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then there are a multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ \Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then obtain results about the coefficients of $D$-finite power series using these facts.