#### Vol. 15, No. 7, 2021

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

### Jason P. Bell, Shaoshi Chen and Ehsaan Hossain

Vol. 15 (2021), No. 7, 1699–1728
DOI: 10.2140/ant.2021.15.1699
##### Abstract

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form ${a}_{n}:=f\left({\phi }^{n}\left({x}_{0}\right)\right)$, where $\phi :X--\to X$ and $f:X--\to {ℙ}^{1}$ are rational maps defined over $K$ and ${x}_{0}\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\phi$ 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 upper Banach density zero. In addition, we show that if ${a}_{n}\in G$ for every $n$ and $X$ is irreducible and the $\phi$ orbit of $x$ is Zariski dense in $X$ then there is 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 ${a}_{n}=\left(g\circ {\Psi }^{n}\right)\left(y\right)$ for some $y\in {\mathbb{𝔾}}_{m}^{d}$. We then obtain results about the coefficients of $D$-finite power series using these facts.

##### Keywords
$S$-units, $D$-finite series, arithmetic dynamics, algebraic groups, orbits
##### Mathematical Subject Classification
Primary: 14E05, 37P55
Secondary: 12H05
##### Milestones
Received: 18 May 2020
Revised: 6 January 2021
Accepted: 4 February 2021
Published: 1 November 2021
##### Authors
 Jason P. Bell Department of Pure Mathematics University of Waterloo Waterloo, ON Canada Shaoshi Chen Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Ehsaan Hossain Department of Mathematical & Computational Sciences University of Toronto Mississauga Mississauga, ON Canada