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

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