The Skolem–Mahler–Lech conjecture with dynamically varying coefficients

Let $l\geq 1$, let $g,a_i\in\mathbb{C}(x)$ for $i=0,\dots,l-1$, and let $\alpha\in\mathbb{C}$ satisfy $a_i(g^n(\alpha))\neq\infty$ for every $i=0,\dots,l-1$ and $n\geq 0$. Let $\{A_n\}_{n\geq 0}$ be a recurrence sequence satisfying

$$A_{n+l}=\sum_{i=0}^{l-1}a_i(g^n(\alpha))A_{n+i}$$

for all $n\geq 0$. The dynamically varying-coefficient Skolem–Mahler–Lech conjecture. The zero set

$$\{n\geq 0\mid A_n=0\}$$

is a union of at most finitely many arithmetic progressions. This generalizes the classical theorem by allowing coefficients generated along an orbit of a rational function; its status is left open in the source.