Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor. 

Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an involution $\tau\_i:C\to C$. Let $G\simeq \mathbb Z/2 \* \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

Are there points on $C$ with finite orbit under $G$?
['https://mathoverflow.net/questions/219984', 'https://mathoverflow.net', 'https://mathoverflow.net/users/68722/']
No. Letting $\sigma$ and $\tau$ denote the two involutions, $\sigma \circ \tau$ is a translation by an element of $\mathrm{Pic}^0(C)$. In general, this translation will not be torsion, so its orbit through any point is infinite. (In fact, if the translation IS torsion, then $G$ is a finite dihedral group, not $\mathbb{Z}/2 \ast \mathbb{Z}/2$.)
Regarding K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, you might find the following two articles relevant. (Also forward trace them as references for further articles.)

1. Baragar, A. Rational points on K3 surfaces in $\mathbb
 P^1\times\mathbb P^1\times\mathbb P^1$. *Math. Ann.* **305** (1996),
no. 3, 541-558. (MR1397435)
2. Wang, Lan Rational points and canonical heights on K3-surfaces in
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. *Recent developments
in the inverse Galois problem* (Seattle, WA, 1993), 273-289,
*Contemp. Math.*, **186**, Amer. Math. Soc., Providence, RI, 1995
(MR1352278)

**Addendum** Since you mention that you're interested in higher dimensional analogous, I'll mention the following. The involutions on a $(2,2,2)$-surface in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ are morphisms. But if $n\ge4$, then the involutions on a $(2,2,\ldots,2)$ hypersurface in $\underbrace{\mathbb P^1\times\mathbb P^1\times\cdots\times\mathbb P^1}\_{\text{$n$ factors}}$ have non-trivial indeterminacy locus, so some points have orbits that terminate.