Preimages Question for Surjective Endomorphisms on $(\mathbb{P}^1)^n$

Abstract

Let $K$ be a number field and let $f : (\mathbb{P}^1)^n \to (\mathbb{P}^1)^n$ be a dominant endomorphism defined over $K$. We show that if $V$ is an $f$-invariant subvariety (that is, $f(V)=V$) then there is a positive integer $s_0$ such that $ (f^{-s-1}(V)\setminus f^{-s}(V))(K) = \emptyset$ for every integer $s \geq s_0$, answering the Preimages Question of Matsuzawa, Meng, Shibata, and Zhang in the case of $(\mathbb{P}^1)^n$.