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 $(f^{-s-1}(V)\setminus f^{-s}(V))(K)=\varnothing$ for every integer $s\ge s_0$, answering the Preimages Question of Matsuzawa, Meng, Shibata, and Zhang in the case of $({\mathbb{P}}^1{)}^n$.