Very ample polarized self maps extend to projective space

Abstract

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into $\mb{P}^m$ and let $\phi: X \to X$ be a morphism such that $\phi^*L \cong L^{\otimes q}, q\geq 2$. Then there exists an integer $r>0$ extending $\phi^r$ to $\mb{P}^m$.