Very ample polarized self maps extend to projective space

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