Surjective rational maps and del Pezzo surfaces

We study surjective (not necessarily regular) rational endomorphisms $f$ of smooth del Pezzo surfaces $X$. We prove that under certain natural non,-,degeneracy condition $f$ can have degree bigger than $1$ only when $(-K_X^2)>5$. Some structural properties of $f$ in the case $X={\text{p}}^2$ are also established.