  We describe a general program for studying the dynamics of surjective
endomorphisms of algebraic varieties that are amenable to techniques from the
minimal model program. We obtain density results on the pre-periodic points of
surjective endomorphisms of varieties admitting an int-amplified endomorphism,
and reduce certain cases of the Medvedev-Scanlon conjecture to so called
Q-abelian varieties using our approach. We also provide a connection between
the existence of an automorphism with positive entropy and group of connected
components of a variety. In particular, we show that if $X$ is normal and
projective with finitely generated nef cone then $X$ has an automorphism of
positive entropy if and only if the group of connected components
$\pi_0\Aut(X)$ has an element of infinite order.
