  Let $X$ be a projective variety over an algebraically closed field $k$ of
arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is
$q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and
integer $q > 1$.
  Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We
show that the anti-canonical divisor $-K_X$ is numerically equivalent to an
effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion
of Boucksom, de Fernex and Favre (Theorem C) and also covering singular
varieties over an algebraically closed field of arbitrary characteristic.
  Suppose $f$ is separable and $X$ is normal. We show that the Albanese
morphism of $X$ is an algebraic fibre space and $f$ induces polarized
endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$
being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion
$\mathbb{Q}$-divisor.
  Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if
$p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal model
program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a
mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical
divisor (and also with $Y$ being a quasi-\'etale quotient of an abelian variety
when $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as a
scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when
$X$ is a smooth and rationally chain connected projective variety of dimension
at most three.

Algebraic Geometry;Dynamical Systems