On endomorphisms of projective varieties with numerically trivial canonical divisors

Let X be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism f: X → X is amplified (respectively, quasi-amplified) if fa- D - D is ample (respectively, big) for some Cartier divisor D. We show that after iteration and equivariant birational contractions, a quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when X is Hyperkähler, f is quasi-amplified if and only if it is of positive entropy. In both cases, f has Zariski dense periodic points. When X is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).