  We characterize possible periodic subvarieties for surjective endomorphisms
of complex abelian varieties in terms of the eigenvalues of the cohomological
actions induced by the endomorphisms, extending previous work in this direction
by Pink and Roessler. By applying our characterization to induced endomorphisms
on Albanese varieties, we draw conclusions about the dynamics of surjective
endomorphisms for a broad class of projective varieties. We also analyze
several classes of surjective endomorphisms that are distinguished by
properties of their cohomological actions.
