  Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X
\rightarrow X$ be a finite surjective morphism of degree at least two. We
establish a close relation between the irreducible components of the locus of
singularities that are not log-canonical and the dynamics of the endomorphism
f. As a consequence we prove that if X is projective and f polarised, then X
has at most log-canonical singularities.
