  In this paper we investigate fixed-point numbers and entropies of
endomorphisms on abelian varieties. It was shown quite recently that the number
of fixed-points of an iterated endomorphism on a simple complex torus is either
periodic or grows exponentially. Criteria to decide whether a given
endomorphism is of the one type or the other are still missing. Our first
result provides such criteria for simple abelian varieties in terms of the
possible types of endomorphism algebras. The number of fixed-points depends on
the eigenvalues and we exactly show which analytic eigenvalues occur. This
insight is also the starting point to ask for the entropy of an endomorphism.
Our second result offers criteria for an endomorphism to be of zero or positive
entropy. The entropy is computed as the logarithm of a real number and our
third result characterizes the algebraic structure of this number.
