Simple abelian varieties and primitive automorphisms of null entropy of   surfaces
  We characterize simple complex abelian varieties and simple abelian surfaces in terms of primitivity of translation automorphisms. Applying this together with a result due to Diller and Favre, we then classify all primitive birational automorphisms with trivial first dynamical degree of smooth projective surfaces over an algebraically closed field of any characteristic. 
Keiji Oguiso
math.AG