Dynamics on abelian varieties in positive characteristic

We show that the dynamics of endomorphisms of abelian varieties (e.g., elliptic curves) over field of characteristic $p>0$ is sharply divided according to two possible scenario’s, reflected in properties of the Artin-Mazur zeta function (transcendental/rational) and orbit growth, somewhat similar to the mixing/non-mixing dichotomy in measurable dynamics. The two scenario’s are distinguished by the action on the $p$-torsion subgroup scheme.