Dynamics on abelian varieties in positive characteristic

We study periodic points and orbit length distribution for endomorphisms of abelian varieties in characteristic p>0. .We study rationality, algebraicity and the natural boundary property for the dynamical zeta function (the latter using a general result on power series proven by Royals and Ward in the appendix), as well as analogues of the prime number theorem, also for tame dynamics, ignoring orbits whose order is divisible by p. The behavior is governed by whether or not the action on the local p-torsion group scheme is nilpotent.