  We investigate some general questions in algebraic dynamics in the case of
generic endomorphisms of projective spaces over a field of characteristic zero.
The main results that we prove are that a generic endomorphism has no
non-trivial preperiodic subvarieties, any infinite set of preperiodic points is
Zariski dense and any infinite subset of a single orbit is also Zariski dense,
thereby verifying the dynamical "Manin--Mumford" conjecture of Zhang and the
dynamical "Mordell--Lang" conjecture of Denis and Ghioca--Tucker in this case.

Dynamical Systems