Simple automorphisms of complex projective varieties

Start with a regular automorphism f of a complex projective
variety X. Then, iterating f and looking at its orbits, one gets a dynamical
system; there are examples with a rich dynamical behavior, for instance
on specific K3 surfaces or hyperkähler manifolds, and these examples have
been intensively studied during the last decades. During this talk, I will
look in a somewhat opposite direction, and ask for a description of
automorphisms with a dynamics of low complexity.