  We study the iterative behavior of the family of 3-step linear fractional
recurrences and the family of birational maps they define. We determine all the
possible periodicities within this family or, equivalently, the birational maps
of finite order. This family also contains pseudo-automorphisms of infinite
order. One such family consists of completely integrable maps, and another
family consists of maps of positive entropy. Both of these families have
invariant families of $K3$ surfaces.
