  A system of four globally coupled doubling maps is studied in this paper. It
is known that such systems have a unique absolutely continuous invariant
measure (acim) for weak interaction, but the case of stronger coupling is still
unexplored. As in the case of three coupled sites, we prove the existence of a
critical value of the coupling parameter at which multiple acims appear. Our
proof has several new ingredients in comparison to the one presented in our
previous paper regarding the system of three sites. We strongly exploit the
symmetries of the dynamics in the course of the argument. This simplifies the
computations considerably, and gives us a precise description of the geometry
and symmetry properties of the arising asymmetric invariant sets. Some new
phenomena are observed which are not present in the case of three sites. In
particular, the asymmetric invariant sets arise in areas of the phase space
which are transient for weaker coupling and a nontrivial symmetric invariant
set emerges, shaped by an underlying centrally symmetric Lorenz map. We state
some conjectures on further invariant sets, indicating that unlike the case of
three sites, ergodicity breaks down in many steps, and not all of them are
accompanied by symmetry breaking.
