  Motivated by proving the loss of ergodicity in expanding systems of piecewise
affine coupled maps with arbitrary number of units, all-to-all coupling and
inversion symmetry, we provide ad-hoc substitutes - namely inversion-symmetric
maps of the simplex with arbitrary number of vertices - that exhibit several
asymmetric absolutely continuous invariant measures when their expanding rate
is sufficiently small. In a preliminary study, we consider arbitrary maps of
the multi-dimensional torus with permutation symmetries. Using these
symmetries, we show that the existence of multiple invariant sets of such maps
can be obtained from their analogues in some reduced maps of a smaller phase
space. For the coupled maps, this reduction yields inversion-symmetric maps of
the simplex. The subsequent analysis of these reduced maps show that their
systematic dynamics is intractable because some essential features vary with
the number of units; hence the substitutes which nonetheless capture the
coupled maps common characteristics. The construction itself is based on a
simple mechanism for the generation of asymmetric invariant union of polytopes,
whose basic principles should extend to a broad range of maps with permutation
and inversion symmetries.
