  There are many tools for studying local dynamics. An important problem is how
this information can be used to obtain global information. We present examples
for which local stability does not carry on globally. To this purpose we
construct, for any natural n>1, planar maps whose symmetry group is Zn having a
local attractor that is not a global attractor. The construction starts from an
example with symmetry group Z4. We show that although this example has
codimension 3 as a Z4-symmetric map-germ, its relevant dynamic properties are
shared by two 1-parameter families in its universal unfolding. The same
construction can be applied to obtain examples that are also dissipative. The
symmetry of these maps forces them to have rational rotation numbers.
