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One well known question in equivariant bifurcation is usually attributed to
Ize and it reads: in a one parameter family of nonlinear maps which are
equivariant with respect to an absolutely irreducible group action, a loss
of stability implies bifurcation of equilibria.
One strategy to prove such a statement is to prove that absolutely
irreducible group actions have at least one subgroup with an odd
dimensional fixed point space.
We show that this last assertion is false, by constructing absolutely
irreducible group actions in R4 and R8
which have no odd dimensional fixed point space.
We state a new conjecture on absolutely irreducible group actions and
present some evidence. We discuss bifurcation for most of these groups and
note that there are interesting aspects concerning equivariant Hamiltonian
systems.
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