Dynamical Systems Seminar

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.