Axial subgroup

For a steady-state equivariant dynamical system, an axial subgroup is a stabilizer subgroup whose real fixed-point subspace of a group action has dimension one. Requiring an actual stabilizer subgroup matters: a smaller subgroup can have the same fixed line without being the full symmetry of any point on that line. The fixed line reduces the steady-state bifurcation equation to one real amplitude.

In an equivariant Hopf bifurcation, a complex axial isotropy subgroup is instead a stabilizer subgroup for the spatial symmetry group combined with the common phase circle action, with one-dimensional complex fixed-point subspace of a group action. Distinct conjugacy classes of such subgroups can describe distinct types of bifurcating branches.

For the spatial group combined with the temporal phase circle in an equivariant Hopf problem, a complex axial isotropy subgroup is an actual stabilizer whose fixed-point space is one complex line. Its real dimension is two. One must include the kernel of the combined action when writing the full stabilizer, not just a generator having the right fixed space.

// Target: dynamical-systems.bigb