Spatiotemporal symmetry of a periodic orbit ($gx(t)=x(t+\theta T/(2\pi ))$)

A spatial group element combined with a time shift may preserve a periodic orbit even if the spatial element alone does not fix it at each instant. The temporal circle acts complex linearly on Hopf amplitudes. Complex axial isotropy subgroups identify the primary branches guaranteed by the equivariant Hopf theorem.

// Target: dynamical-systems.bigb