SPLITTING EQUIVARIANT DYNAMICS

We prove that any dynamical system on a G manifold M which is equivari ant under the G action, can be decomposed into the semi-direct product of an autonomous dynamics in the G orbit space Omega = M/G, and a dyn amics (depending on the G orbit) on G. This result is actually a corol lary of Michel theorem (L. Michel, C. R. Acad. Sci. Paris A, 272 (1971 ) 433) on the geometry of symmetry breaking, and uses the same ingredi ents for the proof. It permits to unify a number of known and useful r esults in the literature, as discussed here.