On eigenvalues on the orbit space

A map P equivariant with respect to a compact Lie group induces a map Q on the orbit space, which is differentiable provided that P is sufficiently sm ooth. An equilibrium of Q corresponds to an invariant orbit of P. We compar e the eigenvalues of the linearization at the equilibrium with the eigenval ues of a kind of linearization at the invariant orbit. It turns out that th e former are products of the latter. Furthermore, the eigenvalues on the or bit space suffice to determine asymptotic stability or instability of the i nvariant orbit. This generalizes a similar result for vector fields by Koen ig, Math. Proc. Cambridge, Philos. Sec. 121 (1997) 401-424 and applies also to relative periodic points. (C) 2001 Elsevier Science B.V. All rights res erved.