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.