  We obtain some general restrictions on the continuous endomorphisms of a
profinite group G under the assumption that G has only finitely many open
subgroups of each index (an assumption which automatically holds, for instance,
if G is finitely generated). In particular, given such a group G and a
continuous endomorphism phi we obtain a semidirect decomposition of G into a
'contracting' normal subgroup and a complement on which phi induces an
automorphism; both the normal subgroup and the complement are closed. If G is
isomorphic to a proper open subgroup of itself, we show that G has an infinite
abelian normal pro-p subgroup.
