  We introduce an extension of the (tame) polynomial automorphism group over
finite fields: the profinite (tame) polynomial automorphism group, which is
obtained by putting a natural topology on the automorphism group. We show that
most known candidate non-tame automorphisms are inside the profinite tame
polynomial automorphism group, giving another result showing that tame maps are
potentially "dense" inside the set of automorphisms. We study the profinite
tame automorphism group and show that it is not far from the set of bijections
obtained by endomorphisms.
