  In this paper, we study the actions of profinite groups on Cantor sets which
arise from representations of Galois groups of certain fields of rational
functions. Such representations are associated to polynomials, and they are
called profinite iterated monodromy groups. We are interested in a topological
invariant of such actions called the asymptotic discriminant. In particular, we
give a complete classification by whether the asymptotic discriminant is stable
or wild in the case when the polynomial generating the representation is
quadratic. We also study different ways in which a wild asymptotic discriminant
can arise.
