  The issues of scaling symmetry and critical point behavior are studied for
fluctuations about extremal charged black holes. We consider the scattering and
capture of the spherically symmetric mode of a charged, massive test field on
the background spacetime of a black hole with charge $Q$ and mass $M$. The
spacetime geometry near the horizon of a $|Q|=M$ black hole has a scaling
symmetry, which is absent if $|Q|<M$, a scale being introduced by the surface
gravity. We show that this symmetry leads to the existence of a self-similiar
solution for the charged field near the horizon, and further, that there is a
one parameter family of discretely self-similiar solutions . The scaling
symmetry, or lack thereof, also shows up in correlation length scales, defined
in terms of the rate at which the influence of an external source coupled to
the field dies off. It is shown by constructing the Greens functions, that an
external source has a long range influence on the extremal background, compared
to a correlation length scale which falls off exponentially fast in the $|Q|<M$
case. Finally it is shown that in the limit of $\Delta \equiv (1-{Q^2 \overM^2}
)^{1\over 2} \rightarrow 0$ in the background spacetime, that infinitesimal
changes in the black hole area vary like $\Delta ^{1\over 2}$.
