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The issues of scaling symmetry and critical point behavior are studied forfluctuations about extremal charged black holes. We consider the scattering andcapture of the spherically symmetric mode of a charged, massive test field onthe background spacetime of a black hole with charge $Q$ and mass $M$. Thespacetime geometry near the horizon of a $|Q|=M$ black hole has a scalingsymmetry, which is absent if $|Q|<M$, a scale being introduced by the surfacegravity. We show that this symmetry leads to the existence of a self-similiarsolution for the charged field near the horizon, and further, that there is aone parameter family of discretely self-similiar solutions . The scalingsymmetry, or lack thereof, also shows up in correlation length scales, definedin terms of the rate at which the influence of an external source coupled tothe field dies off. It is shown by constructing the Greens functions, that anexternal source has a long range influence on the extremal background, comparedto 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 infinitesimalchanges in the black hole area vary like $\Delta ^{1\over 2}$.Comment: 12 pages (harvmac), UMHEP-40

Discrete self-similarity and critical point behavior in fluctuations about extremal black holes