  We discuss the late-time behaviour of a dynamically perturbed Kerr black
hole. We present analytic results for near extreme Kerr black holes that show
that the large number of virtually undamped quasinormal modes that exist for
nonzero values of the azimuthal eigenvalue m combine in such a way that the
field oscillates with an amplitude that decays as 1/t at late times. This
prediction is verified using numerical time-evolutions of the Teukolsky
equation. We argue that the observed behaviour may be relevant for
astrophysical black holes, and that it can be understood in terms of the
presence of a ``superradiance resonance cavity'' immediately outside the black
hole.
