  We provide an in-depth investigation of quasinormal-mode oscillations of Kerr
black holes with nearly extremal angular momenta. We first discuss in greater
detail the two distinct types of quasinormal mode frequencies presented in a
recent paper (arXiv:1212.3271). One set of modes, that we call "zero-damping
modes", has vanishing imaginary part in the extremal limit, and exists for all
corotating perturbations (i.e., modes with azimuthal index m being
nonnegative). The other set (the "damped modes") retains a finite decay rate
even for extremal Kerr black holes, and exists only for a subset of corotating
modes. As the angular momentum approaches its extremal value, the frequency
spectrum bifurcates into these two distinct branches when both types of modes
are present. We discuss the physical reason for the mode branching by
developing and using a bound-state formulation for the perturbations of generic
Kerr black holes. We also numerically explore the specific case of the
fundamental l=2 modes, which have the greatest astrophysical interest. Using
the results of these investigations, we compute the quasinormal mode response
of a nearly extremal Kerr black hole to perturbations. We show that many
superimposed overtones result in a slow power-law decay of the quasinormal
ringing at early times, which later gives way to exponential decay. This
exceptional early-time power-law decay implies that the ringdown phase is
long-lived for black holes with large angular momentum, which could provide a
promising strong source for gravitational-wave detectors.
