  Recent observations of z~6 quasars and the cosmic microwave background imply
a complex history to cosmic reionization. Such a history requires some form of
feedback to extend reionization over a long time interval, but the nature of
the feedback and how rapidly it operates remain highly uncertain. Here we focus
on one aspect of this complexity: which physical processes can cause the global
ionized fraction to evolve non-monotonically with cosmic time? We consider a
range of mechanisms and conclude that double reionization is much less likely
than a long, but still monotonic, ionization history. We first examine how
galactic winds affect the transition from metal-free to normal star formation.
Because the transition is actually spatially inhomogeneous and temporally
extended, this mechanism cannot be responsible for double reionization given
plausible parameters for the winds. We next consider photoheating, which causes
the cosmological Jeans mass to increase in ionized regions and hence suppresses
galaxy formation there. In this case, double reionization requires that small
halos form stars efficiently, that the suppression from photoheating is strong
relative to current expectations, and that ionizing photons are preferentially
produced outside of previously ionized regions. Finally, we consider H_2
photodissociation, in which the buildup of a soft ultraviolet background
suppresses star formation in small halos. This can in principle cause the
ionized fraction to temporarily decrease, but only during the earliest stages
of reionization. Finally, we briefly consider the effects of some of these
feedback mechanisms on the topology of reionization.
