  We find that in nonhermitian PT-symmetric systems (as realized in resonators
with balanced absorption and amplification), a mechanism based on
quantum-to-classical correspondence reduces the occurrence of strongly
amplified states. The reduction arises from semiclassically emerging
hierarchical phase-space structures that are associated with the coupling of
the amplifying and absorbing regions (forward and backward-trapped sets and
their complements), and amounts to a generalization of the fractal Weyl law,
earlier proposed for ballistically open systems. In the context of the recently
introduced class of PT-symmetric laser-absorbers, this phenomenon reduces the
number of states participating in the mode competition.
