  We study the dynamical Chern-Simons gravity as an effective quantum field
theory, and discuss a broad range of its parameter space where the theory is
valid. Within that validity range, we show that slowly rotating black holes
acquire novel geometric structures due to the gravitational dynamical
Chern-Simons term. In particular, the rotating black hole solutions get endowed
with two, cap-like domains, emanating from the north and south poles in the
standard Boyer-Lindquist coordinates. The domains extend out to a distance that
is approximately a few percent of the black hole's size. The cap-like domains
have an unusual equation of state, pointing to non-standard dynamics within the
caps. In particular, the focusing condition for geodesics is violated in those
domains. This in turn implies that the Hawking-Penrose singularity theorem
cannot be straightforwardly applied to hypothetical probe matter placed within
the Chern-Simons caps.
