  Aims: In this paper we study density cusps that may contain central black
holes. The actual co-eval self-similar growth would not distinguish between the
central object and the surroundings. Methods: To study the environment of a
growing black hole we seek descriptions of steady `cusps' that may contain a
black hole and that retain at least a memory of self-similarity. We refer to
the environment in brief as the `bulge' and on smaller scales, the `halo'.
Results: We find simple descriptions of the simulations of collisionless matter
by comparing predicted densities, velocity dispersions and distribution
functions with the simulations. In some cases central point masses may be
included by iteration. We emphasize that the co-eval self-similar growth allows
an explanation of the black hole bulge mass correlation between approximately
similar collisionless systems. Conclusions: We have derived our results from
first principles assuming adiabatic self-similarity and either self-similar
virialisation or normal steady virialisation. We conclude that distribution
functions that retain a memory of self-similar evolution provide an
understanding of collisionless systems. The implied energy relaxation of the
collisionless matter is due to the time dependence. Phase mixing relaxation may
be enhanced by clump-clump interactions.
