  We consider a family of Cuntz-Pimsner algebras associated to self-similar
group actions, and their Toeplitz analogues. Both families carry natural
dynamics implemented by automorphic actions of the real line, and we
investigate the equilibrium states (the KMS states) for these dynamical
systems.
  We find that for all inverse temperatures above a critical value, the KMS
states on the Toeplitz algebra are given, in a very concrete way, by traces on
the full group algebra of the group. At the critical inverse temperature, the
KMS states factor through states of the Cuntz-Pimsner algebra; if the
self-similar group is contracting, then the Cuntz-Pimsner algebra has only one
KMS state. We apply these results to a number of examples, including the
self-similar group actions associated to integer dilation matrices, and the
canonical self-similar actions of the basilica group and the Grigorchuk group.
