  Recent results of Laca, Raeburn, Ramagge and Whittaker show that any
self-similar action of a groupoid on a graph determines a 1-parameter family of
self-mappings of the trace space of the groupoid C*-algebra. We investigate the
fixed points for these self-mappings, under the same hypotheses that Laca et
al. used to prove that the C*-algebra of the self-similar action admits a
unique KMS state. We prove that for any value of the parameter, the associated
self-mapping admits a unique fixed point, which is in fact a universal
attractor. This fixed point is precisely the trace that extends to a KMS state
on the C*-algebra of the self-similar action.
