We consider linear iterated function systems (IFS) with a constant
contraction ratio in the plane for which the ``overlap set'' $\Ok$ is finite,
and which are ``invertible'' on the attractor $A$, the sense that there is a
continuous surjection $q: A\to A$ whose inverse branches are the contractions
of the IFS. The overlap set is the critical set in the sense that $q$ is not a
local homeomorphism precisely at $\Ok$. We suppose also that there is a
rational function $p$ with the Julia set $J$ such that $(A,q)$ and $(J,p)$ are
conjugate.