  If $X$ is the attractor set of a conformal IFS in dimension two or three, we
prove that there exists a quasiregular semigroup $G$ with Julia set equal to
$X$. We also show that in dimension two, with a further assumption similar to
the open set condition, the same result can be achieved with a semigroup
generated by one element. Consequently, in this case the attractor set is
quasiconformally equivalent to the Julia set of a rational map.
