  If $F$ and $G$ are iterated function systems, then any infinite word $W$ in
the symbols $F$ and $G$ induces a limit set. It is natural to ask whether this
Cantor set can also be realized as the limit set of a single $C^{1 + \alpha}$
iterated function system $H$. We prove that under certain assumptions on $F$,
$G$ and $H$, the answer is no. This problem is motivated by the spectral theory
of one-dimensional quasicrystals.
