  Feng and Wang showed that two homogeneous iterated function systems in
$\mathbb{R}$ with multiplicatively independent contraction ratios necessarily
have different attractors. In this paper, we extend this result to graph
directed iterated function systems in $\mathbb{R}^n$ with contraction ratios
that are of the form $\frac{1}{\beta}$, for integers $\beta$. By using a result
of Boigelot et al., this allows us to give a proof of a conjecture of
Adamczewski and Bell. In doing so, we link the graph directed iterated function
systems to B\"uchi automata. In particular, this link extends to real numbers
$\beta$. We introduce a logical formalism that permits to characterize sets of
$\mathbb{R}^n$ whose representations in base $\beta$ are recognized by some
B\"uchi automata. This result depends on the algebraic properties of the base:
$\beta$ being a Pisot or a Parry number. The main motivation of this work is to
draw a general picture representing the different frameworks where an analogue
of Cobham's theorem is known.
