An analogue of Cobham's theorem for graph directed iterated function systems

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 {\em 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 Büchi 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 Büchi 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.