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

Feng and Wang showed that two homogeneous iterated function systems in R with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend this result to graph directed iterated function systems in R with contraction ratios that are of the form 1 β , for integers β. 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 Büchi automata. In particular, this link extends to real numbers β. We introduce a logical formalism that permits to characterize sets of R whose representations in base β are recognized by some Büchi automata. This result depends on the algebraic properties of the base: β 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.