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

作者：Charlier Emilie; Leroy Julien; Rigo Michel^*

来源：Advances in Mathematics, 2015, 280: 86-120.

DOI：10.1016/j.aim.2015.04.008

• Iterated function systems
• Graph-directed iterated function systems
• Self-similar set
• Cobham's theorem
• beta-Numeration
• Buchi automaton

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-n with contraction ratios that are of the form 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 Buchi automata. In particular, this link extends to real numbers beta. We introduce a logical formalism that permits to characterize sets of R-n whose representations in base beta are recognized by some Buchi 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.

• 出版日期2015-8-6