Given an IFS (iterated function system), the question whether its attractor can be obtained as the attractor of another IFS was addressed by Feng and Wang in 2009. In the case of homogeneous IFS of $\mathbb R$ whose contraction maps are affine maps and satisfying the open set condition, they proved that if a subset of $\mathbb R$ is not a finite union of intervals and is the attractor of two different IFS, then the contraction ratios of these two IFS have to be strongly linked. Such a statement resembles very much the theorem of Cobham of 1969, although the two results concern a priori different areas of mathematics. Surprisingly enough, a quite similar result was obtained by Adamczewski and Bell in 2011: A compact subset of $[0, 1]$ is simultaneously $b-$ and $b'-$self-similar if and only if it is a finite union of intervals with rational endpoints. Even more surprisingly, a third similar result was obtained by Boigelot, Brusten and Bruyère in 2010 in a theoretical computer science setting. It concerns subsets of R d whose sets of representations in some integer base are accepted by weak Büchi automata. The aim of this talk is to show the connections between these three results. This connection is achieved by using GDIFS (graph directed iterated functions systems) and allows us to provide extensions of the results in the three frameworks: Adamcweski and Bell’s result extends to $\mathbb{R}^d$ , Feng and Wang’s result extends to a large class of GDIFS, and the logical characterization of recognizable sets used by Boigelot, Brusten and Bruyère extends to the so-called Pisot real bases.
