Geometric Models of Pisot Substitutions and Non-commutative Arithmetic

Unimodular Substitutions on 2 letters. Conjecture: the dynamical system associated with a primitive substitution on 2 letters, with matrix in SL(2,Z), is measurably isomorphic to a circle rotation. There is a very convenient criterium, due to B.Host: Definition: the substitution σ has strong coincidence if there exists n and k such that σ(0) and σ(1) have same letter of index k, and the 2 prefixes of order k have the same number of 0 and 1 (in other word, these two prefixes have the same abelianization). Host proved that every unimodular substitution on 2 letters with strong coincidence generates a dynamical system that is isomorphic to a circle rotation. No example of unimodular substitution without strong coincidence is known, and the above conjecture says that such an example does not exist. Remark that there are primitive non unimodular substitutions without coincidences, the simplest being Morse substitution (0 7→ 01, 1 7→ 10). It is tempting to generalize for more letters; the simplest natural setting is that of unimodular Pisot substitution, generalized strong coincidence conditions are easy to define, and again no example of a unimodular Pisot substitution without strong coincidences is known. In that case, strong coincidence condition implies that the dynamical system is isomorphic to an exchange of pieces by translation, and an additional condition implies that it is isomorphic to a toral translation. One can again conjecture that every unimodular Pisot substitution is isomorphic to a toral rotation, although a counterexample seems more likely in 3 or more letters. See [S] and the forthcoming papers of Anne Siegel for more information on the subject. The above conjecture has been recently proved by Barge and Diamond, see [BD]. They proved that all Pisot substitutions, even not unimodular, and on d letters, have at least one strong coincidence. This proves that all Pisot substitutions on 2 letters have discrete spectrum, see [HS]. Extension of this result to 3 or more letters seems difficult.