Imbalances in Arnoux-Rauzy sequences

In a 1982 paper Rauzy showed that the subshift (X,T) generated by the morphism 1 --> 12, 2 --> 13 and 3 --> 1 is a natural coding of a rotation on the two-dimensional torus T-2, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in R-2, each domain being translated by the same vector module a lattice. It was believed more generally that each sequence of block complexity 2n + 1 satisfying a combinatorial criterion known as the (*) condition of Arnoux and Rauzy codes the orbit of a point under a rotation on T-2. In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence w(*) which is unbalanced in the following sense: for each N > 0 there exist two factors of w(*) of equal length, with one having at least N more occurrences of a given letter than the other. We then invoke a result due to Rauzy on bounded remainder sets to establish the existence of an Arnoux-Rauzy sequence which is not a natural coding of a rotation on T-2.