Abstract
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 modulo 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 ω * which is unbalanced in the following sense: for each N>0 there exist two factors of ω * 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 .
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