Pisot substitutions and Rauzy fractals. (English) Zbl 1007.37001

The Rauzy fractal is a domain in \(\mathbb{R}^2\) with fractal boundary introduced in 1982 [P. Arnoux and G. Rauzy, Bull. Soc. Math. Fr. 119, No. 2, 199-215 (1991; Zbl 0789.28011)], which is related to the substitution \(1\to 12\), \(2\to 13\), \(3\to 1\). The authors study a generalization of this fractal by showing that the dynamical system generated by a Pisot primitive unimodular substitution on \(d\) letters satisfying an extra combinatorial condition is measurably isomorphic to a domain exchange in \(\mathbb{R}^{d-1}\), and is a finite extension of a translation on the torus \(\mathbb{T}^{d-1}\). Note that the system is actually isomorphic to a translation on the torus in all known examples: the question whether this is always true is open.
Note that Reference [AISO1] has appeared with authors in a different order [J. Anal. 83, 183-206 (2001; Zbl 0987.11013)]. Also note that Reference [IH 94] is given without page numbers: it should probably be replaced by a paper of S. Ito with the same title reviewed as [Aoki, N. (ed.) et al., Dynamical systems and chaos. Vol. 1: Mathematics, engineering and economics, Proceedings of the international conference, Hochioji, Japan, May 23-27, 1994, Singapore: World Scientific, 101-102 (1995; Zbl 0990.37506)].


37B10 Symbolic dynamics
28A80 Fractals
11B85 Automata sequences