Symmetries in Rauzy Fractals

In the present paper we study geometrical symmetries of the Rauzy fractals and their relation to symbolic symmetries, i.e. symmetries of the languages that define the fractals. The geometrical symmetries studied here are reflections through a point, i.e. its center of symmetry. We show that for unimod- ular Pisot substitutions so that the abelianization of the set of proper prefixes is symmetric in R, there is a symmetrical subset of the Rauzy fractal. This subset corresponds to the maximal symbolic system in the paths of the prefix automaton that is invariant under the involution defined by the symmetries of the prefixes. We also give a generalization of this construction when the abelianization of the set of proper prefixes is not symmetrical. We apply some of these techniques to show that the Rauzy fractal of substi- tutions of type 1 → 1 · · · 1 2, 2 → 1 · · · 1 3, . . . , (k − 1) → 1 · · · 1 k, k → 1; | {z } | {z } | {z } n n n with n ≥ 1 and k ≥ 3, are symmetric.