%0 Journal Article
%T Symmetries in Rauzy fractals
%A Victor F. Sirvent
%J Uniform Distribution Theory
%D 2012
%I Mathematical Institute of the Slovak Academy of Sciences
%X 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 unimodular Pisot substitutions so thatthe 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 whenthe abelianization of the set of proper prefixes is not symmetrical.We apply some of these techniques to show that the Rauzy fractal of substitutions of type$$egin{array}{cccccc}1 a underbrace{1cdots 1}_{n}2, &2 a underbrace{1cdots 1}_{n}3, && ldots, & (k-1) a underbrace{1cdots 1}_{n}k, &k a 1;end{array}$$with $ngeq 1$ and $kgeq 3$, are symmetric.
%K Rauzy fractals
%K substitution dynamical systems
%K symmetry groups
%K finite automata
%K Kolakoski substitution.
%U http://www.boku.ac.at/MATH/udt/vol07/no1/08sirvent24-11.pdf