Arnoux-Rauzy interval exchange transformations

Frequencies of Factors in Arnoux-rauzy Sequences

V. Berthé showed that the frequencies of factors in a Sturmian word of slope α, as well as the number of factors with a given frequency, can be expressed in terms of the continued fraction expansion of α. In this paper we describe a multi-dimensional continued fraction process associated with a class of sequences of (block) complexity kn+1 originally introduced by P. Arnoux and G. Rauzy. This v...

Balance properties of Arnoux-Rauzy words

The paper deals with balances and imbalances in Arnoux-Rauzy words. We provide sufficient conditions for C-balancedness, but our results indicate that even a characterization of 2-balanced Arnoux-Rauzy words on a 3-letter alphabet is not immediate.

Balancedness of Arnoux-Rauzy and Brun Words

We study balancedness properties of words given by the Arnoux-Rauzy and Brun multi-dimensional continued fraction algorithms. We show that almost all Brun words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely balanced; in particular, boundedness of the strong partial quotients implies balancedness. On the other hand, we provide examples of unbalanced Brun words on 3 le...

Connectedness of fractals associated with Arnoux-Rauzy substitutions

Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact planar sets, where each set is the projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the fractal associated with any...

Arnoux-rauzy Subshifts: Linear Recurrence, Powers, and Palindromes