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

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.

Arnoux-Rauzy 词中的均匀因子平衡性

Uniform Factor-Balancedness in Arnoux-Rauzy Words

本文刻画了 Arnoux-Rauzy 词的均匀因子平衡性，将其从三元字母表推广到任意有限字母表，并证明该性质等价于有界弱部分商。

On non-repetitive complexity of Arnoux-Rauzy words

The non-repetitive complexity $nr\mathcal{C}_{\bf u}$ and the initial non-repetitive complexity $inr\mathcal{C}_{\bf u}$ are functions which reflect the structure of the infinite word ${\bf u}$ with respect to the repetitions of factors of a given length. We determine $nr\mathcal{C}_{\bf u}$ for the Arnoux-Rauzy words and $inr\mathcal{C}_{\bf u}$ for the standard Arnoux-Rauzy words. Our main tools are $S$-adic representation of Arnoux-Rauzy words and description of return words to their factors. The formulas we obtain are then used to evaluate $nr\mathcal{C}_{\bf u}$ and $inr\mathcal{C}_{\bf u…

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 sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux-Rauzy substitutions.

Open and closed factors of Arnoux-Rauzy words

A finite word $u$ is called closed if its longest repeated prefix has exactly two occurrences in $u,$ once as a prefix and once as a suffix. We study the function $f_x^c:\mathbb N \rightarrow \mathbb N$ which counts the number of closed factors of each length in an infinite word $x.$ We derive an explicit formula for $f_x^c$ in case $x$ is an Arnoux-Rauzy word. As a consequence we prove that $\liminf _{n\rightarrow \infty}f_x^c(n)=+\infty.$

Substitutive Arnoux-Rauzy sequences have pure discrete spectrum

We prove that the symbolic dynamical system generated by a purely substitutive Arnoux-Rauzy sequence is measurably conjugate to a toral translation. The proof is based on an explicit construction of a fundamental domain with fractal boundary (a Rauzy fractal) for this toral translation.