In a 1982 paper Rauzy showed that the subshift $(X,T)$ generated by the morphism $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$ is a natural coding of a rotation on the two-dimensional torus ${𝕋}^2$, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ${ℝ}^2,$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2n+1$ satisfying a combinatorial criterion known as the $☆$ condition of Arnoux and Rauzy codes the orbit of a point...

We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.

Richomme asked the following question: what is the infimum of the real numbers α > 2 such that there exists an infinite word that avoids α-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is α = 7/3.

We consider numeration systems with base β and − β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Zβ and Z− β of numbers with integer expansion in base β, resp. − β. Our main result is the comparison of languages of infinite words uβ and u− β coding the ordering of distances between consecutive β- and (− β)-integers. It turns out that for a class of roots β of x2 − mx − m, the languages coincide, while for other quadratic Pisot numbers the language...

This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by a substitutional system, and dynamical properties are considered...

Nous montrons que le tracé d’un kolam indien classique, que l’on retrouve aussi dans la tradition des dessins sur le sable aux îles Vanuatu, peut être engendré par un morphisme de monoïde. La suite infinie morphique ainsi obtenue est reliée à la célèbre suite de Prouhet-Thue-Morse, mais elle n’est $k$-automatique pour aucun entier $k\ge 1$.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

This paper consists of three parts. In the first part we prove a general theorem on the image of a language $K$ under a substitution, in the second we apply this to the special case when $K$ is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth.

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of sturmian words.

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.

We prove that for each positive integer $n,$ the finite commutative language $E_n=c(a_1{a}_2\cdots a_n)$ possesses a test set of size at most $5n.$ Moreover, it is shown that each test set for $E_n$ has at least $n-1$ elements. The result is then generalized to commutative languages $L$ containing a word $w$ such that (i) $\text{alph}\left(w\right)=\text{alph}\left(L\right);$ and (ii) each symbol $a\in \text{alph}\left(L\right)$ occurs at least twice in $w$ if it occurs at least twice in some word of $L$: each such $L$ possesses a test set of size $11n$, where $n=\text{Card}\left(\text{alph}\left(L\right)\right)$. The considerations rest on the analysis of some basic types of word equations....

We prove that for each positive integer n, the finite commutative language En = c(a1a2...an) possesses a test set of size at most 5n. Moreover, it is shown that each test set for En has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L))....

Motivated by striking properties of the well known Fibonacci word we consider pictures which are defined by this word and its variants via so-called turtle graphics. Such a picture can be bounded or unbounded. We characterize when the picture defined by not only the Fibonacci recurrence, but also by a general recurrence formula, is bounded, the characterization being computable.