Integers in number systems with positive and negative quadratic Pisot base

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets $\Z_\beta$ and $\Z_{-\beta}$ of numbers with integer expansion in base $\beta$, resp. $-\beta$. Our main result is the comparison of languages of infinite words $u_\beta$ and $u_{-\beta}$ coding the ordering of distances between consecutive $\beta$- and $(-\beta)$-integers. It turns out that for a class of roots $\beta$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_\beta$ can be identified only with the language of a morphic image of $u_{-\beta}$. We also study the group structure of $(-\beta)$-integers.Comment: 19 pages, 5 figure

On the number of return words in infinite words with complexity 2n+1

In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the number of return words over a given word w for these infinite words is exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm

Occurrences of palindromes in characteristic Sturmian words

This paper is concerned with palindromes occurring in characteristic Sturmian words $c_\alpha$ of slope $\alpha$, where $\alpha \in (0,1)$ is an irrational. As $c_\alpha$ is a uniformly recurrent infinite word, any (palindromic) factor of $c_\alpha$ occurs infinitely many times in $c_\alpha$ with bounded gaps. Our aim is to completely describe where palindromes occur in $c_\alpha$. In particular, given any palindromic factor $u$ of $c_\alpha$, we shall establish a decomposition of $c_\alpha$ with respect to the occurrences of $u$. Such a decomposition shows precisely where $u$ occurs in $c_\alpha$, and this is directly related to the continued fraction expansion of $\alpha$.Comment: 17 page

On a generalization of Abelian equivalence and complexity of infinite words

In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ of length less than or equal to $k,$ the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v.$ This defines a family of equivalence relations $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $k=+\infty).$ We show that the number of $k$-Abelian equivalence classes of words of length $n$ grows polynomially, although the degree is exponential in $k.$ Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of $k$-Abelian equivalence classes of factors of $\omega$ of length $n.$ We show that the complexity function $\mathcal {P}^{(k)}$ is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if $\mathcal {P}^{(k)}_{\omega}(n)<q^k(n)$ for some k \in \ints ^+ \cup {+\infty} and $n\geq 0,$ the $\omega$ is ultimately periodic. Moreover if $\omega$ is aperiodic, then $\mathcal {P}^{(k)}_{\omega}(n)=q^k(n)$ if and only if $\omega$ is Sturmian. We also study $k$-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if $\omega$ has bounded $k$-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer $N,$ there exists a $k$-Abelian $N$ power occurring in $\omega$ at some position $j\in D.

Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $\beta$-fractional positions, which may arise as a result of addition of two $\beta$-integers. For the infinite word $u_\beta$ coding distances between consecutive $\beta$-integers, we determine precisely also the balance. The word $u_\beta$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_\beta$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $\beta$-integers are related.Comment: 15 page

Episturmian words: a survey

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more reference

Factor-balanced SS-adic languages

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue-Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two notions essentially coincide when the sequence of substitutions is proper. For the example of Thue-Morse-Sturmian languages, we give a full characterisation of factor-balancedness

Generalizations of Sturmian sequences associated with NN-continued fraction algorithms

Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired by Sturmian sequences, we introduce the $N$-continued fraction sequences $\omega(x,N)$ and $\hat{\omega}(x,N)$, which are related to the $N$-continued fraction expansion of $x$. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are $S$-adic sequences. When $N=1$, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that $\omega(x,N)$ and $\hat{\omega}(x,N)$ are $C$-balanced for some explicit values of $C$ and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for $N$-continued fraction expansions, which provides an additive version of $N$-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.Comment: 23 pages, 2 figure

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced