On the expansions of real numbers in two multiplicative dependent bases

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, and we show that this bound is best possible.Comment: 15pages. arXiv admin note: substantial text overlap with arXiv:1512.0693

On the expansions of real numbers in two integer bases

Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.Comment: 11 pages, to appear at Annales de l'Institut Fourie

Directive words of episturmian words: equivalences and normalization

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.Comment: 15 page

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense" (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.Comment: 18 pages; to appear in the European Journal of Combinatoric

A Coloring Problem for Infinite Words

In this paper we consider the following question in the spirit of Ramsey theory: Given $x\in A^\omega,$ where $A$ is a finite non-empty set, does there exist a finite coloring of the non-empty factors of $x$ with the property that no factorization of $x$ is monochromatic? We prove that this question has a positive answer using two colors for almost all words relative to the standard Bernoulli measure on $A^\omega.$ We also show that it has a positive answer for various classes of uniformly recurrent words, including all aperiodic balanced words, and all words $x\in A^\omega$ satisfying $\lambda_x(n+1)-\lambda_x(n)=1$ for all $n$ sufficiently large, where $\lambda_x(n)$ denotes the number of distinct factors of $x$ of length $n.$Comment: arXiv admin note: incorporates 1301.526

Open and closed complexity of infinite words

In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $w$ of an infinite word $x=x_1x_2x_3\cdots$ with each $x_i$ belonging to a fixed finite set $\mathbb{A},$ we say $w$ is closed if either $w\in \mathbb{A}$ or if $w$ is a complete first return to some factor $v$ of $x.$ Otherwise $w$ is said to be open. We show that for an aperiodic word $x\in \mathbb{A}^\mathbb{N},$ the complexity functions $Cl_x$ (resp. $Op_x)$ that count the number of closed (resp. open) factors of $x$ of each given length are both unbounded. More precisely, we show that if $x$ is aperiodic then $\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty$ and $\limsup_{n\in S} Cl_x(n)=+\infty$ for any syndetic subset $S$ of $\mathbb{N}.$ However, there exist aperiodic infinite words $x$ verifying $\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty.$ Keywords: word complexity, periodicity, return words

Canonical Representatives of Morphic Permutations

An infinite permutation can be defined as a linear ordering of the set of natural numbers. In particular, an infinite permutation can be constructed with an aperiodic infinite word over $\{0,\ldots,q-1\}$ as the lexicographic order of the shifts of the word. In this paper, we discuss the question if an infinite permutation defined this way admits a canonical representative, that is, can be defined by a sequence of numbers from [0, 1], such that the frequency of its elements in any interval is equal to the length of that interval. We show that a canonical representative exists if and only if the word is uniquely ergodic, and that is why we use the term ergodic permutations. We also discuss ways to construct the canonical representative of a permutation defined by a morphic word and generalize the construction of Makarov, 2009, for the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on Words: 10th International Conference. arXiv admin note: text overlap with arXiv:1503.0618