On a Family of Morphic Images of Arnoux-Rauzy Words

In this paper we prove the following result. Let s be an infinite word on a finite alphabet, and N ≥ 0 be an integer. Suppose that all left special factors of s longer than N are prefixes of s, and that s has at most one right special factor of each length greater than N. Then s is a morphic image, under an injective morphism, of a suitable standard Arnoux-Rauzy word

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

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

Inverse problems of symbolic dynamics

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 substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can get a morphic word appearing in interval exchange transformation such that frequencies of letters are algebraic numbers of an arbitrary degree. Concerning multydimensional systems, our main result is the following. Let P(n) be a polynomial, having an irrational coefficient of the highest degree. A word $w$ (w=(w_n), n\in \nit) consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote the number of different subwords of $w$ of length $k$ by $T(k)$ . \medskip {\bf Theorem.} {\it There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently great $k$.

Aperiodic pseudorandom number generators based on infinite words

In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the well distributed occurrences (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WELLDOC property if, for each factor w of u, positive integer m, and vector v in (Z_d)^m, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. (The Parikh vector of a finite word v over an alphabet A has its i-th component equal to the number of occurrences of the i-th letter of A in v.) We prove that Sturmian words, and more generally Arnoux–Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 and PractRand statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property

Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials

We consider discrete one-dimensional Schrödinger operators whose potentials belong to minimal subshifts of low combinatorial complexity and prove for a large class of such operators that the spectrum is a Cantor set of zero Lebesgue measure. This is obtained through an analysis of the frequencies of the subwords occurring in the potential. Our results cover most circle map and Arnoux–Rauzy potentials

Combinatorics on Words 10th International Conference

This volume contains the Local Proceedings of the Tenth International Conference on WORDS, that took place at the Kiel University, Germany, from the 14th to the 17th September 2015. WORDS is the main conference series devoted to the mathematical theory of words, and it takes place every two years. The first conference in the series was organised in 1997 in Rouen, France, with the following editions taking place in Rouen, Palermo,Turku, Montreal, Marseille, Salerno, Prague, and Turku. The main object in the scope of the conference, words, are finite or infinite sequences of symbols over a finite alphabet. They appear as natural and basic mathematical model in many areas, theoretical or applicative. Accordingly, the WORDS conference is open to both theoretical contributions related to combinatorial, algebraic, and algorithmic aspects of words, as well as to contributions presenting application of the theory of words, for instance, in other fields of computer science, inguistics, biology and bioinformatics, or physics. For the second time in the history of WORDS, after the 2013 edition, a refereed proceedings volume was published in Springer’s Lecture Notes in Computer Science series. In addition, this local proceedings volume was published in the Kiel Computer Science Series of the Kiel University. Being a conference at the border between theoretical computer science and mathematics, WORDS tries to capture in its two proceedings volumes the characteristics of the conferences from both these worlds. While the Lecture Notes in Computer Science volume was dedicated to formal contributions, this local proceedings volume allows, in the spirit of mathematics conferences, the publication of several contributions informing on current research and work in progress in areas closely connected to the core topics of WORDS. All the papers, the ones published in the Lecture Notes in Computer Science proceedings volume or the ones from this volume, were refereed to high standards by the members of the Program Committee. Following the conference, a special issue of the Theoretical Computer Science journal will be edited, containing extended versions of papers from both proceedings volumes. In total, the conference hosted 18 contributed talks. The papers on which 14 of these talks were based, were published in th LNCS volume; the other 4 are published in this volume. In addition to the contributed talks, the conference program included six invited talks given by leading experts in the areas covered by the WORDS conference: Jörg Endrullis (Amsterdam), Markus Lohrey (Siegen), Jean Néraud (Rouen), Dominique Perrin (Paris), Michaël Rao (Lyon), Thomas Stoll (Nancy). WORDS 2015 was the tenth conference in the series, so we were extremely happy to welcome, as invited speaker at this anniversary edition, Jean Néraud, one of the initiators of the series and the main organiser of the first two editions of this conference. We thank all the invited speakers and all the authors of submitted papers for their contributions to the the success of the conference. We are grateful to the members of the Program Committee for their work that lead to the selection of the contributed talks, and, implicitly, of the papers published in this volume. They were assisted in their task by a series of external referees, gratefully acknowledged below. The submission and reviewing process used the Easychair system; we thank Andrej Voronkov for this system which facilitated the work of the Programme Committee and the editors considerably. We grateful thank Gheorghe Iosif for designing the logo, poster, and banner of WORDS 2015; the logo of the conference can be seen on the front cover of this book. We also thank the editors of the Kiel Computer Science Series, especially Lasse Kliemann, for their support in editing this volume. Finally, we thank the Organising Committee of WORDS 2015 for ensuring the smooth run of the conference