Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

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

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

Distances and automatic sequences in distinguished variants of Hanoi graphs

In this thesis three open problems concerning Hanoi-type graphs are addressed. I prove a theorem to determine all shortest paths between two arbitrary vertices s and t in the General Sierpiński graph S_p^n with base p ≥ 3 and exponent n ≥ 0 and find an algorithm based on this theorem which gives us the index of the potential auxiliary subgraph, the distance between s and t and the best first move(s). Using the isomorphism between S_3^n and the Hanoi graphs H_3^n, this algorithm also determines the shortest paths in H_3^n. The results are also used in order to simplify proofs of already known metric properties of S_p^n. Additionally, I compute the average number of input pairs (s_i, t_i) for i ϵ{1,...,n} to be read by the algorithm. The Theorem and the algorithm for S_p^n are modified for the Sierpiński triangle graphs, which are deeply connected to the well-known Sierpiński triangle and the Sierpiński graphs, with the result that the shortest paths in the Sierpiński triangle graphs can be determined for the first time. The Hanoi graphs H_3^n are then considered as directed graphs by differentiating the directions of the disc moves between the pegs of the corresponding Tower of Hanoi. For the problem to transfer a tower from one peg to another peg there are five different solvable variants. Here, the variants TH(C_3^+) and TH(K_3^-) are discussed concerning the infinite sequences of moves which arise from the solutions as n tends to infinity. The Allouche-Sapir Conjecture says that these sequences are not d-automatic for any d. I prove this for the TH(C_3^+) sequence with the aid of the frequency of a letter and its rationality in automatic sequences. For the TH(K_3^-) sequence I employ Cobham’s Theorem about multiplicative independence, automatic sequences and ultimate periodicity. I show that this sequence is the image, under a 1-uniform morphism, of an iterative fixed point of a primitive prolongable endomorphism. F. Durand’s methodᵃ is then used for the decision about the question whether the sequence is ultimately periodic. The method of I. V. Mitrofanovᵇ, which works with subword schemata,is applied to the problem as well. Using the theory of recognisable sets, a sufficient condition for deciding the question about the automaticity of the TH(K_3^-) sequence is deduced. Finally, a yet not studied distance problem on the so-called Star Tower of Hanoi, which is based on the star graph S t(4), is considered. Assuming that the Frame-Stewart type strategy is optimal, a recurrence for the length of the resulting paths is deduced and solved up to n = 12. ᵃ F. Durand, HD0L ω-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012 ᵇ I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 2015Ich beweise ein Theorem zur Bestimmung aller kürzesten Wege zwischen zwei beliebigen Ecken s und t in den allgemeinen Sierpiński-Graphen S_p^n mit Basis p ≥ 3 und Exponent n ≥ 0 und erstelle auf diesem Theorem beruhend einen Algorithmus, der den Index des allfälligen Hilfsuntergraphen, den Abstand zwischen s und t und einen besten ersten Schritt liefert. Unter Verwendung des Isomorphismus zwischen S_3^n und den Hanoi-Graphen H_3^n bestimmt dieser Algorithmus auch die kürzesten Wege in H_3^n. Die Ergebnisse werden benutzt, um Beweise bereits bekannter metrischer Eigenschaften der S_p^n zu vereinfachen. Zusätzlich berechne ich die durchschnittlich benötigte Anzahl von Eingabepaaren (s_i, t_i) für i ϵ{1,...,n} in den Algorithmus. Das Theorem und der Algorithmus für S_p^n werden für die Klasse der Sierpiński-Dreiecksgraphen, welche in direktem Zusammenhang mit dem berühmten Sierpiński-Dreieck und den Sierpiński-Graphen stehen, modifiziert, sodass erstmals auch die kürzesten Wege in diesen Graphen bestimmt werden können. Die Hanoi-Graphen H_3^n werden dann als gerichtete Graphen betrachtet, indem man die Richtungen der Bewegungen zwischen den Stäben des entsprechenden Turms von Hanoi differenziert. Für das Problem des Versetzens eines Turms von einem Stab auf einen anderen gibt es fünf verschiedene lösbare Varianten. Die Varianten TH(C_3^+) und TH(K_3^-) werden bezüglich der unendlichen Folgen von Bewegungen betrachtet, die sich durch die Lösung für n gegen Unendlich strebend ergeben. Die Allouche-Sapir-Vermutung besagt, dass für kein d diese Folgen d-automatisch erzeugt sind. Ich beweise dies für die TH(C_3^+) Folge mit Hilfe der Theorie über die Häufigkeit eines Buchstabens und deren Rationalität in automatisch erzeugten Folgen. Für die TH(K_3^-) Folge wird Cobhams Theorem über multiplikative Unabhängigkeit, automatisch erzeugte Folgen und ultimative Periodizität verwendet. Ich zeige, dass diese Folge das Bild, unter einem 1-uniformen Morphismus, eines iterativen Fixpunktes eines primitiven verlängerbaren Endomorphismus ist. Die Methode von F. Durandᵃ wird dann für die Entscheidung über die Frage, ob die Folge ultimativ periodisch ist, verwendet. Ebenso wird die Methode von I. V. Mitrofanovᵇ, welche mit Teilwortschemata arbeitet, auf das Problem angewandt. Unter Verwendung der Theorie über erkennbare Mengen wird eine hinreichende Bedingung für die Frage der Automatizität der TH(K_3^-) Folge hergeleitet. Zuletzt wird ein bislang nicht untersuchtes Abstandsproblem im sogenannten Stern-Turm-von- Hanoi betrachtet, welcher auf dem Stern-Graphen St(4) beruht. Unter der Annahme, dass die Frame-Stewart-Strategie optimal sei, wird eine Rekursionsvorschrift für die Länge der so gewonnenen Wege entwickelt und bis n = 12 gelöst. ᵃ F. Durand, HD0L ω-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012 ᵇ I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 201

Conferences WORDS, years 1997-2017: Open Problems and Conjectures

International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017

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

On the k-Abelian Equivalence Relation of Finite Words

This thesis is devoted to the so-called k-abelian equivalence relation of sequences of symbols, that is, words. This equivalence relation is a generalization of the abelian equivalence of words. Two words are abelian equivalent if one is a permutation of the other. For any positive integer k, two words are called k-abelian equivalent if each word of length at most k occurs equally many times as a factor in the two words. The k-abelian equivalence defines an equivalence relation, even a congruence, of finite words. A hierarchy of equivalence classes in between the equality relation and the abelian equivalence of words is thus obtained. Most of the literature on the k-abelian equivalence deals with infinite words. In this thesis we consider several aspects of the equivalence relations, the main objective being to build a fairly comprehensive picture on the structure of the k-abelian equivalence classes themselves. The main part of the thesis deals with the structural aspects of k-abelian equivalence classes. We also consider aspects of k-abelian equivalence in infinite words. We survey known characterizations of the k-abelian equivalence of finite words from the literature and also introduce novel characterizations. For the analysis of structural properties of the equivalence relation, the main tool is the characterization by the rewriting rule called the k-switching. Using this rule it is straightforward to show that the language comprised of the lexicographically least elements of the k-abelian equivalence classes is regular. Further word-combinatorial analysis of the lexicographically least elements leads us to describe the deterministic finite automata recognizing this language. Using tools from formal language theory combined with our analysis, we give an optimal expression for the asymptotic growth rate of the number of k-abelian equivalence classes of length n over an m-letter alphabet. Explicit formulae are computed for small values of k and m, and these sequences appear in Sloane’s Online Encyclopedia of Integer Sequences. Due to the fact that the k-abelian equivalence relation is a congruence of the free monoid, we study equations over the k-abelian equivalence classes. The main result in this setting is that any system of equations of k-abelian equivalence classes is equivalent to one of its finite subsystems, i.e., the monoid defined by the k-abelian equivalence relation possesses the compactness property. Concerning infinite words, we mainly consider the (k-)abelian complexity function. We complete a classification of the asymptotic abelian complexities of pure morphic binary words. In other words, given a morphism which has an infinite binary fixed point, the limit superior asymptotic abelian complexity of the fixed point can be computed (in principle). We also give a new proof of the fact that the k-abelian complexity of a Sturmian word is n + 1 for length n 2k. In fact, we consider several aspects of the k-abelian equivalence relation in Sturmian words using a dynamical interpretation of these words. We reprove the fact that any Sturmian word contains arbitrarily large k-abelian repetitions. The methods used allow to analyze the situation in more detail, and this leads us to define the so-called k-abelian critical exponent which measures the ratio of the exponent and the length of the root of a k-abelian repetition. This notion is connected to a deep number theoretic object called the Lagrange spectrum