Arnoux-Rauzy Words Research Articles

On abelian closures of infinite non-binary words

Two finite words u and v are called abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). In this paper, we investigate how this property extends to non-binary words. We consider the abelian closures of most natural generalizations of Sturmian words to non-binary alphabets, such as balanced words and minimal complexity words. We characterize the abelian closures of words in these families and show that in both families, there exist both words which satisfy the property A(x)=Ω(x) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion Ω(x)⊂A(x). We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

Clustering and Arnoux-Rauzy words

We characterize the clustering of a word under the Burrows-Wheeler transform in terms of the resolution of a bounded number of bispecial factors belonging to the language generated by all its powers. We use this criterion to compute, in every given Arnoux-Rauzy language on three letters, an explicit bound K such that each word of length at least K is not clustering; this bound is sharp for a set of Arnoux-Rauzy languages including the Tribonacci one. In the other direction, we characterize all standard Arnoux-Rauzy clustering words, and all perfectly clustering Arnoux-Rauzy words. We extend some results to episturmian languages, characterizing those which produce infinitely many clustering words, and to larger alphabets.

Sequences of symmetry groups of infinite words

In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup Gn of the symmetric group Sn, it acts on the set of finite words of length n by permutation. We associate to an infinite word w a sequence (Gn(w))n≥1 of its symmetry groups: For each n, a symmetry group of w is a subgroup Gn(w) of the symmetric group Sn such that g(v) is a factor of w for each permutation g∈Gn(w) and each factor v of length n of w. We study general properties of the symmetry groups of infinite words and characterize the sequences of symmetry groups of several families of infinite words. We show that for each subgroup G of Sn there exists an infinite word w with Gn(w)=G. On the other hand, the structure of possible sequences (Gn(w))n≥1 is quite restrictive: we show that they cannot contain for each order n certain cycles, transpositions and some other permutations. The sequences of symmetry groups can also characterize a generalized periodicity property. We prove that symmetry groups of Sturmian words and more generally Arnoux-Rauzy words are of order two for large enough n; on the other hand, symmetry groups of certain Toeplitz words have exponential growth.

A Rauzy fractal unbounded in all directions of the plane

We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to $\mathbb{Z}^3$. In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.

On non-repetitive complexity of Arnoux–Rauzy words

The non-repetitive complexity nrCu and the initial non-repetitive complexity inrCu are functions which reflect the structure of the infinite word u with respect to the repetitions of factors of a given length. We determine nrCu for the Arnoux–Rauzy words and inrCu for the standard Arnoux–Rauzy words. Our main tools are the S-adic representation of Arnoux–Rauzy words and a description of return words to their factors. The formulas we obtain are then used to evaluate nrCu and inrCu for the d-bonacci word.

Open and closed factors in Arnoux-Rauzy words

Given a finite non-empty set A, let A+ denote the free semigroup generated by A consisting of all finite words u1u2⋯un with ui∈A. A word u∈A+ is said to be closed if either u∈A or if u is a complete first return to some factor v∈A+, meaning u contains precisely two occurrences of v, one as a prefix and one as a suffix. We study the function fxc:N→N which counts the number of closed factors of each length in an infinite word x. We derive an explicit formula for fxc in case x is an Arnoux-Rauzy word. As a consequence we prove that lim infn→∞fxc(n)=+∞.

Balancedness and coboundaries in symbolic systems

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.

Rigidity and Substitutive Dendric Words

Dendric words are infinite words that are defined in terms of extension graphs. These are bipartite graphs that describe the left and right extensions of factors. Dendric words are such that all their extension graphs are trees. They are also called tree words. This class of words includes classical families of words such as Sturmian words, codings of interval exchanges, or else, Arnoux–Rauzy words. We investigate here the properties of substitutive dendric words and prove some rigidity properties, that is, algebraic properties on the set of substitutions that fix a dendric word. We also prove that aperiodic minimal dendric subshifts (generated by dendric words) cannot have rational topological eigenvalues, and thus, cannot be generated by constant length substitutions.

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∈Zmd, 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 [11] and PractRand [5] 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.

BALANCE PROPERTIES OF ARNOUX–RAUZY WORDS

This paper deals with balances and imbalances in Arnoux–Rauzy words. We provide sufficient conditions for C-balancedness, but our results indicate that even a characterization of 2-balanced Arnoux–Rauzy words on a 3-letter alphabet is not immediate.

ALMOST RICH WORDS AS MORPHIC IMAGES OF RICH WORDS

We focus on Θ-rich and almost Θ-rich words over a finite alphabet [Formula: see text], where Θ is an involutive antimorphism over [Formula: see text]. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is also an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one special case of almost Θ-rich words: we show that every Θ-standard word with seed is an image of an Arnoux-Rauzy word.

A generalized palindromization map in free monoids

The palindromization map ψ in a free monoid A∗ was introduced in 1997 by the first author in the case of a binary alphabet A, and later extended by other authors to arbitrary alphabets. Acting on infinite words, ψ generates the class of standard episturmian words, including standard Arnoux–Rauzy words. In this paper, we generalize the palindromization map, starting with a given code X over A. The new map ψX maps X∗ to the set PAL of palindromes of A∗. In this way, some properties of ψ are lost and some are saved in a weak form. When X has a finite deciphering delay, one can extend ψX to Xω, generating a class of infinite words much wider than standard episturmian words. For a finite and maximal code X over A, we give a suitable generalization of standard Arnoux–Rauzy words, called X-AR words. We prove that any X-AR word is a morphic image of a standard Arnoux–Rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity.For any code X, we say that ψX is conservative when ψX(X∗)⊆X∗. We study conservative maps ψX and conditions on X assuring that ψX is conservative. We also investigate the special case of morphic-conservative maps ψX, i.e., maps such that φ∘ψ=ψX∘φ for an injective morphism φ. Finally, we generalize ψX by replacing palindromic closure with ϑ-palindromic closure, where ϑ is any involutory antimorphism of A∗. This yields an extension of the class of ϑ-standard words introduced by the authors in 2006.

A palindromization map on free monoids

This paper is a survey of several results of combinatorial nature which have been obtained starting from a palindromization map on a free monoid A* introduced by the author in 1997 in the case of a binary alphabet and, successively, generalized by other authors for arbitrary finite alphabets. If one extends the action of the palindromization map to infinite words, one can generate the class of all standard episturmian words, which includes standard Sturmian words and Arnoux-Rauzy words. In this framework, an essential role is played by the class of palindromic prefixes of all standard episturmian words called epicentral words. These words are precisely the images of A* under the palindromization map. Epicentral words have several different representations and satisfy interesting combinatorial properties. A further extension of the palindromization map to a t9-palindromization map, where t9 is an arbitrary involutory antimorphism of A*, is also briefly discussed.

Transcendence of numbers with an expansion in a subclass of complexity 2n + 1

We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k > 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.

RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS

Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.