Thue-Morse Word Research Articles

Initial non-repetitive complexity of infinite words

The initial non-repetitive complexity function of an infinite word x (first introduced by Moothathu) is the function of n that counts the number of distinct factors of length n that appear at the beginning of x prior to the first repetition of a length-n factor. We examine general properties of the initial non-repetitive complexity function, as well as obtain formulas for the initial non-repetitive complexity of the Thue–Morse word, the Fibonacci word and the Tribonacci word.

Optimal Bounds for the Similarity Density of the Thue-Morse Word with Overlap-Free and 73-Power-Free Infinite Binary Words

We consider a measure of similarity for infinite words that generalizes the usual number-theoretic notion of asymptotic or natural density for subsets of natural numbers. We show that every [Formula: see text]-power-free infinite binary word, other than the Thue-Morse word t and its complement [Formula: see text], has this measure of similarity with t between [Formula: see text] and [Formula: see text], and that this bound is optimal in a strong sense just for the class of overlap-free words. This is a generalization of a classical 1927 result of Kurt Mahler.

Words with many Palindrome Pair Factors

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

Another generalization of abelian equivalence: Binomial complexity of infinite words

The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u. Based on this classical notion, we introduce the m-binomial equivalence of two words refining the abelian equivalence. Two words x and y are m-binomially equivalent, if, for all words v of length at most m, the binomial coefficients of x and v and respectively, y and v are equal. The m-binomial complexity of an infinite word x maps an integer n to the number of m-binomial equivalence classes of factors of length n occurring in x. We study the first properties of m-binomial equivalence. We compute the m-binomial complexity of two classes of words: Sturmian words and (pure) morphic words that are fixed points of Parikh-constant morphisms like the Thue–Morse word, i.e., images by the morphism of all the letters have the same Parikh vector. We prove that the frequency of each symbol of an infinite recurrent word with bounded 2-binomial complexity is rational.

On the 2-abelian complexity of the Thue–Morse word

We show that the 2-abelian complexity of the infinite Thue–Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue–Morse word.

Privileged factors in the Thue–Morse word—A comparison of privileged words and palindromes

In this paper we study the privileged complexity function of the Thue–Morse word. We prove a recursive formula describing this function, and using the formula we show that the function is unbounded and that the values of the function have arbitrarily large gaps of zeros. This demonstrates that the privileged complexity function of an infinite word can drastically differ from its palindromic complexity function, even though there are relations between these functions. Further we study the behavior of palindromes and privileged words in infinite words and the relation between rich words and privileged words.

A New Approach to the 2-Regularity of the $\ell$-Abelian Complexity of 2-Automatic Sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue-Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.

ON THE NUMBER OF ABELIAN BORDERED WORDS (WITH AN EXAMPLE OF AUTOMATIC THEOREM-PROVING)

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

On generating binary words palindromically

We regard a finite word u=u1u2⋯un up to word isomorphism as an equivalence relation on {1,2,…,n} where i is equivalent to j if and only if ui=uj. Some finite words (in particular all binary words) are generated by palindromic relations of the form k∼j+i−k for some choice of 1≤i≤j≤n and k∈{i,i+1,…,j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function μ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that μ(u)≤2 for each factor u of x, then x is ultimately periodic. On the other hand, we establish the existence of non-ultimately periodic words for which μ(u)≤3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue–Morse word, we show that the function μ is unbounded.

Infinite self-shuffling words

In this paper we introduce and study a new property of infinite words: An infinite word x∈AN, with values in a finite set A, is said to be k-self-shuffling(k≥2) if x admits factorizations: x=∏i=0∞Ui(1)⋯Ui(k)=∏i=0∞Ui(1)=⋯=∏i=0∞Ui(k). In other words, there exists a shuffle of k-copies of x which produces x. We are particularly interested in the case k=2, in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non-self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue–Morse word fixed by the morphism 0↦01, 1↦10. As another example we show that all Sturmian words of slope α∈R∖Q and intercept 0<ρ<1 are self-shuffling (while those of intercept ρ=0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model. One important feature of self-shuffling words stems from their morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.

Similarity density of the Thue-Morse word with overlap-free infinite binary words

We consider a measure of similarity for infinite words that generalizes the notion of asymptotic or natural density of subsets of natural numbers from number theory. We show that every overlap-free infinite binary word, other than the Thue-Morse word t and its complement t bar, has this measure of similarity with t between 1/4 and 3/4. This is a partial generalization of a classical 1927 result of Mahler.

Palindromic closures using multiple antimorphisms

A generalized pseudostandard word u, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet A and by a directive sequence of involutory antimorphisms acting on A⁎. Prefixes of u with increasing length are constructed using a pseudopalindromic closure operator.We show that generalized Thue–Morse words tb,m, with b,m∈N and b, m⩾2, are generalized pseudostandard words if and only if tb,m is a periodic word or b⩽m. This extends the result of de Luca and De Luca obtained for the classical Thue–Morse words.

Reconstructing Words from a σ-palindromic Language

We consider words on a finite alphabet Σ and study the structure of its σ-palindromes, i.e. words w satisfying w = σ($\tilde{w}$) for some involution σ on the alphabet. We provide algorithms for the computation of σ-lacunas in w, that is the positions where the longest σ-palindromic suffix is not uni-occurrent. The σ-palindromic defect is explicitly computed for Sturmian words and the Thue-Morse word. Finally, the problem of reconstructing words from a given fixed set of σ-palindromes is decidable.

FACTOR AND PALINDROMIC COMPLEXITY OF THUE-MORSE’S AVATARS

Two infinite words that are connected with some significant univoque numbers are studied. It is shown that their factor and palindromic complexities almost coincide with the factor and palindromic complexities of the famous Thue-Morse word.

Central sets generated by uniformly recurrent words

AbstractA subset $A$ of $ \mathbb{N} $ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $\mathop{({x}_{n} )}\nolimits_{n\in \mathbb{N} } $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly recurrent words, including Sturmian words, the Thue–Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. The results in this paper rely on interactions between different areas of mathematics, some of which have not previously been directly linked. They include the general theory of combinatorics on words, abstract numeration systems, and the beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to the algebraic/topological properties of the Stone-Čech compactification of $ \mathbb{N} $.

Languages invariant under more symmetries: Overlapping factors versus palindromic richness

Factor complexity C and palindromic complexity P of infinite words with language closed under reversal are known to be related by the inequality P(n)+P(n+1)≤2+C(n+1)−C(n) for every n∈N. Words for which the equality is attained for every n are usually called rich in palindromes. We show that rich words contain infinitely many overlapping factors. We study words whose languages are invariant under a finite group G of symmetries. For such words we prove a stronger version of the above inequality. We introduce the notion of G-palindromic richness and give several examples of G-rich words, including the Thue–Morse word as well.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

Introducing privileged words: Privileged complexity of Sturmian words

In this paper we introduce a new class of so-called privileged words which have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian words is given in terms of privileged complexity. The privileged complexity of the Thue–Morse word is also briefly studied.

On additive properties of sets defined by the Thue–Morse word

In this paper we study some additive properties of subsets of the set N of positive integers: A subset A of N is called k-summable (where k∈N) if A contains {∑n∈Fxn|∅≠F⊆{1,2,…,k}} for some k-term sequence of natural numbers 〈xt〉t=1k satisfying uniqueness of finite sums. We say A⊆N is finite FS-big if A is k-summable for each positive integer k. We say A⊆N is infinite FS-big if for each positive integer k, A contains {∑n∈Fxn|∅≠F⊆N and #F⩽k} for some infinite sequence of natural numbers 〈xt〉t=1∞ satisfying uniqueness of finite sums. We say A⊆N is an IP-set if A contains {∑n∈Fxn|∅≠F⊆N and #F<∞} for some infinite sequence of natural numbers 〈xt〉t=1∞. By the Finite Sums Theorem (Hindman, 1974) [5], the collection of all IP-sets is partition regular, i.e., if A is an IP-set then for any finite partition of A, one cell of the partition is an IP-set. Here we prove that the collection of all finite FS-big sets is also partition regular. Let T=011010011001011010010110011010… denote the Thue–Morse word fixed by the morphism 0↦01 and 1↦10. For each factor u of T we consider the set T|u⊆N of all occurrences of u in T. In this note we characterize the sets T|u in terms of the additive properties defined above. Using the Thue–Morse word we show that the collection of all infinite FS-big sets is not partition regular.

PERMUTATION COMPLEXITY AND THE LETTER DOUBLING MAP

Given a countable set X (usually taken to be ℕ or ℤ), an infinite permutation π of X is a linear ordering ≺π of X, introduced in [6]. This paper investigates the combinatorial complexity of infinite permutations on ℕ associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.