Related papers: Abelian Closures of Infinite Binary Words
We introduce a class of fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words. We conjecture that this class contains all fixed points of primitive morphisms among aperiodic binary generalized…
A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal…
We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for…
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,…
An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of a same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words are known to be rigid. In the present paper,…
We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive…
If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at…
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under…
Let u be a cyclic word in a free group F_n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v|=|u| and v= \phi(u) for some \phi \in \text {Aut}F_n}.…
We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…
We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem,…
In their 1938 seminal paper on symbolic dynamics, Morse and Hedlund proved that every aperiodic infinite word $x\in A^N,$ over a non empty finite alphabet $A,$ contains at least $n+1$ distinct factors of each length $n.$ They further showed…
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…
It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian…
The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at…
We study the structure of the language of binary cube-free words. Namely, we are interested in the cube-free words that cannot be infinitely extended preserving cube-freeness. We show the existence of such words with arbitrarily long finite…
We define a word in two positive definite (complex Hermitian) matrices $A$ and $B$ as a finite product of real powers of $A$ and $B$. The question of which words have only positive eigenvalues is addressed. This question was raised some…
Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence.…
The binomial notation (w u) represents the number of occurrences of the word u as a (scattered) subword in w. We first introduce and study possible uses of a geometrical interpretation of (w ab) and (w ba) when a and b are distinct letters.…