On Generating Binary Words Palindromically

We regard a finite word $u=u_1u_2\cdots u_n$ up to word isomorphism as an equivalence relation on $\{1,2,\ldots, n\}$ where $i$ is equivalent to $j$ if and only if $x_i=x_j.$ Some finite words (in particular all binary words) are generated by "{\it palindromic}" relations of the form $k\sim j+i-k$ for some choice of $1\leq i\leq j\leq n$ and $k\in \{i,i+1,\ldots,j\}.$ 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 $\mu(u)$ defined as the least number of palindromic relations required to generate $u.$ We show that every aperiodic infinite word must contain a factor $u$ with $\mu(u)\geq 3,$ and that some infinite words $x$ have the property that $\mu(u)\leq 3$ for each factor $u$ of $x.$ We 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 $\mu$ is unbounded

On prefixal factorizations of words

We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots$ where each $U_i$ is a non-empty prefix of $x.$ With each $x\in {\cal P}_1$ one naturally associates a "derived" infinite word $\delta(x)$ which may or may not admit a prefixal factorization. We are interested in the class ${\cal P}_{\infty}$ of all words $x$ of ${\cal P}_1$ such that $\delta^n(x) \in {\cal P}_1$ for all $n\geq 1$. Our primary motivation for studying the class ${\cal P}_{\infty}$ stems from its connection to a coloring problem on infinite words independently posed by T. Brown in \cite{BTC} and by the second author in \cite{LQZ}. More precisely, let ${\bf P}$ be the class of all words $x\in A^\omega$ such that for every finite coloring $\varphi : A^+ \rightarrow C$ there exist $c\in C$ and a factorization $x= V_0V_1V_2\cdots$ with $\varphi(V_i)=c$ for each $i\geq 0.$ In \cite{DPZ} we conjectured that a word $x\in {\bf P}$ if and only if $x$ is purely periodic. In this paper we show that ${\bf P}\subseteq {\cal P}_{\infty},$ so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of ${\cal P}_{\infty}.$ We establish several results on the class ${\cal P}_{\infty}$. In particular, we show that a Sturmian word $x$ belongs to ${\cal P}_{\infty}$ if and only if $x$ is nonsingular, i.e., no proper suffix of $x$ is a standard Sturmian word

The Ehrenfeucht–Silberger problem

AbstractWe consider repetitions in words and solve a longstanding open problem about the relation between the period of a word and the length of its longest unbordered factor (where factor means uninterrupted subword). A word u is called bordered if there exists a proper prefix that is also a suffix of u, otherwise it is called unbordered. In 1979 Ehrenfeucht and Silberger raised the following problem: What is the maximum length of a word w, w.r.t. the length τ of its longest unbordered factor, such that τ is shorter than the period π of w. We show that, if w is of length 73τ or more, then τ=π which gives the optimal asymptotic bound

Least Periods of Factors of Infinite Words

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a byproduct of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.Work of the first author supported by a Discovery Grant from NSERC. Work of the second author supported by the Finnish Academy under grant 8206039.https://www.rairo-ita.org/articles/ita/abs/2009/01/ita08003/ita08003.htm

On morphisms preserving infinite Lyndon words

In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order