The Overlap Gap Between Left-Infinite and Right-Infinite Words

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

Anti-power $j$-fixes of the Thue-Morse word

Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as a word of the form $w^{(1)} w^{(2)} \cdots w^{(k)}$, where $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the same length. For an infinite word $w$ and a positive integer $k$, define $AP_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1} w_{j+2} \cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$, $\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k \in \mathbb{Z}^+$ sufficiently large, $\Gamma_j(k) = \sup ((2 \mathbb{Z}^+ -1) \setminus \mathcal{F}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that $\gamma_0(k)$ and $\Gamma_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that $\gamma_j(k)$ and $\Gamma_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $\displaystyle 1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and $\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$. Comment: 21 pages

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

Upper bound for palindromic and factor complexity of rich words

A finite word w of length n contains at most n + 1 distinct palindromic factors. If the bound n + 1 is attained, the word w is called rich. An infinite word w is called rich if every finite factor of w is rich. Let w be a word (finite or infinite) over an alphabet with q > 1 letters, let Facw(n) be the set of factors of length n of the word w, and let Palw(n) ⊆ Facw(n) be the set of palindromic factors of length n of the word w. We present several upper bounds for |Facw(n)| and |Palw(n)|, where w is a rich word. Let δ = [see formula in PDF]. In particular we show that |Facw(n)| ≤ (4q2n)δ ln 2n+2. In 2007, Baláži, Masáková, and Pelantová showed that |Palw(n)|+|Palw(n+1)| ≤ |Facw(n+1)|-|Facw(n)|+2, where w is an infinite word whose set of factors is closed under reversal. We prove this inequality for every finite word v with |v| ≥ n + 1 and v(n + 1) closed under reversal.

Avoiding conjugacy classes on the 5-letter alphabet

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].

Complexity and fractal dimensions for infinite sequences with positive entropy

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].

Characterization of Infinite LSP Words and Endomorphisms Preserving the LSP Property

Answering a question of G. Fici, we give an [Formula: see text]-adic characterization of the family of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes. More precisely we provide a finite set of morphisms [Formula: see text] and an automaton [Formula: see text] such that an infinite word is LSP if and only if it is [Formula: see text]-adic and one of its directive words is recognizable by [Formula: see text]. Then we characterize the endomorphisms that preserve the property of being LSP for infinite words. This allows us to prove that there exists no set [Formula: see text] of endomorphisms for which the set of infinite LSP words corresponds to the set of [Formula: see text]-adic words. This implies that an automaton is required no matter which set of morphisms is used.

The Markoff Property

The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.

Entropy ratio for infinite sequences with positive entropy

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

Greedy Palindromic Lengths

In [A. Frid, S. Puzynina and L. Q. Zamboni, On palindromic factorization of words, Adv. in Appl. Math. 50 (2013) 737–748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately periodic. We introduce variants of this conjecture and prove this conjecture when the bound is 2. Especially we introduce left and right greedy palindromic lengths. These lengths are always greater than or equals to the initial palindromic length. When the greedy left (or right) palindromic lengths of prefixes of a word are bounded then this word is ultimately periodic.