Decimations and sturmian words

Quasiperiods of biinfinite words

We study infinite words u over an alphabet $\mathcal{A}$ satisfying the property $\mathcal{P} :~\mathcal{P}(n)+ \mathcal{P}(n+1) = 1+ \#\mathcal{A} {\rm any} n \in \mathbb{N}$ , where $\mathcal{P}(n)$ denotes the number of palindromic factors of length n occurring in the language of u . We study also infinite words satisfying a stronger property $\mathcal{PE}$ : every palindrome of u has exactly one palindromic extension in u . For binary words, the properties $\mathcal{P}$ and $\mathcal{PE}$ coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n) = n + 1 for any $n \in \mathbb{N}$ . In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u , we prove that if C(n) = 2n + 1 for any $n \in \mathbb{N}$ , then u satisfies the property $\mathcal{P}$ and moreover u is rich in palindromes. Also a sufficient condition for the property $\mathcal{PE}$ is given. We construct a word demonstrating that $\mathcal{P}$ on a ternary alphabet does not imply $\mathcal{PE}$ .

Prefix normal words are binary words that have no factor with more 1s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Liptak, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.

Given an infinite word with values in a finite alphabet that admit frequencies for all its factors, the smallest frequency function associates with a given positive integer the smallest frequency for factors of this length, and the recurrence function measures the gaps between successive occurrences of factors. We develop in this paper the relations between the growth orders of the smallest frequency and of the recurrence functions. As proved by M. Boshernitzan, linearly recurrent words are characterized in terms of the smallest frequency function. In this paper, we see how to extend this result beyond the case of linearly recurrent words. We first establish a general relation between the smallest frequency and the recurrence functions. Given a lower bound for the smallest frequency function, this relation provides an upper bound for the recurrence function which involves the primitive of this lower bound. We then see how to improve this relation in the case of Sturmian words where the product of the smallest frequency and the recurrence function is bounded for most of the lengths of factors. We also indicate how to construct Sturmian words having a prescribed behaviour for the smallest frequency function, and for the product of the smallest frequency and the recurrence function.

In this paper we give a broad unified framework via group actions for constructing complexity functions of infinite words x = x 0 x 1 x 2 ⋯ ∈ A N x=x_0x_1x_2\cdots \in \mathbb {A}^{\mathbb {N}} with values in a finite set A . \mathbb {A}. Factor complexity, Abelian complexity and cyclic complexity are all particular cases of this general construction. We consider infinite sequences of permutation groups ω = ( G n ) n ≥ 1 \omega = (G_n)_{n\geq 1} with each G n ⊆ S n . G_n \subseteq S_n. Associated with every such sequence is a complexity function p ω , x : N → N p_{\omega ,x}:\mathbb {N} \rightarrow \mathbb {N} which counts, for each length n , n, the number of equivalence classes of factors of x x of length n n under the action of G n G_n on A n \mathbb {A}^n given by g ∗ ( u 1 u 2 ⋯ u n ) = u g − 1 ( 1 ) u g − 1 ( 2 ) ⋯ u g − 1 ( n ) . g* (u_1u_2\cdots u_n)=u_{g^{-1}(1)}u_{g^{-1}(2)}\cdots u_{g^{-1}(n)}. Each choice of ω = ( G n ) n ≥ 1 \omega =(G_n)_{n\geq 1} defines a unique complexity function which reflects a different combinatorial property of a given infinite word. For instance, an infinite word x x has bounded Abelian complexity if and only if x x is k k -balanced for some positive integer k , k, while bounded cyclic complexity is equivalent to x x being ultimately periodic. A celebrated result of G.A. Hedlund and M. Morse states that every aperiodic infinite word x ∈ A N x\in \mathbb {A}^{\mathbb {N}} contains at least n + 1 n+1 distinct factors of each length n . n. Moreover x ∈ A N x\in \mathbb {A}^{\mathbb {N}} has exactly n + 1 n+1 distinct factors of each length n n if and only if x x is a Sturmian word, i.e., binary, aperiodic and balanced. We prove that this characterisation of aperiodicity and Sturmian words extends to this general framework.

In our paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: \(s = X_1^2 X_2^2 \cdots \). The square root \(\sqrt{s}\) of s is the infinite word \(X_1 X_2 \cdots \) obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

On a combinatorial property of Sturmian words

Standard sturmian morphisms

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let b ≥ 2 b \ge 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b b -ary digits is a Sturmian sequence over { 0 , 1 , … , b − 1 } \{0, 1, \ldots , b-1\} and we prove that this lower bound is best possible. As an application, we derive some information on the b b -ary expansion of log ⁡ ( 1 + 1 a ) \log (1 + \frac {1}{a}) for any integer a ≥ 34 a \ge 34 .

Standard Sturmian words and automata minimization algorithms

The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starting point for the development of an infinite class of right infinite Lambda words, Λθ. The word is formed by encoding differences in the sequence {Mθ}i = {a + bθ}i, a, b ∈ N. Although the word is on an infinite alphabet, it is traversable via environments containing no more than three letters. When θ = ∂ = log2 3, the word encodes all well-formed scales and regions generated by the intervals octave and perfect twelfth. The study sheds additional light on the role of palindromes in musical tone structures. Regions are palindromes on two letters, and form the largest palindromes in the Lambda word, as it develops. The regions have a significant dual representation, connecting them to the palindromic prefixes of a characteristic Sturmian word. The Lambda word is rich in palindromes beyond regions. In particular, a palindrome is formed between any two successive appearances of the same letter. Although Λ∂ is of particular importance musically, Lambda words are interesting in their own right as word theoretic objects. The paper ends with a brief look at the Fibonacci Lambda word, Λφ.

Words of slow growth, which are a natural generalization of Sturmian words, are studied in the paper. An infinite word W = (w n ) n∈ℤ is called a word of slow growth if its complexity function satisfies the relation T W (n + 1) − T W (n) = 1 for all sufficiently large n. The aim of this paper is to describe all words of slow growth in terms of rotations of a circle.

On generating binary words palindromically

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The Abelian subshift \(\mathcal A_{\varvec{x}}\) of an infinite word \(\varvec{x}\) is the set of infinite words \(\varvec{y}\) such that, for each factor u of \(\varvec{y}\), there exists a factor v of \(\varvec{x}\) which is Abelian equivalent to u. The notion of Abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which \(\mathcal A_{\varvec{x}}\) equals the shift orbit closure \(\varOmega _{\varvec{x}}\). On the other hand, the Abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this paper we undertake a general study of Abelian subshifts. In particular, we characterize the Abelian subshifts of recurrent aperiodic balanced words and the Abelian subshifts of ternary words having factor complexity \(n+2\) for all \(n\ge 1\).