Words derivated from Sturmian words

AbstractA return word of a factor of a Sturmian word starts at an occurrence of that factor and ends exactly before its next occurrence. Derivated words encode the unique decomposition of a word in terms of return words. Vuillon has proved that each factor of a Sturmian word has exactly two return words. We determine these two return words, as well as their first occurrence, for the prefixes of characteristic Sturmian words. We then characterize words derivated from a characteristic Sturmian word and give their precise form. Finally, we apply our results to obtain a new proof of the characterization of characteristic Sturmian words which are fixed points of morphisms

Derivated sequences of complementary symmetric Rote sequences

Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence $\mathbf{v}$ which is associated with a standard Sturmian sequence $\mathbf{u}$. We show that any non-empty prefix of $\mathbf{v}$ has three return words. We prove that any derivated sequence of $\mathbf{v}$ is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that $\mathbf{v}$ is primitive substitutive if and only if $\mathbf{u}$ is primitive substitutive. Moreover, if the sequence $\mathbf{u}$ is a fixed point of a primitive morphism, then all derivated sequences of $\mathbf{v}$ are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms

On substitutions closed under derivation: examples

We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.Comment: 10 pages, 1 figures, submitted to Words 201

Asymptotic Abelian Complexities of Certain Morphic Binary Words

We study asymptotic Abelian complexities of morphic binary words. We completethe classification of upper Abelian complexities of pure morphic binary words initiatedrecently by F. Blanchet-Sadri, N. Rampersad, and N. Fox. We also study a class ofmorphic binary words having different asymptotic factor complexities despite havingthe same asymptotic Abelian complexity.</p

The complexity of tangent words

In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341

On the k-Abelian Equivalence Relation of Finite Words

This thesis is devoted to the so-called k-abelian equivalence relation of sequences of symbols, that is, words. This equivalence relation is a generalization of the abelian equivalence of words. Two words are abelian equivalent if one is a permutation of the other. For any positive integer k, two words are called k-abelian equivalent if each word of length at most k occurs equally many times as a factor in the two words. The k-abelian equivalence defines an equivalence relation, even a congruence, of finite words. A hierarchy of equivalence classes in between the equality relation and the abelian equivalence of words is thus obtained. Most of the literature on the k-abelian equivalence deals with infinite words. In this thesis we consider several aspects of the equivalence relations, the main objective being to build a fairly comprehensive picture on the structure of the k-abelian equivalence classes themselves. The main part of the thesis deals with the structural aspects of k-abelian equivalence classes. We also consider aspects of k-abelian equivalence in infinite words. We survey known characterizations of the k-abelian equivalence of finite words from the literature and also introduce novel characterizations. For the analysis of structural properties of the equivalence relation, the main tool is the characterization by the rewriting rule called the k-switching. Using this rule it is straightforward to show that the language comprised of the lexicographically least elements of the k-abelian equivalence classes is regular. Further word-combinatorial analysis of the lexicographically least elements leads us to describe the deterministic finite automata recognizing this language. Using tools from formal language theory combined with our analysis, we give an optimal expression for the asymptotic growth rate of the number of k-abelian equivalence classes of length n over an m-letter alphabet. Explicit formulae are computed for small values of k and m, and these sequences appear in Sloane’s Online Encyclopedia of Integer Sequences. Due to the fact that the k-abelian equivalence relation is a congruence of the free monoid, we study equations over the k-abelian equivalence classes. The main result in this setting is that any system of equations of k-abelian equivalence classes is equivalent to one of its finite subsystems, i.e., the monoid defined by the k-abelian equivalence relation possesses the compactness property. Concerning infinite words, we mainly consider the (k-)abelian complexity function. We complete a classification of the asymptotic abelian complexities of pure morphic binary words. In other words, given a morphism which has an infinite binary fixed point, the limit superior asymptotic abelian complexity of the fixed point can be computed (in principle). We also give a new proof of the fact that the k-abelian complexity of a Sturmian word is n + 1 for length n 2k. In fact, we consider several aspects of the k-abelian equivalence relation in Sturmian words using a dynamical interpretation of these words. We reprove the fact that any Sturmian word contains arbitrarily large k-abelian repetitions. The methods used allow to analyze the situation in more detail, and this leads us to define the so-called k-abelian critical exponent which measures the ratio of the exponent and the length of the root of a k-abelian repetition. This notion is connected to a deep number theoretic object called the Lagrange spectrum