Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

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