Lyndon factorization of the Thue-Morse word and its relatives

We compute the Lyndon factorization of the Thue-Morse word. We also compute the Lyndon factorization of two related sequences involving morphisms that give rise to new presentations of these sequences

Automatic Sequences and Decidable Properties: Implementation and Applications

In 1912 Axel Thue sparked the study of combinatorics on words when he showed that the Thue-Morse sequence contains no overlaps, that is, factors of the form ayaya. Since then many interesting properties of sequences began to be discovered and studied. In this thesis, we consider a class of infinite sequences generated by automata, called the k-automatic sequences. In particular, we present a logical theory in which many properties of k-automatic sequences can be expressed as predicates and we show that such predicates are decidable. Our main contribution is the implementation of a theorem prover capable of practically characterizing many commonly sought-after properties of k-automatic sequences. We showcase a panoply of results achieved using our method. We give new explicit descriptions of the recurrence and appearance functions of a list of well-known k-automatic sequences. We define a related function, called the condensation function, and give explicit descriptions for it as well. We re-affirm known results on the critical exponent of some sequences and determine it for others where it was previously unknown. On the more theoretical side, we show that the subword complexity p(n) of k-automatic sequences is k-synchronized, i.e., the language of pairs (n, p(n)) (expressed in base k) is accepted by an automaton. Furthermore, we prove that the Lyndon factorization of k-automatic sequences is also k-automatic and explicitly compute the factorization for several sequences. Finally, we show that while the number of unbordered factors of length n is not k-synchronized, it is k-regular

Lyndon factorization of generalized words of Thue

The i-th symbol of the well-known infinite word of Thue on the alphabet \ 0,1\ can be characterized as the parity of the number of occurrences of the digit 1 in the binary notation of i. Generalized words of Thue are based on counting the parity of occurrences of an arbitrary word w∈\ 0,1\^+-0^* in the binary notation of i. We provide here the standard Lyndon factorization of some subclasses of this class of infinite words

Some properties of the Tribonacci sequence

AbstractIn this paper, we consider the factor properties of the Tribonacci sequence. We define the singular words, and then give the singular factorization and the Lyndon factorization. As applications, we study the powers of the factors and the overlap of the factors. We also calculate the free index of the sequence

Systems of iterative functional equations : theory and applications

Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015We formulate a general theoretical framework for systems of iterative functional equations between general spaces. We find general necessary conditions for the existence of solutions such as compatibility conditions (essential hypotheses to ensure problems are well-defined). For topological spaces we characterize continuity of solutions; for metric spaces we find sufficient conditions for existence and uniqueness. For a number of systems we construct explicit formulae for the solution, including affine and other general non-linear cases. We provide an extended list of examples. We construct, as a particular case, an explicit formula for the fractal interpolation functions with variable parameters. Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. When conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties. We provide a general construction for finding solutions, including an explicit formula showing how, in certain cases, a solution can be constructively determined. We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description/enumeration of periodic and pre-periodic points and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects such as necklaces and words. We provide some examples. We show the relevance of this for the representation of rational numbers. There are many possible proofs of Fermat's little theorem. We exemplify those using necklaces and dynamical systems. Both methods lead to generalizations. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps. The representation of numbers plays an important role in much of this work. Starting from the classical base p representation we present other type of representation of numbers: signed base p representation, Q-representation and finite base p representation of rationals. There is an extended p representation that generalizes some of the listed representations. We consider the concept of bold play in gambling, where the game has a unique win pay-off. The probability that a gambler reaches his goal using the bold play strategy is the solution of a functional equation. We compare with the timid play strategy and extend to the game with multiple pay-offs.Centro de Matemática e Aplicações Fundamentais da ULisboa; Fundação da Faculdade de Ciências da ULisbo

Lyndon factorization of the Thue-Morse word and its relatives

this paper, we concentrate on the Thue--Morse word and give the computation of its Lyndon factorization (Theorem 3.1) and describe some of its properties (Corollary 3.2, Remark 3.3 and Corollary 3.4). Incidentally, we are able to compute the factorization for the `dual' Thue--Morse word in which appears an infinite Lyndon word (cf Theorem 3.7). We also look at relatives (Equations (4) and (6)) of the Thue--Morse word from the same point of view; these were first studied in [7] and [4], and later in [1]. The factorizations given here for these infinite words (cf Theorems 4.6 and 4.7) use morphisms having special properties with respect to Lyndon words. Moreover, we give identities involving these morphisms for these infinite words. 2 Basic Results and Notation

Lyndon factorization of the Thue-Morse word and its relatives

We compute the Lyndon factorization of the Thue-Morse word. We also compute the Lyndon factorization of two related sequences involving morphisms that give rise to new presentations of these sequences

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum