Almost everywhere balanced sequences of complexity 2n+12n+1

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\mu$-almost every $\mathcal{C}$-adic sequence is balanced, where $\mu$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $\mu$ is negative.Comment: 42 pages, 9 figures. Extended and augmented version of arXiv:1707.0274

Conferences WORDS, years 1997-2017: Open Problems and Conjectures

International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017

Functional Stepped Surfaces, Flips and Generalized Substitutions

Remplace N° 06014 (2006) 23 [lirmm-00102710 ? version 1]International audienceA substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words is proved to be well-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated with any usual unimodular substitution. The aim of this paper is to extend the domain of definition of such multidimensional substitutions to functional stepped surfaces. One central tool for this extension is the notion of flips acting on tilings by lozenges of the plane

Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

Rich Words and Balanced Words

This thesis is mostly focused on palindromes. Palindromes have been studied extensively, in recent years, in the field of combinatorics on words.Our main focus is on rich words, also known as full words. These are words which have maximum number of distinct palindromes as factors.We shed some more light on these words and investigate certain restricted problems. Finite rich words are known to be extendable to infinite rich words. We study more closely how many different ways, and in which situations, rich words can be extended so that they remain rich.The defect of a ord is defined to be the number of palindromes the word is lacking.We will generalize the definition of defect with respect to extending the word to be infinite.The number of rich words, on an alphabet of size $n$, is given an upper and a lower bound. Hof, Knill and Simon presented (Commun. Math. Phys. 174, 1995) a well-known question whether all palindromic subshifts which are enerated by primitive substitutions arise from substitutions which are in class P. Over the years, this question has transformed a bit and is nowadays called the class P conjecture. The main point of the conjecture is to attempt to explain how an infinite word can contain infinitely many palindromes.We will prove a partial result of the conjecture. Rich square-free words are known to be finite (Pelantov\'a and Sarosta, Discrete Math. 313, 2013). We will give another proof for that result. Since they are finite, there exists a longest such word on an $n$-ary alphabet.We give an upper and a lower bound for the length of that word. We study also balanced words. Oliver Jenkinson proved (Discrete Math., Alg. and Appl. 1(4), 2009) that if we take the partial sum of the lexicographically ordered orbit of a binary word, then the balanced word gives the least partial sum. The balanced word also gives the largest product. We will show that, at the other extreme, there are the words of the form $0^{q-p}1^p$ ($p$ and $q$ are integers with $1\leq p<q$), which we call the most unbalanced words. They give the greatest partial sum and the smallest product.Tässä väitöskirjassa käsitellään pääasiassa palindromeja. Palindromeja on tutkittu viime vuosina runsaasti sanojen kombinatoriikassa.Suurin kiinnostuksen kohde tässä tutkielmassa on rikkaissa sanoissa. Nämä ovat sanoja joissa on maksimaalinen määrä erilaisia palindromeja tekijöinä.Näitä sanoja tutkitaan monesta eri näkökulmasta. Äärellisiä rikkaita sanoja voidaan tunnetusti jatkaa äärettömiksi rikkaiksi sanoiksi.Työssä tutkitaan tarkemmin sitä, miten monella tavalla ja missä eri tilanteissa rikkaita sanoja voidaan jatkaa siten, että ne pysyvät rikkaina.Sanan vajauksella tarkoitetaan puuttuvien palindromien lukumäärää.Vajauksen käsite yleistetään tapaukseen, jossa sanaa on jatkettava äärettömäksi sanaksi.Rikkaiden sanojen lukumäärälle annetaan myös ylä- ja alaraja. Hof, Knill ja Simon esittivät kysymyksen (Commun. Math. Phys. 174, 1995), saadaanko kaikki äärettömät sanat joissa on ääretön määrä palindromeja tekijöinä ja jotka ovat primitiivisen morfismin generoimia, morfismeista jotka kuuluvat luokkaan P. Nykyään tätä ongelmaa kutsutaan luokan P konjektuuriksi ja sen tarkoitus on saada selitys sille,millä tavalla äärettömässä sanassa voi olla tekijöinä äärettömän monta palindromia. Osittainen tulos tästä konjektuurista todistetaan. Rikkaiden neliövapaiden sanojen tiedetään olevan äärellisiä (Pelantov\'a ja Starosta, Discrete Math. 313, 2013). Tälle tulokselle annetaan uudenlainen todistus.Koska kyseiset sanat ovat äärellisiä, voidaan selvittää mikä niistä on pisin.Ylä- ja alaraja annetaan tällaisen pisimmän sanan pituudelle. Työssä tutkitaan myös tasapainotettuja sanoja.Tasapainotetut sanat antavat pienimmän osittaissumman binäärisille sanoille (Jenkinson, Discrete Math., Alg. and Appl. 1(4), 2009).Lisäksi ne antavat suurimman tulon.Muotoa $0^{q-p}1^p$ ($p$ ja $q$ ovat kokonaislukuja joille $1\leq p<q$) olevien sanojen todistetaan vastaavasti antavan suurimman osittaissumman ja pienimmän tulon.Ne muodostavat täten toisen ääripään tasapainotetuille sanoille, ja asettavat kaikki muut sanat näiden väliin.Siirretty Doriast

From combinatorial games to shape-symmetric morphisms

The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words. In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff's game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review some of their characterizations (in terms of morphisms, finiteness of their k-kernel,...). These sequences take values in a finite set but the Sprague-Grundy function of a game, such as Nim of Wythoff, is usually unbounded. This provides a motivation to introduce k-regular sequences (in the sense of Allouche and Shallit) whose k-kernel is not finite, but finitely generated. In the second lecture, games played on several piles of token naturally give rise to a multidimensional setting. Thus, we reconsider k-automatic and k-regular sequences in this extended framework. In particular, determining the structure of the bidimensional array encoding the (loosing) P-positions of the Wythoff's game is a long-standing and challenging problem in CGT. Wythoff's game is linked to non-standard numeration system: P-positions can be determined by writing those in the Fibonacci system. Next, we present the concept of shape-symmetric morphism: instead of iterating a morphism where images of letters are (hyper-)cubes of a fixed length k, one can generalize the procedure to images of various parallelepipedic shape. The shape-symmetry condition introduced twenty years ago by Maes permits to have well-defined fixed point. In the last lecture, we move to generalized numeration systems: abstract numeration systems (built on a regular language) and link them to morphic (multidimensional) words. In particular, pictures obtained by shape-symmetric morphisms coincide with automatic sequences associated with an abstract numeration system. We conclude these lectures with some work in progress about games with a finite rule-set. This permits us to discuss a bit Presburger definable sets