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# Documents  68R15 | enregistrements trouvés : 11

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## Post-edited  Applications of algebra to automatic sequences and pattern avoidance - Lecture 1 Bell, Jason P. (Auteur de la Conférence) | CIRM (Editeur )

We will cover some of the more important results from commutative and noncommutative algebra as far as applications to automatic sequences, pattern avoidance, and related areas. Well give an overview of some applications of these areas to the study of automatic and regular sequences and combinatorics on words.

## Post-edited  Ergodic measures for subshifts with eventually constant growth Fickenscher, Jon (Auteur de la Conférence) | CIRM (Editeur )

We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron. We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss ...

## Multi angle  Palindromes patterns Brlek, Srecko (Auteur de la Conférence) | CIRM (Editeur )

The study of palindromes and their generalizations in a word has gained a lot of interest in the last 20 years, motivated by applications in physics, biology, discrete geometry, to name only a few. Using Sebastien Ferenczi as an example, we illustrate the computation of its palindromic complexity and its relation with the usual factor complexity, via an identity attributed to Brlek and Reutenauer involving also the palindromic defect. Periodic infinite words as well as the family of words with language closed by reversal also satisfy the identity. The identity remains valid when palindromic is replaced by $\sigma$-palindromic, and we also discuss some other patterns. The study of palindromes and their generalizations in a word has gained a lot of interest in the last 20 years, motivated by applications in physics, biology, discrete geometry, to name only a few. Using Sebastien Ferenczi as an example, we illustrate the computation of its palindromic complexity and its relation with the usual factor complexity, via an identity attributed to Brlek and Reutenauer involving also the palindromic defect. Periodic ...

## Multi angle  Logic, decidability and numeration systems - Lecture 1 Charlier, Émilie (Auteur de la Conférence) | CIRM (Editeur )

The theorem of Büchi-Bruyère states that a subset of $N^d$ is $b$-recognizable if and only if it is $b$-definable. As a corollary, the first-order theory of $(N,+,V_b)$ is decidable (where $V_b(n)$ is the largest power of the base $b$ dividing $n$). This classical result is a powerful tool in order to show that many properties of $b$-automatic sequences are decidable. The first part of my lecture will be devoted to presenting this result and its applications to $b$-automatic sequences. Then I will move to $b$-regular sequences, which can be viewed as a generalization of $b$-automatic sequences to integer-valued sequences. I will explain bow first-order logic can be used to show that many enumeration problems of $b$-automatic sequences give rise to corresponding $b$-regular sequences. Finally, I will consider more general frameworks than integer bases and (try to) give a state of the art of the research in this domain. The theorem of Büchi-Bruyère states that a subset of $N^d$ is $b$-recognizable if and only if it is $b$-definable. As a corollary, the first-order theory of $(N,+,V_b)$ is decidable (where $V_b(n)$ is the largest power of the base $b$ dividing $n$). This classical result is a powerful tool in order to show that many properties of $b$-automatic sequences are decidable. The first part of my lecture will be devoted to presenting this result and its ...

## Multi angle  Independence of normal words Becher, Verónica (Auteur de la Conférence) | CIRM (Editeur )

Recall that normality is a elementary form of randomness: an infinite word is normal to a given alphabet if all blocks of symbols of the same length occur in the word with the same asymptotic frequency. We consider a notion of independence on pairs of infinite words formalising that two words are independent if no one helps to compress the other using one-to-one finite transducers with two inputs. As expected, the set of independent pairs has Lebesgue measure 1. We prove that not only the join of two normal words is normal, but, more generally, the shuffling with a finite transducer of two normal independent words is also a normal word. The converse of this theorem fails: we construct a normal word as the join of two normal words that are not independent. We construct a word x such that the symbol at position n is equal to the symbol at position 2n. Thus, x is the join of x itself and the subsequence of odd positions of x. We also show that selection by finite automata acting on pairs of independent words preserves normality. This is a counterpart version of Agafonov's theorem for finite automata with two input tapes.
This is joint work with Olivier Carton (Universitéé Paris Diderot) and Pablo Ariel Heiber (Universidad de Buenos Aires).
Recall that normality is a elementary form of randomness: an infinite word is normal to a given alphabet if all blocks of symbols of the same length occur in the word with the same asymptotic frequency. We consider a notion of independence on pairs of infinite words formalising that two words are independent if no one helps to compress the other using one-to-one finite transducers with two inputs. As expected, the set of independent pairs has ...

## Multi angle  $k$-abelian singletons and Gray codes for Necklaces Whiteland, Markus (Auteur de la Conférence) | CIRM (Editeur )

$k$-abelian singletons in connection with Gray codes for Necklaces. This work is based on [1]. We are interested in the equivalence classes induced by $k$-abelian equivalence, especially in the number of the classes containing only one element, $k$-abelian singletons. By characterizing $k$-abelian equivalence with $k$-switchings, a sort of rewriting operation, we are able to obtain a structural representation of $k$-abelian singletons. Analyzing this structural result leads, through rather technical considerations, to questions of certain properties of sets of vertex-disjoint cycles in the de Bruijn graph $dB_\Sigma(k-1)$ of order $k-1$. Some problems turn out to be equivalent to old open problems such as Gray codes for necklaces (or conjugacy classes). We shall formulate the problem in the following.
Let $\mathcal{C} = \lbrace V_1, . . . , V_n\rbrace$ be a cycle decomposition of $dB_\Sigma(n)$, that is, a partition of the vertex set $\Sigma^n$ into sets, each inducing a cycle or a loop in $dB_\Sigma(n)$. Let us then define the quotient graph $dB_\Sigma/\mathcal{C}$ as follows. The set of points are the sets in $\mathcal{C}$. For distinct sets $X, Y \in \mathcal{C}$, we have and edge from $X$ to $Y$ if and only if there exists $x{\in}X,y{\in}Y$ such that $(x,y){\in}dB_\Sigma(n)$. An old result shows that the size of a cycle decomposition of $dB_\Sigma(n)$ is at most the number of necklaces of length $n$ over $\Sigma$ (see [2]). We call a cycle decomposition maximal, if its size is maximal. In particular, the cycle decomposition given by necklaces is maximal.
Conjecture 1. For any $\Sigma$ and $n{\in}\mathbb{N}$, there exist a maximal cycle decomposition $\mathcal{C}$ of $dB_\Sigma(n)$ such that $dB_\Sigma(n)/\mathcal{C}$ contains a hamiltonian path.
A natural candidate to study here is the cycle decomposition given by necklaces. This has been studied in the literature in the connection of Gray codes for necklaces. Concerning this, there is an open problem since $1997$ $[3]$ : Let $\Sigma = \lbrace0, 1\rbrace$, $n$ odd, and $\mathcal{C}$ be the cycle decomposition given by necklaces of length $n$ over $\lbrace0,1\rbrace$. Does $dB(n)/\mathcal{C}$ contain a hamiltonian path ?
The answer to the above has been verified to be "yes" for $n \le 15$ $([1]$). The case of $n \ge 4$ and $n$ even, the graph is bipartite with one partition larger than the other. On the other hand, we can find other maximal cycle decompositions of $dB_\Sigma(4)$, $dB_\Sigma(6)$, and $dB_\Sigma(8)$ for the binary alphabet which all admit hamiltonian quotient graphs.
We concluded in $[1]$ that Conjecture $1$ is equivalent to the following $\Theta$-estimation of the number of $k$-abelian singletons of length $n$.
Conjecture 2. The number of $k$-abelian singletons of length $n$ over alphabet $\Sigma$ is of order $\Theta(n^{N_{\Sigma}(k-1)-1})$, where $N_\Sigma(l)$ is the number of necklaces of length $l$ over $\Sigma$.
$k$-abelian singletons in connection with Gray codes for Necklaces. This work is based on [1]. We are interested in the equivalence classes induced by $k$-abelian equivalence, especially in the number of the classes containing only one element, $k$-abelian singletons. By characterizing $k$-abelian equivalence with $k$-switchings, a sort of rewriting operation, we are able to obtain a structural representation of $k$-abelian singletons. Analyzing ...

## Multi angle  $k$-abelian complexity and fluctuation Saarela, Aleksi (Auteur de la Conférence) | CIRM (Editeur )

Words $u$ and $v$ are defined to be $k$-abelian equivalent if every factor of length at most $k$ appears as many times in $u$ as in $v$. The $k$-abelian complexity function of an infinite word can then be defined so that it maps a number $n$ to the number of $k$-abelian equivalence classes of length-$n$ factors of the word. We consider some variations of extremal behavior of $k$-abelian complexity.

First, we look at minimal and maximal complexity. Studying minimal complexity leads to results on ultimately periodic and Sturmian words, similar to the results by Morse and Hedlund on the usual factor complexity. Maximal complexity is related to counting the number of equivalence classes. As a more complicated topic, we study the question of how much k-abelian complexity can fluctuate between fast growing and slowly growing values. These questions could naturally be asked also in a setting where we restrict our attention to some subclass of all words, like morphic words.
Words $u$ and $v$ are defined to be $k$-abelian equivalent if every factor of length at most $k$ appears as many times in $u$ as in $v$. The $k$-abelian complexity function of an infinite word can then be defined so that it maps a number $n$ to the number of $k$-abelian equivalence classes of length-$n$ factors of the word. We consider some variations of extremal behavior of $k$-abelian complexity.

First, we look at minimal and maximal ...

## Multi angle  $k$-abelian equivalence: an equivalence relation in between the equality and the abelian equivalence Karhumäki, Juhani (Auteur de la Conférence) | CIRM (Editeur )

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