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This mini-event, organized by [[start|One World Combinatorics on Words Seminar]], is designed to take the place of a small and very informal conference in our field, since many of us miss direct communication of this kind. | This mini-event, organized by [[start|One World Combinatorics on Words Seminar]], is designed to take the place of a small and very informal conference in our field, since many of us miss direct communication of this kind. | ||
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Today, we hardly know anything about the geometrical and topological properties of these unbalanced Rauzy fractals. The Oseledets theorem suggests that these fractals are contained in a strip of the plane: indeed, if the Lyapunov exponents of the matricial product associated with the word exist, one of these exponents at least is nonpositive since their sum equals zero. | Today, we hardly know anything about the geometrical and topological properties of these unbalanced Rauzy fractals. The Oseledets theorem suggests that these fractals are contained in a strip of the plane: indeed, if the Lyapunov exponents of the matricial product associated with the word exist, one of these exponents at least is nonpositive since their sum equals zero. | ||
The study of the pairs of abelianized factors of Arnoux-Rauzy words disproves this belief. | The study of the pairs of abelianized factors of Arnoux-Rauzy words disproves this belief. | ||
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- | **Célia Cisternino** //Two applications of the composition of a $2$-tape automaton and a weighted automaton// | ||
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- | Starting with a $2$-tape automaton ${\mathcal A}$ and a weighted automaton ${\mathcal B}$, we can obtain a new $K$-automaton using an ad hoc operation that can be considered as the composition ${\mathcal B}\circ {\mathcal A}$. First, I will define and illustrate this operation. Next, I will present an application of this operation on automata in terms of synchronized relation and formal series. In particular, using the characterization of synchronized sequences in terms of synchronized relations and that of regular sequences in terms of formal series, I will present two consequences of this composition of automata in the theory of regular sequences. | ||
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- | This is a joint work with Émilie Charlier and Manon Stipulanti based on two recent papers: [[https:// | ||
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- | **Lubka Dvořáková** //On balanced sequences with the minimal asymptotic critical exponent// | ||
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- | We study balanced sequences over a $d$-letter alphabet, $d\geq 3$. Any balanced sequence is defined by a Sturmian sequence and two periodic constant gap sequences $y$ and $y‘$. The minimal value $E(d)$ of the critical exponent of balanced sequences over a $d$-letter alphabet has been recently determined by Baranwal, Rampersad, Shallit, and Vandomme for $d\leq 8$. We focus on the minimal asymptotic critical exponent $E^*(d)$. We show that $E(d)$ and $E^*(d)$ coincide if $d\leq 5$ and differ if $6\leq d\leq 8$. We determine $E^*(6) $ and give an upper bound on $E^*(d)$, $d\leq 15$. | ||
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- | Our method is based on the construction of a finite directed labeled graph $\Gamma$ which depends on the period lengths of $y$ and $y'$ and on a parameter $\beta> | ||
- | On the other hand, if $\Gamma$ contains an oriented cycle labeled by $p$, then we inspect all balanced sequences associated to Sturmian sequences whose slopes have the continued fraction expansion with the period | ||
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- | This is a joint work with Edita Pelantová. | ||
** Anna Frid** //The semigroup of trimmed morphisms// | ** Anna Frid** //The semigroup of trimmed morphisms// | ||
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To achieve these results, we introduce a novel data-structure, | To achieve these results, we introduce a novel data-structure, | ||
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- | ** Christophe Reutenauer ** //An arithmetical characterization of Christoffel words// | ||
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- | Christoffel word have many combinatorial characterizations. | ||
- | I propose a new one, of arithmetical nature: a primitive word on the alphabet $11$, $22$ is a Christoffel word if and only if for any of its conjugates, $u=a_1\cdots a_{2n}$ say, one has | ||
- | $p(a_1, | ||
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- | Here $p(x_1, | ||
- | $p(x_1, | ||
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- | These polynomials are well-known in the theory of continued fractions. One has for example $p(x, | ||
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- | The characterization above is motivated by the theory of Markoff. As intermediate result, I prove another characterization: | ||
- | a primitive word $w$ on the alphabet $\{a< | ||
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** Jeffrey | ** Jeffrey |