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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: arXiv: | ||
| ** Anna Frid** //The semigroup of trimmed morphisms// | ** Anna Frid** //The semigroup of trimmed morphisms// | ||