Combinatorical and Algebraic Structures Seminar

Session details

Date: 23.10.2018
Speaker: Valérie Goyheneche, LAMFA, Université de Picardie Jules Verne
Title: Does a substitutive sequence admit a letter in arithmetical progression?
Abstract: The object of this talk is the following question: given a primitive substitutive sequence $x$, does there exist a letter that occurs in arithmetical progression? In other words, does there exist a letter $a$ and two integers $k$ and $p$ such that $x_{k+np} = a$ for all $n\in\mathbb{N}$? Our method mainly relies on the relationship between constant arithmetical subsequence and eigenvalues associated to the underlying dynamical system. Its study leads to a necessary condition for the existence of constant arithmetical subsequence. We will then explain a method to compute algorithmically the set of rational eigenvalues associated to a substitution. We can then deduce, given an integer $p$, if the sequence $x$ contains a letter in arithmetical progression of period $p$. At the end of the talk, we will focus on the case of primitive constant-length substitutions. In this case, we construct a graph that will provide the set of periods for letters.

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