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Valeurs propres des automates cellulaires

Abstract : We investigate properties of one-dimensional cellular automata. This category of cellular automata has been widely studied but many questions are still open. Among them the spectral theory of unidimensional cellular automata is an open field with few indirect results. We want a better understanding of both ergodic and topological aspect by investigating the existence of eigenvalues of cellular automata, in particular irrational ones, i.e., those of the form e^{2Iπα} where α is irrationnal and I the complex root of -1. The last question seems not to have been studied yet.In the topological field the results of Kůrka & Blanchard and Tisseur about equicontinuous cellular automata have as direct consequence that any equicontinuous CA has rational eigenvalues. Density of shift periodic points leads to the impossibility for CA to have topological irrational eigenvalues. The density of periodic points of cellular automata seems to be related with the question of eignevalues. If the CA has equicontinuity points without being equicontinuous, the density of periodic points implies the fact that the spectrum contains all rational roots of the unity, i.e., all numbers of the form e^{2Iπα} with α∈Q .In the measurable field the question becomes harder. We assume that the cellular automaton is surjective, which implies that the uniform measure is invariant. Most results are still available in more general conditions. We first prove that cellular automata with equicontinuity points never have irrational measurable eigenvalues. This result is then generalized to cellular automata with μ-equicontinuous points according to Gilman's classification. We also prove that cellular automata with μ-equicontinuous points have rational eigenvalues
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Rezki Chemlal. Valeurs propres des automates cellulaires. Mathématiques générales [math.GM]. Université Paris-Est, 2012. Français. ⟨NNT : 2012PEST1067⟩. ⟨tel-00794398⟩

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