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Aspects ergodiques et algébriques des automates cellulaires

Abstract : Ergodic and algebraic aspects of cellular automata The first part of this manuscript falls within the framework of probability theory, and is devoted to the study of filtrations generated by some cellular automata. We study two versions of an algebraic automaton acting on configurations whose states take values in a finite Abelian group: one is deterministic, and consists in adding the states of two consecutive cells, and the second is a random perturbation of the first one. From these automata, random Markovian processes are constructed and the filtrations generated by these processes are studied. Using the I-cosiness criterion, we show that the two filtrations are standard in the sense developed by Vershik. However, cellular automata have the particularity of commuting with the coordinate shift operator. In this thesis, we introduce a new classification of the filtrations called "dynamic" which takes into account the action of this transformation. Filtrations are no longer defined on probability spaces but on dynamical systems, and are in this case "factor" filtrations: each sigma-algebra is invariant by the dynamics of the system. The counterpart of standardity from the dynamic point of view is studied. This creates a necessary criterion for dynamic standardity called "dynamic I-cosiness". The question of whether the dynamic I-cosiness is sufficient remains open, but a first result in this direction is given, showing that a strengthened version of the dynamic I-cosiness leads to dynamic standardity. By establishing that it does not satisfy the criterion of dynamic I-cosiness, it is proved that the factor filtration generated by the deterministic automaton is not dynamically standard, and therefore that the dynamic classification of the filtrations differs from the classification developed by Vershik. The probabilistic automaton depends on an error parameter, and it is shown by a percolation argument that the factor filtration generated by this automaton is dynamically standard for large enough values of this parameter. It is conjectured that it will not be dynamically standard for very small values of this parameter. The second part of this manuscript, more algebraic, has its origin in a musical problem, linked to the calculation of intervals in a periodic melodic line. The work presented here continues the research of the Romanian composer Anatol Vieru and of Moreno Andreatta and Dan Vuza, but in an original way from the point of view of cellular automata. We study the action on periodic sequences of two algebraic cellular automata, one of which is identical to that of the first part. The questions on the characterization of reducible and reproducible sequences as well as the associated times have been deepened and improved for these two automata. The calculation of preimages and images via the two automata was explained. The question of the evolution of the periods was treated with the creation of a tool called "characteristic" which allows to describe and control the evolution of the period in negative times. Simulations show that the evolution of the periods when the preimages are drawn at random follows an almost regular pattern, and the explanation of this phenomenon remains an open question. The mathematical results of this second part have been used in the "Automaton" module of a free composing software called "UPISketch ». This module allows a composer to create melodic lines by iterating images or taking successive preimages of a starting melodic line.
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Submitted on : Tuesday, January 5, 2021 - 11:35:38 AM
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Paul Lanthier. Aspects ergodiques et algébriques des automates cellulaires. Mathématiques [math]. Université de Rouen Normandie, 2020. Français. ⟨tel-03097149⟩

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