Asymptotic behaviour of cellular automata : computation and randomness

L'objet de cette these est l'etude de l'auto-organisation dans les automates cellulaires unidimensionnels.Les automates cellulaires sont un systeme dynamique discret ainsi qu'un modele de calcul massivement parallele, ces deux aspects s'influencant mutuellement. L'auto-organisation est un phenomene ou un comportement organise est observe asymptotiquement, independamment de la configuration initiale. Typiquement, nous considerons que le point initial est tire aleatoirement: etant donnee une mesure de probabilite decrivant une distribution de configurations initiales, nous etudions son evolution sous l'action de l'automate, le comportement asymptotique etant decrit par la(les) mesure(s) limite(s).Notre etude presente deux aspects. D'abord, nous caracterisons les mesures qui peuvent etre atteintes a la limite par les automates cellulaires; ceci correspond aux differents comportements asymptotiques pouvant apparaitre en simulation. Cette approche rejoint divers resultats recents caracterisant des parametres de systemes dynamiques par des conditions de calculabilite, utilisant des outils d'analyse calculable. Il s'agit egalement d'une description de la puissance de calcul des automates cellulaires sur les mesures.Ensuite, nous proposons des outils pour letude de l'auto-organisation dans des classes restreintes. Nous introduisons un cadre d'etude d'automates pouvant etre vus comme un ensemble de particules en interaction, afin d'en deduire des proprietes sur leur comportement asymptotique. Une derniere direction de recherche concerne les automates convergeant vers la mesure uniforme sur une large classe de mesures initiales (phenomene de randomisation).

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