A shift - invariant metric on SZ inducing a non - trivialtopologyG

In this paper we discuss the meaning of sensitivity and its implications in CA behavior. A new shift-invariant metric is given. The metric topology induced by this metric is perfect but not compact. Moreover we prove that the new space is \suitable" for the study of the dynamical behavior of CA. In this context sensitivity assumes a stronger meaning than before (usually S Z is given the product topology). Now cellular automata are sensitive if they are not only capable of \transporting" the information but if they are also able to create new information. Finally we prove that additive CA which presents space-time diagrams with complicated structure are sensitive to initial conditions.ogy R esum e Dans cet article nous parlons de la signiication de la sensibilit e et de ses implications au comportement des AC. Une nouvelle m etrique invariante par translation est donn e. La topologie induite par cette m etrique est parfaite mais elle n'est pas compacte. On prouve que ce nouvel espace est \propice" a l' etude du comportement de la dy-namique des AC. Dans ce contexte, la notion de sensibilit e prend une signiication plus puissante. Pour ^ etre sensibles les AC ne doivent pas seulement ^ etre capables de transporter de l'information mais aussi capables de cr eer de nouvelles informations. Finalement nous prouvons que les AC additifs avec un diagramme d'espace-temps avec une structure complique sont sensibles aux conditions initiales. Mots-cl es: Automates cellulaires, sensibilit e aux conditions initiales, topolo-gie m etrique Abstract In this paper we discuss the meaning of sensitivity and its implications in CA behavior. A new shift-invariant metric is given. The metric topology induced by this metric is perfect but not compact. Moreover we prove that the new space is \suitable" for the study of the dynamical behavior of CA. In this context sensitivity assumes a stronger meaning than before (usually S Z is given the product topology). Now cellular automata are sensitive if they are not only capable of \transporting" the information but if they are also able to create new information. Finally we prove that additive CA which presents fractal space-time diagrams are sensitive to initial conditions.