Periodicity and transitivity for cellular automata in Besicovitch topologies

Abstract

We study cellular automata (CA) behavior in Besicovitch topology. We solve an open problem about the existence of transitive CA. The proof of this result has some interest in its own since it is obtained by using Kolmogorov complexity. At our knowledge it if the first result on discrete dynamical systems obtained using Kolmogorov complexity. We also prove that every CA (in Besicovitch topology) either has a unique fixed point or a countable set of periodic points. This result underlines that CA have a great degree of stability and may be considered a further step towards the understanding of CA periodic behavior. © Springer-Verlag Berlin Heidelberg 2003.