Topological definitions of chaos applied to cellular automata dynamics

Abstract

We apply the two different definitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of 1-dimensional cellular automata. A DTDS is chaotic according to the Devaney's definition of chaos iff it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen's definition of chaos iff it has a dense orbit and it is sensitive to initial conditions. We continue the work initiated in [3], [4], [5], and [14] by proving that an easy-to-check property of local rules on which cellular automata are defined-introduced by Hedlund in [11] and called permutivity-is a sufficient condition for chaotic behavior. Permutivity turns out to be also a necessary condition for chaos in the case of elementary cellular automata while this is not true for general 1-dimensional cellular automata. The main technical contribution of this paper is the proof that permutivity of the local rule either in the leftmost or in the rightmost variable forces the cellular automaton to have dense periodic orbits.