Chaotic Properties of the Elementary Cellular Automaton Rule 40 in Wolfram's Class I

This paper examines the chaotic properties of the elementary cellular automaton rule 40. Rule 40 has been classified into Wolfram’s class I and also into class 1 by G. Braga et al. These classifications mean that the time-space patterns generated by this cellular automaton die out in a finite time and so are not interesting. As such, we may hardly realize that rule 40 has chaotic properties. In this paper we show that the dynamical system defined by rule 40 is Devaney chaos on a class of configurations of some particular type and has every periodic point except prime period one, four, or six. In the process of the proof, it is noticed that the dynamical properties of rule 40 can be related to some interval dynamical systems. These propositions are shown in Theorems 2 and 4.