A Symbolic Dynamics Perspective of Conway's Game of Life

An interesting question is whether the intrinsic complexity of the gliders in D-dimensional cellular automata could be quantitatively analyzed in rigorously mathematical sense. In this paper, by introducing the D-dimensional symbolic space, some fundamental dynamical properties of D-dimensional shift map are explored in a subtle way. The purpose of this article is to present an accurate characterization of complex symbolic dynamics of gliders in Conway's game of life. A series of dynamical properties of gliders on their concrete subsystems are investigated by means of the directed graph representation and transition matrix. More specifically, the gliders here are topologically mixing and possess the positive topological entropy on their subsystems. Finally, it is worth mentioning that the method presented in this paper is also applicable to other gliders in different D dimensions.