Complex Systems

A Single Nonexpansive, Nonperiodic Rational Direction Download PDF

K. M. Madden
Electronic mail address:
Department of Mathematics and Computer Science,
Drew University,
Madison, NJ 07940


A two-dimensional cellular automaton consists of a two-dimensional lattice of sites, each of which takes on a finite number of values, and a cellular automaton map. The cellular automaton map updates the value at each site using a translation invariant rule that only depends on the values at the sites in some finite neighborhood of . A number of global properties of a two-dimensional cellular automata, such as the directional entropies introduced by Milnor, can be studied using the methods of dynamical systems. In this work we consider , the expansive one-dimensional subspaces of as defined by Boyle and Lind. Various properties of cellular automata, including Milnor's directional entropies, vary nicely within connected components of so it is natural to ask what subsets of may occur as expansive one-dimensional subspaces. Boyle and Lind give an almost complete answer, the single unresolved case being when is the complement of a line with irrational slope. In this work we construct a related example with the potential to shed light on the unresolved case.