On Limit Set of Two-Dimensional Two-State Linear Cellular Automaton Rules

In 1990´s existence and self-similarity of the limit set of one-dimensional pk-state linear cellular automata under suitable scaling was deeply studied by Willson and Takahashi. According to the study the partially self-similar structure of limit set of one-dimensional cellular automata defines a transition matrix whose maximum eigen value determines the Hausdorff dimension of the limit set. It was also proved that the limit set of higher dimensional linear cellular automata exists and is partially self-similar. However, structure of limit set of higher dimensional linear cellular automata have not reported yet. In the present paper, the authors estimated Hausdorff dimension of the limit set of two-dimensional two state linear cellular automata with the same method as one-dimensional case. These limit sets are, as in the one-dimensional case, characterized by transition matrix, whose maximum eigen value determines Hausdorff dimension. Furthermore, the authors proved that limit set of two-dimensional two-state linear cellular automata contains the limit set of one-dimensional two-state cellular automata as sub-dynamics.