Abstract
Topological structure of the living space is one of the most important parameters for cellular automata (CA), which can be applied to various scientific areas. In order to study CA in larger extent, we define a new type of cellular automata on countable groups and investigate some general dynamical properties of attractors and periodicity. It is shown that if such cellular automaton on a countable group is transitive, it is either sensitive or composed of a single periodic orbit. By using of the form of attractors, classification of this type of cellular automata is given on countable groups. The set of eventually periodic points of a cellular automaton on a countable residually finite group is proved to be dense.
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