Attractors

Abstract

Classification schemes are the holy grail in the theory of cellular automata. One aspect that may serve as an identifier for different categories of cellular automata is asymptotic behavior. Although transient states are interesting, often enough the dynamics settles quite fast on typical structures. The set consisting of these asymptotic states is the attractor. The first part of this section is devoted to the definition of an attractor and the proof that every cellular automaton has attractors. Attractors are not amorphous sets without further features but can be decomposed into smaller parts. We prove the central structure theorem: Conley’s decomposition theorem. Essential for the relevance of an attractor is the “size” of the set of trajectories that eventually tend to the attractor. This “size” of the set of attracted states can be described in terms of measure theory. Therefore we define the Bernoulli measure on the set of states. Using Conley’s decomposition theorem and measure theory, it is possible to classify cellular automata by the structure of their attractor set: the Hurley classification theorem.