Orbits of Bernoulli Measures in Cellular Automata

Abstract: We discuss how to construct shift-invariant probability measures over the space of bisequences of symbols, and how to describe such measures in terms of block probabilities. We then define cellular automata as maps in the space of measures and discuss orbits of shift-invariant probability measures under these maps. Subsequently, the local structure approximation is discussed as a method to approximate orbits of Bernoulli measures under the action of cellular automata. The final sections presents some known examples of cellular automata, both deterministic and probabilistic, for which elements of the orbit of the Bernoulli measure (probabilities of short blocks) can be determined exactly.