Cellular automata and percolation in groups.

RESUMEN: A famous theorem by Gilman shows that every cellular automaton over A^Z satisfies an important dynamical dichotomy with respect to any Bernoulli measure: either almost every configuration is sensitive to initial conditions, or the system is equicontinuous. We show that there exists a fundamental relationship between the existence of a non-trivial percolation threshold on the Cayley graphs of a given group G and the failure of this dichotomy. We use this to give a characterization of the countable groups where Gilman’s dichotomy is satisfied, which correspond to the class of locally virtually cyclic groups.