Cellular automata and Percolation on groups.

Resumen:  a classical theorem of Gilman shows that every cellular automaton over the integers satisfies a strong dichotomy with respect to any iid Bernoulli process: either almost all configurations are sensitive to initial conditions or it is almost everywhere equicontinuous.

If instead of the integers we consider an arbitrary finitely generated group, we will show that there is a strong connection between the triviality of the percolation threshold and the validity of this dichotomy. Using this connection, we will show that Gilman’s dichotomy holds on a countable group if and only if it is locally virtually cyclic.

Posted on Apr 7, 2025 in Seminario de Probabilidades de Chile, Seminars