Cellular automata over semi-direct product groups: Reduction and invertibility results

Cellular automata are transformations of configuration spaces over finitely generated groups, such that the next state in a point only depends on the current state of a finite neighborhood of the point itself. Many questions arise about retrieving global properties from such local descriptions, and finding algorithms to perform these tasks. We consider the case when the group is a semi-direct product of two finitely generated groups, and show that a finite factor (whatever it is) can be thought of as part of the alphabet instead of the group, preserving both the dynamics and some "finiteness" properties. We also show that, under reasonable hypotheses, this reduction is computable: this leads to some reduction theorems related to the invertibility problem.