Connections between the minimal neighborhood and the activity value of cellular automata

For a group $G$ and a finite set $A$, a cellular automaton is a transformation of the configuration space $A^G$ defined via a finite neighborhood and a local map. Although neighborhoods are not unique, every CA admits a unique minimal neighborhood, which consists on all the essential cells in $G$ that affect the behavior of the local map. An active transition of a cellular automaton is a pattern that produces a change on the current state of a cell when the local map is applied. In this paper, we study the links between the minimal neighborhood and the number of active transitions, known as the activity value, of cellular automata. Our main results state that the activity value usually imposes several restrictions on the size of the minimal neighborhood of local maps.