Topological Dynamics of Pullback Maps on Full Shifts

Let $G$ be a group, let $A$ be a finite alphabet, and let $\phi: G \to G$ be an endomorphism. We study the topological dynamics of the pullback map $\phi^* : A^G \to A^G$, given by $\phi^*(x)=x\circ\phi$, a canonical example of a generalized cellular automaton. In the one-dimensional case, where $G=\mathbb Z$ and $\phi_k(n)=kn$, we prove a sharp dichotomy: $\phi_k^*$ is equicontinuous precisely for $k\in\{-1,0,1\}$, and cofinitely sensitive otherwise. Although the fixed identity coordinate prevents transitivity on the full shift, the restriction to the natural invariant components is topologically mixing exactly when $k\notin\{-1,0,1\}$. We then extend the analysis to countable groups, showing that $\phi^*$ is equicontinuous if and only if every element of $G$ is eventually periodic under $\phi$, while the existence of a non-eventually-periodic element is equivalent to cofinite sensitivity and to the absence of equicontinuous points. Finally, we characterize Bernoulli measure preservation and strong mixing on the punctured configuration space in terms of injectivity and eventual periodicity.