On Gottschalk's surjunctivity conjecture for non-uniform cellular automata
This work addresses a theoretical problem in dynamical systems and group theory, providing incremental extensions to existing theorems for cellular automata.
The paper tackles the surjunctivity conjecture for non-uniform cellular automata (NUCA) over surjunctive groups, showing that reversible NUCA with finite memory and bounded singularity must be invertible, and extends dual-surjunctivity results from CA to NUCA.
Gottschalk's surjunctivity conjecture for a group $G$ states that it is impossible for cellular automata (CA) over the universe $G$ with finite alphabet to produce strict embeddings of the full shift into itself. A group universe $G$ satisfying Gottschalk's surjunctivity conjecture is called a surjunctive group. The surjunctivity theorem of Gromov and Weiss shows that every sofic group is surjunctive. In this paper, we study the surjunctivity of local perturbations of CA and more generally of non-uniform cellular automata (NUCA) with finite memory and uniformly bounded singularity over surjunctive group universes. In particular, we show that such a NUCA must be invertible whenever it is reversible. We also obtain similar results which extend to the class of NUCA a certain dual-surjunctivity theorem of Capobianco, Kari, and Taati for CA.