Decidability of topological transitivity for injective group cellular automata

Determine whether topological transitivity is decidable for injective group cellular automata, i.e., for continuous, shift-commuting automorphisms of G^Z where G is a finite group.

Background

The paper studies one-dimensional group cellular automata (GCAs), proving that expansivity is decidable and that, within GCAs, expansivity implies topological transitivity. For abelian groups, many properties, including topological transitivity for 1-GCAs, are known to be efficiently decidable, but for general (possibly non-abelian) GCAs only partial results are known.

In the conclusions, the authors explicitly note that, even for the injective GCA class they focus on, the decidability of topological transitivity itself is not known. Establishing decidability (or undecidability) of topological transitivity for injective GCAs would close this gap.

References

For this class of automorphisms, we proved that expansivity is a decidable property and implies topological transitivity (that is not known to be decidable), which in turn is equivalent to ergodicity.

— Decidability and Characterization of Expansivity for Group Cellular Automata (2510.14568 - Castronuovo et al., 16 Oct 2025) in Section 6 (Conclusions and further work)