This paper deals with subshifts of finite type indexed by a residually finite group \(G\). The main result states that the set of periodic configurations is dense when the subshift of finite type is strongly irreducible and contains a periodic configuration. (Strong irreducibility is sometimes called strong specification.) As a consequence, the subshift is surjunctive, i.e., each injective cellular automaton on it is surjective, and its automorphism group is residually finite. If in addition \(G\) is countable, then the subshift admits an invariant Borel probability measure with full support.
The special case of \(G = \mathbb{Z}^d\) was treated by S. J. Lightwood [Ergodic Theory Dyn. Syst. 23, No. 2, 587–609 (2003; Zbl 1031.37017)]. The assumption that \(X\) contains a periodic configuration can be dropped when \(G = \mathbb{Z}\) or \(G = \mathbb{Z}^2\), but it remains an open question whether each strongly irreducible subshift of finite type contains a dense set of periodic configurations when \(G = \mathbb{Z}^d\) with \(d \geq 3\).
For \(G = \mathbb{Z}\), the authors also provide a proof due to Benjamin Weiss showing that each subshift with weak specification contains a dense set of periodic configurations. (Subshifts with weak specification are called W-subshifts in this paper.)