Topological structure of isolated points in the space of $\mathbb {Z}^d$-shifts

Abstract R. Pavlov and S. Schmieding [On the structure of generic subshifts. Nonlinearity 36 (2023), 4904–4953] recently provided some results about generic $\mathbb {Z}$ -shifts, which rely mainly on an original theorem stating that isolated points form a residual set in the space of $\mathbb {Z}$ -shifts such that all other residual sets must contain it. As a direction for further research, they pointed towards genericity in the space of $\mathbb {G}$ -shifts, where $\mathbb {G}$ is a finitely generated group. In the present text, we approach this for the case of $\mathbb {Z}^d$ -shifts, where $d \ge 2$ . As it is usual, multidimensional dynamical systems are much more difficult to understand. In light of the result of R. Pavlov and S. Schmieding, it is natural to begin with a better understanding of isolated points. We prove here a characterization of such points in the space of $\mathbb {Z}^d$ -shifts, in terms of the natural notion of maximal subsystems that we also introduce in this article. From this characterization, we recover the result of R. Pavlov and S. Schmieding for $\mathbb {Z}^1$ -shifts. We also prove a series of results that exploit this notion. In particular, some transitivity-like properties can be related to the number of maximal subsystems. Furthermore, we show that the Cantor–Bendixon rank of the space of $\mathbb {Z}^d$ -shifts is infinite for $d>1$ , while for $d=1$ , it is known to be equal to one.