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$\mathbb {Z}^d$-shifts
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.
