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

Abstract

R. Pavlov and S. Schmieding provided recently 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 G-shifts, where 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 which we also introduce in this article. From this characterization we recover the result of R. Pavlov and S. Schmieding's for $\mathbb{Z}$-shifts. We also prove a series of results which 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 $d = 1$ is known to be equal to one.