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

Part of: Topological dynamics

Published online by Cambridge University Press:  18 June 2025

SILVÈRE GANGLOFF  and
ALONSO NÚÑEZ
SILVÈRE GANGLOFF*
Affiliation:
Independent researcher, Palma de Mallorca, Spain
ALONSO NÚÑEZ
Affiliation:
Institut de mathématiques de Toulouse, 1R3, Université Paul Sabatier, 118 Route de Narbonne, 31400 Toulouse, France Instituto de Ingeniería Matemática y Computacional, Pontificia Universidad Católica de Chile, Santiago, Chile (e-mail: alonso.herrera_nunez@math.univ-toulouse.fr)
*
e-mail: sfgangloff@gmail.com
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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.

Keywords

shift spacesgeneric propertiesmultidimensional dynamical systemsCantor–Bendixson rank

MSC classification

Primary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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