Mean dimension of natural extension of algebraic systems. (English) Zbl 1537.37018
Summary: Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of an algebraic cellular automaton coincides with the mean dimension of its natural extension, which strengthens a result of D. Burguet and R. Shi [“Mean dimension of continuous cellular automata”, Preprint, arXiv:2105.09708] with a different proof.
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B15 | Dynamical aspects of cellular automata |
| 37C45 | Dimension theory of smooth dynamical systems |
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