Mean dimension of natural extension of algebraic systems

Bingbing Liang, Ruxi Shi

Published 2022-03-02Version 1

Mean dimension may decrease after taking natural extension. In this paper we show that mean dimension stays the same after taking natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of algebraic cellular automaton coincides with its natural extension, which strengthens a result of Burguet and Shi \cite{BS21} with a different proof.