Mean dimension of natural extension of algebraic systems

TL;DR

This paper proves that mean dimension remains unchanged under natural extension for endomorphisms on compact metrizable abelian groups, and applies this to algebraic cellular automata, strengthening previous results.

Contribution

It establishes the invariance of mean dimension under natural extension for a class of algebraic systems and applies this to cellular automata, providing a new proof of existing results.

Findings

Mean dimension is preserved by natural extension for certain algebraic systems.

The mean dimension of an algebraic cellular automaton equals that of its natural extension.

Provides a new proof of a known result using different methods.

Abstract

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 withthe mean dimension of its natural extension, which strengthens a result of Burguet and Shi \cite{BS21} with a different proof.