Remark on the local nature of metric mean dimension

TL;DR

This paper discusses the local properties of metric mean dimension, a key invariant in dynamical systems, and extends the concept to actions of bR^D, providing clarifications and examples.

Contribution

It clarifies the local nature of metric mean dimension using Bowen's ideas and generalizes the concept to bR^D-actions with illustrative examples.

Findings

Clarification of the local nature of metric mean dimension

Extension of metric mean dimension to bR^D-actions

Providing an example illustrating the concepts

Abstract

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean dimension. We also explain the generalization to $\mathbb{R}^D$-actions and an illustrating example.