Properties of mean dimension and metric mean dimension coming from the topological entropy

TL;DR

This paper investigates properties of mean dimension and metric mean dimension, showing their similarities to topological entropy and revealing discontinuity and non-lower semi-continuity in certain settings.

Contribution

It verifies which properties of topological entropy extend to mean dimension and metric mean dimension, highlighting their discontinuity and semi-continuity characteristics.

Findings

Metric mean dimension map is nowhere continuous on certain spaces.

Metric mean dimension is not lower semi-continuous on the interval and circle.

Properties of topological entropy do not always carry over to mean dimensions.

Abstract

In the late 1990's, M. Gromov introduced the notion of mean dimension for a continuous map, which is, as well as the topological entropy, an invariant under topological conjugacy. The concept of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000. In this paper we will verify which properties coming from the topological entropy map are valid for both mean dimension and metric mean dimension. In particular, we will prove that the metric mean dimension map is not continuous anywhere on the set consisting of continuous maps on both the Cantor set, the interval or the circle. Finally we prove that the metric mean dimension on the set consisting of continuous map on the interval and on the circle is not lower semi-continuous.