Finite mean dimesnion and marker property

TL;DR

This paper explores the relationship between finite mean dimension and the marker property in dynamical systems, introducing the $ ext{Z}_p$-index and demonstrating systems with arbitrary mean dimension lacking the marker property.

Contribution

It develops the theory of the $ ext{Z}_p$-index and shows the existence of dynamical systems with any positive mean dimension that do not possess the marker property.

Findings

Introduces the $ ext{Z}_p$-index theory.

Constructs dynamical systems with arbitrary mean dimension without marker property.

Establishes a link between mean dimension and marker property.

Abstract

In this paper, we develop the theory of $\mathbb{Z}_p$-index which has been introduced by Tsukamoto, Tsutaya and Yoshinaga. As an application, we show that given any positive number, there exists a dynamical system with mean dimension equal to such number such that it does not have the marker property.