$G$-index, topological dynamics and marker property

TL;DR

This paper investigates the $G$-index in relation to topological dynamics, demonstrating linear growth bounds and constructing a free dynamical system lacking the marker property, thus solving an open problem.

Contribution

It establishes bounds on the $bZ_p$-index growth and provides a counterexample to the marker property in free dynamical systems.

Findings

The $bZ_p$-index grows at most linearly with $p$.

Constructed a free dynamical system without the marker property.

Connected $G$-index theory with topological dynamics.

Abstract

Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical system, we study the $\mathbb{Z}_p$-index of the set of $p$-periodic points. We find that its growth is at most linear in $p$. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.