$G$-index, topological dynamics and marker property

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 Gindex theory and topological dynamics. For a fixed-point free dynamical system, we study the 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.

Divergent coindex sequence for dynamical systems

“…If (X, T ) is a fixed-point free dynamical system then (P p (X, T ), T ) becomes a free Z p -space. The paper [TTY20] investigated its index and proved Theorem 1.1 ([TTY20], Theorem 1.2). Let (X, T ) be a fixed-point free dynamical system.…”

“…It had been an open problem for several years whether the converse holds or not. This problem was solved by [TTY20]. They constructed an aperiodic dynamical system which does not have the marker property.…”

“…It is easy to check that the right-hand side set is nonempty and finite (see also [TTY20,Lemma 4.1]). Define…”

“…Tsutaya, Yoshinaga and the second-named author [TTY20] found an application of the Z p -index theory to topological dynamics. (One of their motivations is to solve a problem about the marker property of dynamical systems.…”

“…The purpose of this paper is to continue this investigation. In particular we solve a problem posed by [TTY20].…”