Divergent coindex sequence for dynamical systems

TL;DR

This paper investigates the growth of index and coindex in fixed-point free dynamical systems with group actions, demonstrating the existence of systems with divergent coindex sequences, thus addressing a previously posed open problem.

Contribution

It introduces the first example of a fixed-point free dynamical system with a divergent coindex sequence, linking index theory with topological dynamics.

Findings

Existence of fixed-point free systems with divergent coindex sequences

Growth behavior of index and coindex in dynamical systems

Resolution of a problem posed by TTY20

Abstract

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of $p$-periodic points admits a natural free action of $\mathbb{Z}/p\mathbb{Z}$ for each prime number $p$. We are interested in the growth of its index and coindex as $p\to \infty$. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by [TTY20].