Divergent coindex sequence for dynamical systems

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 [Formula: see text]-periodic points admits a natural free action of [Formula: see text] for each prime number [Formula: see text]. We are interested in the growth of its index and coindex as [Formula: see text]. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, [Formula: see text]-index, topological dynamics and marker property, preprint (2020), arXiv: 2012.15372.