  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].
