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 $mathbb{Z}/pmathbb{Z}$ for each prime number $p$. We are interested in the growth of its index and coindex as $pto infty$. 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].
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