  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 $G$-index theory and topological dynamics. For a fixed-point free dynamical
system, we study the $\mathbb{Z}_p$-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.
