  In this paper, we explore the structure of Rabinowitz--Floer homology $RFH_*$
on contact manifolds whose Reeb flow is periodic (and which satisfy an index
condition such that $RFH_*$ is independent of the filling). The main result is
that $RFH_*$ is a module over the Laurent polynomials $\mathbb{Z}_2[s,s^{-1}]$,
where $s$ is the homology class generated by a principal Reeb orbit and the
module structure is given by the pair-of-pants product. In most cases, this
module is free and finitely generated.
