  Using Floer-theoretic methods, we prove that the non-existence of an exotic
symplectomorphism on the standard symplectic ball, $\mathbb{B}^{2n},$ implies a
rather strict topological condition on the free contact circle actions on the
standard contact sphere, $\mathbb{S}^{2n-1}.$ We also prove an analogue for a
Liouville domain and contact circle actions on its boundary. Applications
include results concerning the symplectic mapping class group and the
fundamental group of the group of contactomorphisms.
