Exotic symplectomorphisms and contact circle actions

Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, 2n, implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, 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.