  We prove that if a contact manifold admits an exact filling then every local
contactomorphism isotopic to the identity admits a translated point in the
interior of its support, in the sense of Sandon [San11b]. In addition we prove
that if the Rabinowitz Floer homology of the filling is non-zero then every
contactomorphism isotopic to the identity admits a translated point, and if the
Rabinowitz Floer homology of the filling is infinite dimensional then every
contactmorphism isotopic to the identity has either infinitely many translated
points, or a translated point on a closed leaf. Moreover if the contact
manifold has dimension greater than or equal to 3, the latter option
generically doesn't happen. Finally, we prove that a generic contactomorphism
on $\mathbb{R}^{2n+1}$ has infinitely many geometrically distinct iterated
translated points all of which lie in the interior of its support.
