We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.
"Examples of nontrivial contact mapping classes for overtwisted contact manifolds in all dimensions." Algebr. Geom. Topol. 19 (3) 1207 - 1227, 2019. https://doi.org/10.2140/agt.2019.19.1207