C^0-rigidity of Legendrian and coisotropic submanifolds

Contact homeomorphisms are points in the closure of the (compactly supported) contactomorphism group in the homeomorphism group under the C^0-topology. Recently, Dimitroglou Rizell and Sullivan showed that the images of closed Legendrians under contact homeomorphisms are still Legendrian if they are smooth. We will show that in the cosphere bundle, the images of coisotropics under contact homeomorphisms are still coisotropic if they are smooth. Moreover, we show that the Maslov data and Floer theory invariants of the Legendrians are preserved under contact homeomorphisms in the identity component. This is joint work with Tomohiro Asano, Yuichi Ike and Chris Kuo.