On Certain <i>C</i> 0-Aspects of Contactomorphism Groups

Abstract We study a number of questions related to the $C^{0}$-topology of contactomorphisms and contact homeomorphisms. In particular, we show a connection between the Rokhlin property of the group of contact homeomorphisms and contact non-squeezing, we define a new conjugation-invariant norm on contactomorphisms and explore its relation to the contact fragmentation norm and we introduce a measure of the size of conjugacy classes that is related to weak conjugacy equivalence. We also show that Sandon’s spectral norm is $C^{0}$-locally bounded and extend its definition to contact homeomorphisms.