On certain $C^0$-aspects of contactomorphism groups

TL;DR

This paper investigates the $C^0$-topology of contactomorphisms, establishing connections between contact properties, defining new norms, and extending spectral norms to contact homeomorphisms.

Contribution

It introduces a new conjugation-invariant norm on contactomorphisms, links contact non-squeezing with the Rokhlin property, and extends Sandon's spectral norm to contact homeomorphisms.

Findings

Connected Rokhlin property with contact non-squeezing

Defined a new conjugation-invariant norm on contactomorphisms

Extended Sandon's spectral norm to contact homeomorphisms

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 Rokhlin property 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 which 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.