Contactomorphism groups of R^{2n+1} admit a dense conjugacy class, those of R^{2n}×S^1 do not, and Sandon's spectral norm is C^0-locally bounded and extendable to the C^0-closure.
New steps in $C^0$ symplectic and contact geometry of smooth submanifolds
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abstract
We provide a $C^0$ counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, we prove a quantitative $h$-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, we give another proof of the Dimitroglou Rizell and Sullivan theorem \cite{RS22} which states that Legendrian knots are preserved by contact homeomorphisms, provided their image is smooth. Moreover, our method gives related examples of rigidity in higher dimensions as well.
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On certain $C^0$-aspects of contactomorphism groups
Contactomorphism groups of R^{2n+1} admit a dense conjugacy class, those of R^{2n}×S^1 do not, and Sandon's spectral norm is C^0-locally bounded and extendable to the C^0-closure.