Associated to a symplectic manifold $(M^{2n},\omega)$ is the group of (compactly supported) Hamiltonian diffeomorphisms $\text{Ham}(M,\omega)$. This group is equipped with the famous Hofer metric $d_H$. Denote by $\mathcal{B}$ the unit ball in $\text{Ham}(M,\omega)$ with respect to $d_H$. Are any examples known where $\mathcal{B}$ is contractible? Are examples known where $\mathcal{B}$ is not contractible?

I know that it is known in some cases if $\text{Ham}(M,\omega)$ is contractible or not (See e.g. McDuff's "A survey of the topological properties of symplectomorphism groups"). My question is closer to the following idea: If a loop $\mathcal{B}$ is contractible, can a contracting disc then be realized inside $\mathcal{B}$?


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