Homotopy type of the group of strict contactomorphisms of the standard tight contact structure on the three-sphere

We prove that the group $Aut_1(xi)$ of strict contactomorphisms, also known as quantomorphisms, of the standard tight contact structure $xi$ on $S^3$ is the total space of a fiber bundle $S^1 to Aut_1(xi) to SDiff(S^2)$ over the group of area-preserving $C^infty$-diffeomorphisms of $S^2$, and that it deformation retracts to its finite-dimensional sub-bundle $S^1 to U(2)cup cU(2) to O(3)$, where $U(2)$ is the unitary group and $c$ is complex conjugation.