Simplicity of the contactomorphism group of finite regularity

For a given coorientable contact manifold $(M^{2n+1},\xi )$, we consider the group ${\mathrm{Cont}}_c^{(r,\delta )}(M,\alpha )$ consisting of $C^{r,\delta }$ contactomorphisms with compact support which is equipped with $C^{r,\delta }$-topology of Hölder regularity $(r,\delta )$ for $r\ge 1$ and $0<\delta \le 1$. We prove that for all Hölder class exponents with $r>n+2$ or $r=n+1,\frac{1}{2}<\delta \le 1$ (resp. $r<n+1$ or $r=n+1$ and $0<\delta <\frac{1}{2}$), the group is a perfect (and so a simple) group. In particular, ${\mathrm{Cont}}_c^r(M,\xi )$ is simple for all integer $r\ge 1$. For the case of ${\mathrm{Cont}}_c^{(r,\delta )}(M,\alpha )$ of general Hölder regularity, we prove the simplicity for all pairs $(r,\delta )$ leaving only the case of $(r,\delta )=(n+1,\frac{1}{2})$ open.