Linear preservers of copositive and completely positive matrices

The objective of this manuscript is to understand the structure of an invertible linear map on the space of real symmetric matrices ${\mathcal{S}}^n$ that leaves invariant the closed convex cones of copositive and completely positive matrices ($CO{P}_n$ and $C{P}_n$). A description of an invertible linear map on ${\mathcal{S}}^2$ such that $L(C{P}_2)\subset C{P}_2$ is completely determined.